Méthodes numériques et modélisation pour certains problèmes ...

Méthodes numériques et modélisation pour certainsproblèmes multi-échelles

Mémoire

Présenté en vue d’obtenir

L’HABILITATION A DIRIGER DES RECHERCHES

parAlexei LOZINSKI

Université Paul Sabatier, Toulouse 3

Spécialité :Mathématiques Appliquées

Soutenance prévue le 8 décembre 2010devant le jury composé de

M. Yves Achdou Professeur Université Paris 7M. Claude Le Bris Professeur ENPCM. Martin Gander Professeur Université de GenèveM. Patrick Laborde Professeur Université Paul SabatierM. Olivier Pironneau Professeur Université Paris 6M. Jean-Paul Vila Professeur INSA de Toulouse

Table des matières

Table des matières i

Avant-propos iii

I Sur la résolution numérique de problèmes elliptiques multi-échelles 1

1 Présentation de l’approche dite "Zoom Numérique" 3

2 La méthode des patchs d’éléments finis 92.1 Les estimations d’erreur a priori pour la méthode des patchs . . . . . . . . . . . . . 92.2 La vitesse de convergence de la méthode itérative . . . . . . . . . . . . . . . . . . . 132.3 Une accélération : patchs harmoniques . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Sur la méthode Chimère 203.1 Convergence de l’algorithme de Schwarz sur des maillages arbitraires non-conformes 203.2 Une extension au cas des problèmes non stationnaires . . . . . . . . . . . . . . . . . 22

4 Une comparaison numérique des algorithmes précédents 25

5 "Zoom Numérique" pour "multi-modèles" 30

6 MsFEM 37

7 Perspectives 46

II Sur la modélisation des fluides viscoélastiques 47

8 Construction du modèle de chaînes bille-ressort 49

i

ii TABLE DES MATIÈRES

8.1 Le modèle prenant en compte l’inertie des billes . . . . . . . . . . . . . . . . . . . . 508.2 La limite sans inertie des billes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.3 Le cas des écoulements localement homogènes . . . . . . . . . . . . . . . . . . . . 56

9 Sur le choix de la force de ressort 589.1 Motivation physiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.2 Des résultats théoriques sur le modèle avec réflexion . . . . . . . . . . . . . . . . . . 609.3 Simulations numériques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

10 Perspectives 69

IIIAutres travaux 71

11 Méthodes numériques pour les écoulements biphasiques 7311.1 Sur la discrétisation en temps en simulation des écoulements avec des particules . . . 7311.2 Une méthode du type domaines fictifs pour des écoulements avec des bulles de gaz . 74

12 Problèmes fortement anisotropes 7512.1 Un schéma du type "Asymptotic-Preserving" pour une équation de diffusion anisotrope 7512.2 Méthodes adaptatives sur des maillages anisotropes pour les problèmes non station-

naires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

13 Méthodes numériques non probabilistes pour le modèle des chaînes 77

14 Travaux pendant la thèse et avant 79

Bibliographie 81

Table des matières i

IVArticles annexés 87

Numerical Zoom for Advection Diffusion Problems with Localized Multiscales 91

On discretization in time in simulations of particulate 105

Motion of gas bubbles, considered as massless bodies, affording deformations within aprescribed family of shapes, in an incompressible fluid under the action of gravitationand surface tension 129

Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations 165

An anisotropic error estimator for the Crank-Nicolson method : application to a parabo-lic problem 195

Sparse tensor-product Fokker-Planck-based methods for nonlinear bead-spring chainmodels of dilute polymer solutions 225

Avant-propos

On présent dans ce travail la problématique "multi-échelles" de deux points de vue :– Les problèmes où l’aspect "multi-échelles" apparaît au niveau de fortes variations dans les

données ou les coefficients des équations. On proposera des méthodes numériques adaptées àces situations : patchs d’éléments finis ou le Zoom Numérique, Chimère, les méthodes de type"éléments finis multi-échelles - MsFEM".

– Les problèmes où l’aspect "multi-échelles" est pris en compte déjà au niveau des équationsdu modèle, comme par exemple dans la théorie cinétique des polymères. On présentera destravaux sur la dérivation des modèle des fluides polymériques qui permettent le couplage entreles échelles microscopique (les molécules du polymère) et les échelle macroscopiques (écou-lement du fluide).

Les problème du premier type évoqués ci-dessus sont très répandus dans les calculs industrielsactuels. On peut citer l’exemple des structures aéronautiques, où le calcul de structure de grande taille(par exemple un avion) doit prendre en compte le comportement des nombreux rivets ou des petitesfissures. Un autre exemple de motivation se trouve dans des problèmes du contrôle de la qualité d’air,dans lesquels les sources d’émission des polluants sont de taille beaucoup plus petite que la taille dela région affectée par la pollution. Pour faire une simulation précise sur une grande échelle, on doitcalculer une solution numérique dans un sous-domaine autour de la source de pollution en utilisantun maillage beaucoup plus fin.

Afin de traiter ce type de difficultés, je présenterai une méthode de résolution numérique pour desproblèmes elliptiques avec des données multi-échelles quiutilise plusieurs niveaux des maillages,pas nécessairement emboîtés (patchs d’éléments finis). J’ai participé au développement de ces mé-thodes qui a été mené conjointement par les équipes à l’EPFL (Jacques Rappaz) et à l’Université deHouston (Roland Glowinski et Jiwen He). On résout le problème dans un domaineΩ à l’intérieurduquel on considère des patchsΛ1, Λ2, . . . où une précision plus élevée est désirable. On calculealors successivement des corrections à la solution globaledans les patchs. Nous avons démontré desestimations d’erreur a priori et étudié la vitesse de convergence de la méthode itérative. Ces résultatsthéoriques sont confirmés par des expériences numériques.

Plus récemment, en collaboration avec Olivier Pironneau (Paris 6) et Patrick Laborde (IMT, Tou-louse 3), on a développé des nouvelles variantes de ces méthodes qui sont plus faciles à mettre en

iii

iv TABLE DES MATIÈRES

œuvre. Elles peuvent être appliquées dans les situations oùles multi-échelles sont incrustées dansles équations aux dérivées partielles du problème (et pas seulement dans les données). Cela ouvrela route au couplage de plusieurs modèles dans une seule simulation. Par exemple, en mécaniquedes structures, on peut superposer un calcul fin local autourd’une petite fissure sur le calcul globalrelativement grossier de toute la structure. Nous avons aussi effectué des comparaisons numériquesavec la méthode de Schwarz traditionnelle et une analyse théorique de cette dernière.

J’ai aussi participé dans le groupe de travail mené par Pierre Degond (IMT, Toulouse 3) sur ledéveloppement des méthode du type MsFEM (Multi-scale Finite Element Method). A la différencedes méthodes du type “Zoom numérique” évoquées plus haut, laMsFEM est destinée aux problèmesoù le comportement multi-échelles n’est pas localisé dans une petite région, mais il est présent partoutdans le domaine du calcul. Le but est de pouvoir faire des calcul assez précis sur un maillage de pasplus grand que la longueur caractéristique de la micro-échelle en se permettant d’effectuer des calculsfins et coûteux sur chaque cellule du maillage grossier une fois pour toutes.

Tous ces travaux sont présentés dans la première partie de cette thèse d’habilitation.La deuxième partie du manuscrit est consacrée aux problèmesoù l’aspect “multi-échelles” est pris

en compte déjà au niveau des équations du modèle. On s’intéresse notamment à quelques questionsautour de la modélisation des polymères au niveau microscopique et autour du couplage avec leniveau macroscopique de l’écoulement d’une solution polymérique. On utilise une approche de lathéorie cinétique où les molécules du polymère sont représentées par des “dumbbells” – deux billesreliés par un ressort. On va présenter une modélisation complète qui prend en compte les effetsd’inertie des billes et en déduire des modèles classiques sans inertie comme cas limite, à l’aide detechniques de développements asymptotiques. La deuxième question traitée dans cette partie porterasur un nouveau choix de la loi de force des ressorts dans le modèle des dumbbells, qui prend encompte l’extensibilité maximale d’un ressort et qui mène aux équations différentielles stochastiquesréfléchies. On établira rigoureusement la formule pour le tenseur des contraintes dans ce modèle.

La troisième partie est un résumé de mes autres travaux de recherche. Certains articles décritsdans cette partie sont annexés à la fin du manuscrit.

Première partie

Sur la résolution numérique de problèmeselliptiques multi-échelles

1

2 TABLE DES MATIÈRES

1 Présentation de l’approche dite"Zoom Numérique"

Dans les simulations éléments finis qui exigent une solutiontrès fine dans une zone localisée, lesingénieurs font assez souvent un calcul grossier global d’abord, suivi par un calcul beaucoup plus finsur un sous-ensemble (zoom)Λ du domaine entierΩ. Evidemment, un calcul sur le zoom ne suffitpas en général, sauf si le problème est non-linéaire ou non-stationnaire et les itérations dans le zoomsont une partie de la boucle non-linéaire ou temporelle. On alterne donc les calculs grossiers et finsdans une boucle. Les questions suivantes se posent alors : comment formuler le problème dansΛ etcomment utiliser la solution correspondante dans le solveur grossier pour corriger la solution globale.On présente dans ce chapitre plusieurs implémentations possibles. Dans les chapitres suivants, onétudiera la convergence et les erreurs relatives à ces approches.

1. La méthode Chimère

Dans la méthode Chimère, proposée originellement par Steger [48] pour des problèmes non sta-tionnaires, on creuse un trouD dans le domaine grossier strictement à l’intérieur de la région du zoomΛ. Par exemple, le calcul de la pression hydrostatiqueu d’un écoulement dans un milieu poreux estgouvernée par la loi de Darcy avec la porositéK,

−∇ · (K∇u) = f dans Ω (1.1)

u = g sur Γ = ∂Ω,

où g est la valeur donnée sur la frontière. On choisitD strictement à l’intérieur deΛ et on construitdeux maillages : une triangulation grossièreTH surΩH = Ω \ D et une triangulation fineTh surΛ,voir Fig. 1.1. On appelleSh la frontière deΛ et SH la partie de la frontière deΩH autre queΓ. Onintroduit aussi des espaces d’éléments finisVH et Vh surTH et Th respectivement. On noteV0H =VH ∩H1

0 (ΩH), VgH = vH ∈ VH telles quevH |Γ est l’interpolé nodal deg surΓ, et finalementV0h =Vh ∩H1

0 (Λ). L’algorithme Chimère consiste à alterner en boucle les calculs grossier surΩH et fin surΛ en prenant pour conditions aux limites surSH et Sh les résultats de derniers calculs disponiblesdans l’autre sous-domaine. Pour passer l’information entre TH et Th on doit utiliser l’interpolation

3

4 CHAPITRE 1. PRÉSENTATION DE L’APPROCHE DITE "ZOOM NUMÉRIQUE"

Entrées : Une fonctionu0h ∈ Vh0.

pour n = 1...N faireTrouverun

H ∈ VgH tel que∫

ΩH

K∇unH · ∇vH =

∫

ΩH

fvH , ∀vH ∈ V0H , (1.2)

unH |SH

= rHun−1h |SH

.

Trouverwnh ∈ V0h tel que

∫

Λ

K∇unh · ∇vh =

∫

Λ

fvh, ∀vh ∈ V0h, (1.3)

unh|Sh

= rhunH|Sh

.

finAlgorithme 1.1: Chimère

d’un maillage à l’autre et on introduit doncrH (respectivementrh) les opérateurs d’interpolationnodale surVH (respectivementVh).

FIGURE 1.1: TriangulationsTH surΩH (à l’extérieur deSH) etTh surΛ (à l’intérieur deSh).

Chimère est ainsi identique à l’algorithme alternatif de Schwarz, mais cette terminologie est uti-lisée dans la communauté de “Computational Fluid Dynamics”. Les deux triangulationsTH etTh nesont pas nécessairement choisies de manière conforme. Par exemple, la frontière du trou∂D peutcouper arbitrairement les triangles deTh. Quoique cela fonctionne bien en général, ni la convergencede cet algorithme, ni la convergence de la solution approchée vers la solution exacte sous le raffine-ment des maillages ne sont toujours pas démontrées en général. La difficulté réside dans l’analyse del’erreur d’interpolation deTH surTh lorsqueΛ n’est pas composé des éléments de la triangulation deΩ. Cependant, on présentera dans Chapitre 3 des éléments d’une analyse théorique de cette méthode

5

Entrées : une fonctionu0h ∈ V0h.

pour n = 1...N faireTrouverun

H ∈ VgH tel que∫

Ω

K∇unH · ∇vH =

∫

Ω

fvH −∫

Ω

K∇un−1h · ∇vH , ∀vH ∈ V0H . (1.5)

Trouverunh ∈ V0h tel que∫

Λ

K∇unh · ∇vh =

∫

Λ

fvh −∫

Λ

K∇unH · ∇vh, ∀vh ∈ V0h. (1.6)

finAlgorithme 1.2: Méthode des patchs d’éléments finis

en suivant [32]. Cette analyse est effectuée dans la norme demaximum et repose sur un principe demaximum fort au niveau discret.

2. La méthode des patchs d’éléments finis

Mis à part le faible taux de convergence de la méthode Chimère, souvent observé en pratique,l’inconvénient de cette méthode est la nécessité de changerla géométrie du problème grossier parrapport au problème original non raffiné. En effet,ΩH dans (1.2) n’est pas le domaineΩ tout entier.Une idée alternative, introduite dans [53] et étudiée dans [39], [35], [16], consiste – au niveau continu– à garder le problème global tel quel (on notera ici sa solution parU) et à chercher une correctionu à la solution grossierU dans l’espace des fonctions qui s’annulent en dehors deΛ, c’est-à-dire àtrouverU ∈ H1(Ω), u ∈ H1

0 (Λ) avecU |Γ = g qui satisfont∫

Ω

(K∇(U + u) · ∇(V + v) − f(V + v)) = 0, ∀V ∈ H10 (Ω), v ∈ H1

0 (Λ) (1.4)

Cette équation est facile à discrétiser, en prenantuH ≈ U, uh ≈ u, et en introduisant un schémaitératif de l’Algorithme 1.2. On utilise ici les mêmes notations qu’avant :VH et Vh sont des espacesd’éléments finis sur des triangulations régulièresTH et Th deΩ et Λ respectivement ;V0H = VH ∩H1

0 (Ω) ; VgH est le sous-ensemble deVH contenant les fonctions qui interpolentg surΓ, et finalementV0h = Vh ∩ H1

0 (Λ). Les approximations successives de la solution exacteu sont obtenues dansl’algorithme 1.2 parun

Hh = unH +un

h. On montrera au chapitre 2, en suivant [16], l’estimation d’erreurpour la solutionuHh = limn→∞(un

H + unh) obtenue par l’algorithme :

‖u− uHh‖H1(Ω) ≤ C(Hr||u||Hq(Ω\Λ) + hs||u||Hq(Λ)

), (1.7)

où r et s sont degrés maximaux des polynômes utilisés dans la construction deV0H et V0h respecti-vement etq = max(r, s) + 1.

Dans la suite on distinguera souvent les deux situations : des maillagesTh etTH emboîtés, c’est-à-dire chaque triangle deTh est contenu dans un seul triangle deTH , et des maillagesTh et TH nonemboîtés. Noter que la majoration (1.7) a lieu même si les deux maillages ne sont pas emboîtés l’undans l’autre.


Entrées : une fonctionu0h ∈ V0h.

pour n = 1...N faireTrouverλn

H ∈ V 0H tel que

∫

Ω

K∇λnH · ∇µH =

∫

Ω

fµH −∫

Ω

K∇un−1h · ∇µH , ∀µH ∈ V 0

H . (1.8)

TrouverunH ∈ VgH tel que

∫

Ω

K∇unH · ∇vH =

∫

Ω

fvH −∫

Ω

K∇un−1h · ∇vH −

∫

Ω

K∇λnH · ∇vH , (1.9)

∀vH ∈ V0H .

Trouverunh ∈ V0h tel que∫

Λ

K∇unh · ∇vh =

∫

Λ

fvh −∫

Λ

K∇unH · ∇vh, ∀vh ∈ V0h. (1.10)

finAlgorithme 1.3: La méthode des patch harmoniques

3. La méthode des patchs harmoniques

L’inconvénient de la méthode des patchs consiste dans le fait que son taux de convergence peutêtre très bas pour certains paires de triangulationsTH , Th non emboîtées. Heureusement, la méthodepeut être accélérée au prix de la résolution d’un problème auxiliaire typiquement peu coûteux àchaque itération. Cet algorithme accéléré a été introduit dans [18]. Il est résumé ci-après sous le nomde la méthode des patchs harmoniques, l’algorithme 1.3. On yutilise la notation suivante

V 0H = vH ∈ V0H : suppvH ⊂ Λ.

V 0H est donc le sous-espace deV0H qui contient les fonctions dont le support est à l’intérieurde

Λ. Comme la taille deΛ n’est pas beaucoup plus grande que le pas du maillage grossier, la di-mension deV 0

H n’est pas trop élevée. Le problème pourλnH est donc pas coûteux et la matrice

correspondante est juste une sous-matrice de celle du problème grossier pourunH . La nouvelle va-

riableλnH est purement auxiliaire. La solution approchée du problèmeglobal est toujours obtenue

paruHh = limn→∞(unH + un

h) comme dans le cas de l’algorithme 1.2. Le problème auxiliaire pourλn

H assure que les parties grossièresunH sont toujoursa-orthogonales àV 0

H , c’est-à-dire elles sontapproximativementa-harmoniques dans le casK = 1, d’où le nom de l’algorithme.

L’idée des patchs harmoniques n’apporte rien de nouveau parrapport à l’algorithme 1.2 lorsquele maillageTh est emboîté dansTH . En effet, l’algorithme 1.2 converge normalement très vitedansce cas. De plus, ces deux algorithmes 1.2 et 1.3 sont identiques (toujours dans le cas des maillagesemboîtés) en ce sens que les sommesun

Hh = unH +un

h sont exactement les mêmes dans les algorithmes1.2 et 1.3 sur toutes les itérationsn ≥ 1, bien queun

H etunh puissent être différents d’un algorithme à

l’autre. En effet, l’intersectionV 0 = V0H ∩ V0h coïncide dans ce cas avecV 0H et on peut décomposer

unH = un,0

H + un,⊥H et un

h = un,0h + un,⊥

h avecun,0H , un,0

h ∈ V 0 et un,⊥H , un,⊥

h qui appartiennent auxcompléments orthogonaux deV 0 dansV0H et V0h respectivement. L’orthogonalité est comprise ici

7

Entrées : fonctionsu0H ∈ VgH etw0

h ∈ Vh.pour n = 1...N faire


∫

Ω

K∇unH · ∇vH =

∫

Ω

fwH +

∫

Λ

K∇(un−1H − wn−1

h ) · ∇vH , ∀vH ∈ V0H . (1.11)

Trouverwnh ∈ Vh tel que

∫

Λ

K∇wnh · ∇vh =

∫

Λ

fvh, ∀vh ∈ V0h, (1.12)

wnh |∂Λ = rhu

nH |∂Λ.

finAlgorithme 1.4: Semi-Schwarz

au sens du produit scalairea(u, v) =∫ΩK∇u · ∇v. En notant paru0 ∈ V 0 l’unique solution de

a(u0, v0) =

∫

Ω

fv0, ∀v0 ∈ V 0

et en choisissant les fonctions testvH et vh dansV 0, on voit tout de suite queun,0H = u0 − u0,0

H ,un,0

h = u0,0h à toutes les itérations de l’algorithme 1.2 et queun,0

H = 0, un,0h = u0 à toutes les itérations

de l’algorithme 1.3. Donc les deux algorithmes donnentun,0H + un,0

h = u0. En choisissant maintenantles fonctions test orthogonales àV 0, on voit que les composantesun,⊥

H et un,⊥h satisfont exactement

les mêmes équations pour les deux algorithmes et doncunH + un

h sont les mêmes.L’algorithme des patchs harmoniques prend toute sa valeur dans le cas général de triangulations

non emboîtées. On verra que la convergence de l’algorithme 1.2 peut être extrêmement lente dansce cas. Par contre, l’algorithme 1.3 conserve les bonnes propriétés de convergence du cas emboîté,même sur des triangulations quelconques. La solution finaleuHh obtenue par l’algorithme 1.3 esten général légèrement différente de celle obtenue par l’Algorithme 1.2. On a toutefois la mêmemajoration d’erreur (1.7).

4. La méthode “Semi-Schwarz”

La différence majeure entre les algorithmes des patchs 1.2,1.3, d’un côté, et l’algorithme Chi-mère, de l’autre, réside dans la façon de récupérer la solution globale à partir des solutions grossièreet fine. En effet, dans les algorithmes des patchs on poseun

Hh = unH+un

h tandis que dans ChimèreunHh

peut être défini commeunh à l’intérieur deΛ, un

H à l’extérieur deΛ. Si la triangulationTh est em-boîtée dansTH , alors il est facile de passer d’une représentation à l’autre en posantwn

h = unH |Λ + un

h.En éliminantun

h de l’algorithme des patchs 1.2, on obtient l’algorithme 1.4(proposé dans [22]). Lesinconnues des itérations ici sontwn

h etunH et on passe l’information deun

H àwnh de la même manière

que dans Chimère, autrement dit l’algorithme alternatif deSchwarz. Par contre, on ne peut plus fairela même chose dans l’autre direction, dewn

h àunH, d’où le nom de cet algorithme "Semi-Schwarz".

Les algorithmes 1.2 et 1.4 sont équivalents par construction dans le cas des maillages emboîtés.Dans une situation générale, les relations entre eux sont assez compliquées à cause de l’interpolation


Entrées : fonctionsu0H ∈ VgH etλ0

h ∈ Mh.pour n = 1...N faire


∫

Ω

K∇unH · ∇vH =

∫

Ω\Λ

fvH −∫

∂Λ

λn−1h vH +

∫

Λ

K∇un−1H · ∇vH , ∀vH ∈ V0H . (1.13)

Trouverwnh ∈ Vh λ

nh ∈ Mh tels que∫

Λ

K∇wnh · ∇vh −

∫

∂Λ

λnhvh =

∫

Λ

fvh, ∀vh ∈ V0h, (1.14)∫

∂Λ

wnhµh =

∫

∂Λ

unHµh, ∀µh ∈ Mh.

finAlgorithme 1.5: Semi-Schwarz-Lagrange

présente maintenant dans la relationwnh ≈ un

H |Λ + unh. On voit, en pratique, que l’algorithme 1.4 est

plus proche de la version harmonique 1.3, sa performance étant située entre celles de l’algorithme1.2 et l’algorithme 1.3, voir [22]. On constate ainsi un certain avantage du "Semi-Schwarz" devantl’algorithme des patchs original et cela sans l’introduction d’un problème auxiliaire comme dans laversion harmonique. Noter aussi que le sous-problème fin estplus facile à mettre en œuvre dans"Semi-Schwarz" que dans l’algorithme original parce que leterme mixte

∫ΛK∇un

H ·∇vh, délicat dupoint de vue de l’implémentation, n’y figure plus.

5. La méthode “Semi-Schwarz-Lagrange”

On finit ce panorama des méthodes du type "Zoom Numérique" encore par une réécriture del’algorithme 1.4. L’idée est d’interpréter le terme

∫ΛK∇wn−1

h · ∇vH comme une approximation de∫∂ΛK(∂nw

n−1h )vH et d’introduire une nouvelle variable auxiliaireλn

h (qui vit sur ∂Λ) comme uneapproximation de la dérivée normale∂nw

nh). On introduit donc l’espaceMh des traces sur∂Λ de

fonctions éléments finis deVh. La valeur deλnh ∈ Mh peut être facilement récupérée en tant que

multiplicateur de Lagrange dans le sous-problème fin. Cela donne l’algorithme 1.5 qu’on nomme"Semi-Schwarz-Lagrange". Pour symétriser le problème fin (1.14), on a choisi d’imposer les condi-tions aux limites pourwn

h de manière faible.Encore une fois, ce nouvel algorithme est équivalent à l’algorithme 1.4 et donc aux algorithmes

1.2, 1.3 dans le cas des maillages emboîtés. Mais en général,c’est une méthode différente de cellesintroduites plus haut. Elle a au moins cet avantage sur "Semi-Schwarz" que

∫∂Λλn−1

h vH est moinscoûteux à évoluer en pratique que

∫ΛK∇wn−1

h · ∇vH . De plus, cette méthode peut être réutiliséedans les problèmes du type "multi-échelles" et "multi-modèles" comme décrit dans Chapitre 5.

2 La méthode des patchs d’élémentsfinis

On s’intéresse ici aux algorithmes des patchs 1.2 et 1.3 présentés dans le chapitre précédent.On commence par une étude des erreurs dans l’algorithme 1.2 et de sa convergence en tant qu’uneméthode itérative. A la fin, on étudiera les mêmes questions pour la version accélérée : l’algorithmedes patches harmoniques 1.3.

2.1 Les estimations d’erreura priori pour la méthode des patchs

D’après les articles [16, 19].

On rappelle d’abord les notations : soitΩ ⊂ R2 un domaine borné polygonal etΛ ⊂⊂ Ω un autredomaine polygonal “beaucoup plus petit” queΩ. Etant donnéf ∈ H−1(Ω) et une fonction positiveK ∈ L∞(Ω), on considère le problème de Poisson-Dirichlet : trouveru tel que

−∇ · (K∇u) = f dansΩ etu = 0 sur∂Ω.

La généralisation au cas des conditions aux limites non-homogènesu = g sur∂Ω est évidente.La formulation faible du problème s’écrit : trouveru ∈ V = H1

0 (Ω) tel que

a(u, v) = (f, v), ∀v ∈ V, (2.1)

oùa : V × V → R est une forme bilinéaire symétrique, continue et coercive

a(u, v) =

∫

Ω

K∇u · ∇v (2.2)

et (f, v) =∫Ωfv. On s’intéresse par une situation "multi-échelles" où la donnéef de (2.1) a une

forte variation dansΛ. Ainsi la discrétisation de (2.1) par une méthode des éléments finis ne peut pas

9

10 CHAPITRE 2. LA MÉTHODE DES PATCHS D’ÉLÉMENTS FINIS

être faite en utilisant des grilles de la même résolution surΩ \Λ et surΛ. C’est pourquoi on introduitdeux espaces d’éléments finis :

V0H = g ∈ C0(Ω) telles queg|K ∈ Pr(K), ∀K ∈ TH et g = 0 sur∂Ω

oùTH est une triangulation régulière deΩ et Pr(K) est l’espace des polynômes de degré≤ r sur untriangleK ∈ TH , et

V0h = g ∈ C0(Ω) tel que g|K ∈ Ps(K), ∀K ∈ Th et g = 0 surΩ \ Λ

où Th est une triangulation régulière deΛ. En posantVHh = V0H + V0h et en supposant que l’al-gorithme 1.2 converge, on voit immédiatement que la solution limite uHh = limn→∞ un

Hh est uneapproximation de la solution exacteu donnée par la méthode de Galerkin

a(uHh, v) = 〈f |v〉, ∀v ∈ VHh. (2.3)

On rappelle queV0H ∩ V0h ne se réduit pas nécessairement à l’élément nul et la matricedu système(2.3) peut donc être non inversible. C’est une des motivations pour introduire l’algorithme itératif 1.2pour calculeruHh. De manière plus générale, l’intérêt de l’algorithme 1.2 est de réduire le problèmeéléments finis non standards (2.3) à des problèmes éléments finis classiques à chaque itération quipeuvent être implémentés à l’aide des logiciels existants.

Notre premier résultat donne une estimation d’erreura priori optimale pour l’approximation (2.3)et donc pour l’algorithme 1.2 en supposant toujours qu’il converge.

Théorème 2.1On poseq = max(r, s) + 1 et on suppose que la solutionu de (2.1) appartient àHq(Ω). Alors, on a l’estimation d’erreura priorisuivante pour la solutionuHh de (2.3) :

||u− uHh||H1(Ω) ≤ C(Hr||u||Hq(Ω\Λ) + hs||u||Hq(Λ)

), (2.4)

avec une constanteC qui ne dépend que deΩ et de régularité des maillages.

Démonstration. D’après les théorèmes bien connus d’extension des fonctions dans les espaces deSobolev, il existe un opérateur borné d’extensionE : Hq(Ω \ Λ) → Hq(Ω) pour lequelEv|Ω\Λ =

v|Ω\Λ, ∀v ∈ Hq(Ω \ Λ). On poseu = Eu si ||Eu||Hq(Λ) ≤ ||u||Hq(Λ) et u = u sinon. On a de toutefaçonu = u dansΩ \ Λ et doncu− u ∈ Hq

0(Λ). De plus

||u||Hq(Ω) ≤ C||u||Hq(Ω\Λ), (2.5)

avec une constante génériqueC, et

||u||Hq(Λ) ≤ ||u||Hq(Λ). (2.6)

Notons parrH et rh les opérateurs standards d’interpolation aux espacesV0H etV0h respectivement.On introduituH = rH u, uh = rh(u− u) et uHh = uH + uh. D’après le lemme de Céa

||u− uHh|| ≤ ||u− uHh|| (2.7)

où || · || est la norme d’énergie|| · ||2 = a(·, ·). En écrivantu− uHh = (u− uH) + [(u− u) − uh], eten utilisant des résultats standards d’interpolation par les éléments finis, on voit que

||u− uHh|| ≤ ||u− uH|| + ||(u− u) − uh|| ≤ C(Hr||u||Hq(Ω) + hs||u− u||Hq(Λ)

).

2.1. LES ESTIMATIONS D’ERREURA PRIORI POUR LA MÉTHODE DES PATCHS 11

0−L L0

l

Γ2 Γ1

Γ3Ω

(0, 0)

(−L,−L) (L,−L)

(−L, L)

Ω

Γ

(a) (b)

FIGURE 2.1: Le domaineΩ : (a) cas du problème avec des conditions aux limites mixtes Dirichlet-Neuman ; (b) cas du domaine avec coin rentrant.

(2.6) implique||u− u||Hq(Λ) ≤ 2||u||Hq(Λ). Ainsi, en invoquant (2.5), on obtient

||u− uHh|| ≤ C(Hr||u||Hq(Ω\Λ) + hs||u||Hq(Λ)

). (2.8)

D’où le résultat du lemme.L’algorithme des patchs peut s’avérer utile aussi dans les situations où la solution exacte n’a pas

la régularité requise par le théorème 2.1. En visant par exemple les éléments finisP1, on est dans cettesituation quandu n’est pas dansH2(Ω) même si le second membref est dansL2(Ω). C’est le cas, parexemple, pour le problème de Poisson-Dirichlet dans un domaine polygonale avec un coin rentrant(Fig. 2.1b) ou le problème de Poisson avec les conditions auxlimites mixtes de Dirichlet-Neumann(Fig. 2.1a, avec condition de Dirichlet surΓ1 et celle de Neumann surΓ2). L’erreur de la méthoded’éléments finis standardP1 n’est pas donc d’ordre optimalCH par rapport au pas de maillageH.On peut remarquer néanmoins que le caractère singulier est bien localisé dans ces deux exemples,autour du coin rentrant ou du point de changement du type des conditions aux limites. On peut doncaméliorer le taux de convergence par rapport àH en appliquant un petit patch autour d’un point desingularité et en y introduisant un maillage fin de pash << H.

Illustrons cette idée plus en détail sur l’exemple du problème de Poisson avec les conditions auxlimites mixtes de Dirichlet-Neumann : soit le domaineΩ comme en Fig. 2.1a. Pour une fonctiondonnéef ∈ L2(Ω), on chercheu ∈ H1(Ω) telle que

−∆u = f dansΩ,u = 0 surΓ1,∂u∂n

= 0 surΓ2,u = 0 surΓ3.

La formulation faible de ce problème s’écrit toujours comme(2.1) avec

V = u ∈ H1(Ω) / u|Γ1∪Γ3 = 0etK = 1 dans (2.2). Il est bien connu que la solutionu contient en général une partie singulière

us = c0√r sin

θ

2,


c’est-à-direu = ur + us

où ur ∈ H2(Ω) est la partie régulière. Ici(r, θ) sont les coordonnées polaires et on sous-entendque le point de changement du type des conditions aux limitescoïncide avec l’origine du systèmedes coordonnées. Evidemmentus 6∈ H2(Ω), mais seulementus ∈ Hσ(Ω) pour toutσ < 1/2. Pouraméliorer le taux de convergence de l’approximation par leséléments finis, on pose un patchΛ autourde l’origine, voir Fig. 2.2. Comme précédemment, on construit une triangulation régulièreTH de pasH surΩ et une triangulation plus fineTh surΛ de pash << H. La proposition suivante donne uneestimation d’erreur en fonction des pas des maillagesH eth et de la tailleε du patch. On va énoncerce résultat pour une classe de solutions singulières qui inclut le cas ci-dessus, mais aussi le cas duproblème avec un coin rentrant. Par exemple, la solution du problème de Poisson-Dirichlet sur ledomaine dessiné en Fig. 2.1b contient la partie singulière de la formec0r2/3 sin 2

3θ.

TH

Th

0−L L

0

l

−ε ε

Ω

Cε

Λε

FIGURE 2.2: Position du patch.

Proposition 2.2 Soit la solution exacte du problème (2.1) de la forme

u = ur + rαg(θ)

avecur ∈ H2(Ω), 0 < α < 1, g(θ) ∈ H2([θ1, θ2]) oùθ1, θ2 sont des angles tels que le domaineΩ estcontenu dans le secteurθ1 < θ < θ2. On suppose queCε ⊂ Λ oùCε est la partie du cercle de rayonε centré à0 à l’intérieur deΩ. Alors, on a l’estimation d’erreura priori suivante pour la solutionuHh du problème (2.3) avec des élément finisP1 (r = s = 1) :

|u− uHh|H1(Ω) ≤ C

(H

ε1−α+ hα

)||g||H2 + CH||ur||H2(Ω)

oùC > 0 est une constante qui ne dépend que deΩ et de régularité des maillages.

On renvoie à [19] pour la preuve complète de cette proposition qui est assez similaire à celle duthéorème 2.1. La seule différence est dans la construction de l’extensionus ∈ H2(Ω) deus|Ω\Λ avecus = rαg(θ), la partie régulièreur étant déjà dans l’espace convenable. On ne dispose pas ici d’unthéorème d’extension connu mais on peut facilement construire cette extension en posant

us =

us, dansΩ \ Cε,εαA( r

ε)g(θ), dansCε

2.2. LA VITESSE DE CONVERGENCE DE LA MÉTHODE ITÉRATIVE 13

avec une fonction appropriéeA ∈ C2([0, 1]). La norme deus dansH2(Ω) donne le terme d’ordreH/ε1−α dans l’estimation d’erreur. Remarquer que c’est vraiment le meilleur ordre possible parceque la norme deus dansH2(Ω \ Λ) est aussi proportionnelle à1/ε1−α.

Le résultat de la Proposition 2.2 est important d’un point devue pratique parce qu’il permetde choisir la taille du patchε et le pash de maillage associé de manière équilibrée. En revenant surl’exemple du problème de Poisson avec les conditions aux limites mixtes de Dirichlet-Neumann (Fig.2.1a), on a l’estimation d’erreur pour la méthode des patchs(en ne gardant que les termes essentiels)

|u− uHh|H1(Ω) ≤ C

(H√ε

+√h

).

Afin d’équilibrer les deux termes ici, on devra poserε ∼ H2/h. Pour avoir une idée du bon choixde h en fonction deH, on note que le nombre d’inconnues dans le problème grossiersur TH estproportionnel à1/H2, tandis que celui du problème fin surTh est proportionnel à(ε/h)2 ∼ (H/h)4.Pour avoir un coût de calcul sur le patch de même ordre que le coût pour le problème grossier, onchoisira donch tel que1/H2 ∼ (H/h)4, c’est-à-dire

h ∼ H3/2, ε ∼ H1/2.

Cela améliore l’erreur àO(H3/4) par rapport à l’erreur de la discrétisation standard surTH qui estd’ordreO(H1/2).

2.2 La vitesse de convergence de la méthode itérative

D’après l’article [16].

On procède maintenant à l’étude de convergence de la méthodeitérative dans l’algorithme 1.2.Notons d’abord que cet algorithme s’inscrit dans le cadre général des méthodes de correction parsous-espaces. L’article [66] contient une preuve de convergence de telles méthodes. On établi ici desestimations plus précises de la vitesse de convergence, quinous seront utiles dans la suite.

Le paramètre qui intervient habituellement dans les estimations de vitesse de convergence deméthodes de correction par sous-espaces est l’angle (ou plutôt son cosinus) entre les deux sous-espacesV0H etV0h deVHh. Il s’agit du nombreγ ∈ [0; 1] défini comme

γ = supvh∈V0h,vh 6=0

vH∈V0H,vH 6=0

a(vh, vH)

||vh||||vH|| .

Cependant, on rappelle qu’en généralV 0 = V0H ∩ V0h ne se réduit pas à l’élément zéro. Dans ce cason aγ = 1 et le paramètreγ ne donne donc pas d’information pertinente sur la relation entreV0H etV0h. Une idée pour remédier à ce problème consiste à enlever l’intersectionV0 du calcul de l’angle.On introduit ainsi le paramètre suivant (cf. Fig. 2.3)

γ = supvh∈Vh,vh 6=0

vH∈VH ,vH 6=0

a(vh, vH)

||vh||||vH|| ,

où Vh = V0h ∩ V ⊥0 , VH = V0H ∩ V ⊥

0 etV ⊥0 est le complément orthogonal (dans le produit scalairea)

deV 0 dansVHh.


x1

x2

x3

V0H V0h

V0 = V0H ∩ V0h

α

FIGURE 2.3: Une illustration géométrique de la définition du paramètre γ = cosα. On note queγ = 1 dans ce cas.

On introduit l’opérateur d’itérationB : VHh → VHh pour exprimer l’évolution de l’erreur dans laméthode itérative 1.2

un+1Hh − uHh = B(un

Hh − uHh).

Alors, on voit immédiatement que

B = (I − PH)(I − Ph)

où PH : VHh → V0H et Ph : VHh → V0h sont les projecteurs orthogonaux dans le produit scalairea(·, ·) sur les sous-espacesV0H etV0h.

Proposition 2.3 L’algorithme 1.2 converge, sa vitesse de convergence asymptotique (le rayon spec-tral de son opérateur d’itérationsB) est égale àρ(B) = γ2 < 1. De plus,||B|| = γ < 1.

Cette proposition découle du lemme sur la décomposition d’un espace vectoriel en une sommede plans invariants par rapport aux deux projecteurs orthogonaux donnés :

Lemme 2.4 SoitV un espace vectoriel de dimension finie muni d’un produit scalaire (·, ·). SoientV1

etV2 deux sous-espaces tels queV = V1 + V2. Il existent2p (p ≥ 0) vecteursv(m)1 ∈ V1 etv(m)

2 ∈ V2,m = 1, . . . , p, tels que

||v(m)1 || = ||v(m)

2 || = 1, (v(m)1 , v

(m)2 ) = γm, m = 1, . . . , p,

avec1 > γ1 ≥ γ2 ≥ · · · ≥ γp > 0, (2.9)

et les plansLm = Vecv(m)1 , v

(m)2 ,m = 1, . . . , p sont invariants par rapport aux projecteurs ortho-

gonauxP1 : V → V1 etP2 : V → V1.L’espaceV peut être décomposé en une somme directe

V = V0 ⊕⊥ (V1 ∩ V ⊥0 ) ⊕⊥ (V2 ∩ V ⊥

0 ) ⊕⊥ L1 ⊕⊥ · · · ⊕⊥ Lp, (2.10)

où V0 = V1 ∩ V2, m = 1, . . . , p. On observe que tous les termes dans (2.10) sont invariants parrapport àP1, P2 et

γ1 = supv1∈(V1∩V ⊥

0),v1 6=0

v2∈(V2∩V ⊥0

),v2 6=0

a(v1, v2)

||v1||||v2||.

2.2. LA VITESSE DE CONVERGENCE DE LA MÉTHODE ITÉRATIVE 15

On renvoie à [16] pour les détails de la preuve de ce lemme. Elle consiste en une récurrence surk, 0 ≤ k ≤ p dans laquelle on construit successivement les décompositions

V = V0 ⊕⊥ Wk ⊕⊥ L1 ⊕⊥ · · · ⊕⊥ Lk (2.11)

oùWk sont des sous-espaces deV ⊥0 invariants par rapport aux opérateursP1 etP2. Par exemple, pour

k = 0, on poseW0 = V ⊥0 et, pourk = 1, on poseV (1)

1 = V1 ∩Wk−1, V(1)2 = V2 ∩Wk−1,

γ1 = maxv1∈V

(1)1 ,v2∈V

(1)2 ||v1||=||v2||=1

(v1, v2). (2.12)

On prend pourv(1)1 et v(1)

2 les vecteurs sur lesquels le maximum dans (2.12) est atteint. On peutvérifier que les projecteursP1 etP2 renvoient les vecteursv(1)

1 etv(1)2 l’un à l’autre et l’on procède de

la même manière pourk > 1 jusqu’à l’épuisement des paires de vecteurs qui donnent un maximumpositif dans les formules du type (2.12). On peut identifier alors ce qui reste de l’espaceV avec(V1 ∩ V ⊥

0 ) ⊕⊥ (V2 ∩ V ⊥0 ).

Avec le lemme 2.4, il est facile maintenant de démontrer la proposition 2.3. En effet, on pose

V = VHh = V0H + V0h, V1 = V0H , V2 = V0h.

La décomposition (2.10) montre alors que l’opérateur d’itérationB agit sur chaque planLk séparé-ment. Ainsi, son action surLk dans la base(v(k)

1 , v2) peut être représentée par la matrice

Bk =

(0 −γk

0 γ2k

)(2.13)

dont le rayon spectral estγ2k et la normeγk. En parcourant tous les plansLk ainsi que les autres sous-

espaces dans (2.10) sur lesquels l’opérateurB se réduit à zéro, on obtient le résultat de la proposition2.3.

Remarque 2.5 L’algorithme 1.2 peut être considéré comme une méthode de Gauss-Seidel par blocspour résoudre (2.3). Il était présenté dans [16] sous une forme légèrement différente qui permet d’yinclure une utilisation possible de sous- ou sur-relaxation ce qui le rend similaire à la méthode SORpar blocs. Cela donne l’algorithme 2.1. On retrouve l’algorithme des patchs original 1.2 en posantω = 1.

L’analyse de cet algorithme est semblable au précedent. En effet, son opérateur d’itérationss’écrit aussi à l’aide des projecteurs

B = (I − ωPH)(I − ωPh).

On peut donc appliquer encore une fois le lemme 2.4 pour obtenir, après quelques calculs détaillésen [16], les formules pour le rayon spectral

ρ(B) =

ω2γ2

2− ω + 1 + ωγ

2

√ω2γ2 − 4ω + 4, si ω ≤ ω0(γ),

ω − 1, sinon,

avec

ω0(γ) =2 − 2

√1 − γ2

γ2


Entrées : une fonctionu0h ∈ V0h et un nombreω ∈ (0, 2).

pour n = 1...N faireTrouverwh ∈ V0h tel que

a(wh, vH) = (f, vH) − a(un−1Hh , v) ∀vh ∈ V0h.

Poseru

n− 12

Hh = un−1Hh + ωwh

. TrouverwH ∈ V0H tel que

a(wH , v) = 〈f |v〉 − a(un− 12 , v), ∀vH ∈ V0H .

Poserun

Hh = un− 1

2Hh + ωwH

finAlgorithme 2.1: Méthode des patchs relaxée

et la norme deB

||B|| =1

2ω (2 − ω) γ +

√1

4ω2 (2 − ω)2 γ2 + (ω − 1)2.

En particulier,||B|| < 1 pour tousγ ∈ [0, 1[ etω ∈]0, 2[.

La vitesse de convergence asymptotiqueρ(B) ne dépend donc que deγ et du paramètre de re-laxationω. Cette dépendance est illustrée par Fig. 2.4.

FIGURE 2.4: Le rayon spectralρ(B) en fonction deω pour deux valeurs fixée deγ.

2.3. UNE ACCÉLÉRATION : PATCHS HARMONIQUES 17

2.3 Une accélération : patchs harmoniques

D’après la note [18].

On a vu que l’algorithme 1.2 converge et son taux de convergence asymptotique est donné parγ(V0H , V0h)

2. On peut constater ainsi que la convergence est lente si le paramètreγ(V0H , V0h) estproche de 1, c’est-à-dire lorsque il existe des paires des fonctionsvH ∈ V0H et vh ∈ V0h telles quevh ≈ vH . Noter que même une version plus générale de l’algorithme 1.2 utilisant une sur-relaxationpeut permettre d’améliorer le taux de convergence de seulement1 −

√8(1 − γ) + O(1 − γ) quand

γ → 1. Cette très mauvaise convergence a été en effet observée dans la pratique, et c’est presqueinévitable dans les situations des maillages non-emboîtées quelconques. En effet, n’importe quellefonctionvH ∈ V0H avec supp(vH) ∈ Λ sera très proche d’une fonctionvh ∈ V0h en prenant pourvh

comme l’interpolé nodal devH surTh. On aura ainsivh ≈ vH bien quevH 6∈ V0h

Il y a toutefois au moins une situation dans laquelle l’algorithme 1.2 converge assez rapidement, àsavoir lorsque les triangulationsTH etTh sont emboîtées. Cette observation expérimentale est facile àinterpréter au vu de la proposition 2.3 et du lemme 2.4. En effet, pour chaque fonctionvH ∈ V0H avecsupp(vH) ∈ Λ, on trouve une fonctionvh ∈ V0h telle quevh = vH . Comme l’égalité ici est exactecette paire de fonctions n’intervient pas dans la formule pour γ contrairement au cas non emboîté.Pour voir cela plus en détail, on définit

V 0H = vH ∈ V0H : vH(x) = 0 ∀x ∈ Ω \ Λ, (2.14)

de sorte queV 0H = V0 = V0H ∩V0h. On introduit aussiVH comme le complément orthogonal (au sens

du produit scalairea) deV0 dansV0H :

VH = vH ∈ V0H : a(vH , φH) = 0 ∀φH ∈ V 0H. (2.15)

Cet ensemble se compose de fonctions éléments finis qui sont approximativementa-harmoniquesà l’intérieur deΛ sur la triangulationTH . L’espaceVH est ainsi approximativementa-orthogonal àl’espaceV0h, parce quea(φH , φh) doit être petit pour toutφH ∈ VH et pour toutφh ∈ V0h (φh s’annuleà la frontière∂Λ par définition deV0h). C’est pourquoi on peut s’attendre à ce que le paramètreγ(V0H , V0h) = γ(VH , V0h) soit petit dans le cas des triangulations emboîtés. Notons queV 0

H est unsous-espace deV0h. Ainsi l’approximation GalerkinuHh donnée par (2.3) ne changera pas si on yremplaceVHh = V0H + V0h parVHh = VH + V0h. De plus, on pourra remplacer l’espaceV0H parVH

sur toutes les itérations de l’algorithme 1.2 et cela ne changera pas les itérésunHh.

On s’inspire de la dernière remarque pour modifier l’algorithme 1.2 en remplaçant l’espaceV0H

par son sous-espaces ne contenant que les fonctions approximativement harmoniques à l’intérieur deΛ (et c’est dans le cadre général des maillages non emboîtés) dans l’espoir de transférer les bonnespropriétés de convergence du cas emboîtés au cadre général de triangulations arbitraires. Plus préci-sément, on conserve les notations (2.14) et (2.15) pour définir aussiVHh = VH +V0h dans un cas nonemboîtés. Notons queV 0

H se compose désormais de toutes les fonctions deV0H avec des supportsà l’intérieur deΛH où ΛH est une union des triangles (fermés) deTH qui se trouvent entièrement àl’intérieur deΛ. Soitu la solution de (2.1). On cherche maintenant son approximation uHh ∈ VHh

telle que

a(uHh, v) = (f, v), ∀v ∈ VHh. (2.16)


Pour calculeruHh on va utiliser l’algorithme 1.2 en remplacant partoutV0H par VH . Ainsi, on ren-contrera des problèmes du type : étant donnég ∈ V ′

0H , trouveruH ∈ VH tel que

a(uH , v) = 〈g|v〉, ∀v ∈ VH . (2.17)

Dans la pratique, on n’a pas de base explicite des éléments finis pour l’espaceVH de sorte qu’uneimplémentation directe de (2.17) est impossible. Pour contourner cette difficulté, on considère deuxproblèmes éléments finis standard :

(a) TrouverλH ∈ V 00H tel quea(λH , µ) = 〈g|µ〉, ∀µ ∈ V 0

H .

(b) TrouveruH ∈ VH tel quea(uH , v) = 〈g|v〉 − a(λH , v), ∀v ∈ VH .On peut voir facilement queuH dans (b) donne une solution de (2.17). En utilisant les problèmes dutype (a), (b) à chaque itération, on arrive à l’algorithme 1.3 des patchs harmoniques.

En appliquant le lemme 2.4 avecV1 = VH , V2 = V0h on voit immédiatement que l’algorithme 1.3converge vers la solutionuHh de (2.16). Bien que l’espaceVHh dans (2.16) soit plus pauvre queVHh

dans (2.3), la solutionuHh est une approximation deu à l’ordre optimal :

Proposition 2.6 Soitq = max(r, s) + 1 , on suppose que la solutionu de (2.1) est dansHq(Ω). Sile patchΛ est convexe alors la solutionuHh de (2.16) satisfait

|u− uHh|H1(Ω) ≤ C(Hr||u||Hq(Ω\Λ) + hs||u||Hq(Λ)

), (2.18)

où H et h sont les diamètres maximaux des triangles deTH et Th respectivement etC est uneconstante indépendante deH et deh.

Démonstration. Par le lemme de Céa, il suffit de trouver un élémentuHh ∈ VHh satisfaisant àl’inégalité (2.18) avecuHh remplacé paruHh. Pour cela, on choisitu ∈ H1

0 (Ω) égal àu dansΩ \ Λeta-harmonique dansΛ. Alors

||u||Hq(Λ) ≤ C||u||Hq−1

2 (∂Λ)≤ C||u||Hq(Ω\Λ)

et

||u− u||Hq(Λ) ≤ ||u||Hq(Λ) + C||u||Hq−1

2 (∂Λ)≤ C||u||Hq(Λ)

oùC désigne des constantes génériques. On introduit maintenant uH comme la projectiona-orthogonaledeu surV0H . 0n a alors

a(uH , vH) = a(u, vH) = 0, ∀vH ∈ V 0H ,

c’est-à-direuH ∈ VH . En introduisantuh ∈ V0h comme l’interpolé deu−u surV0H etuHh = uH+uh,on conclut la preuve de la même façon que la preuve du théorème2.1.

L’algorithme 1.3 est particulièrement efficace dans le casΛ = ΛH . Cette situation sera dénommée“maillages conformes au bord” parce que le bord∂Λ est composé d’arêtes de triangles deTh ouTH .Le taux de convergence de l’algorithme 1.3 peut être estimé de façon indépendante deH et h dansce cas.

Proposition 2.7 Si Λ = ΛH , alors la vitesse de convergence de l’algorithme 1.3 est majorée par uneconstanteκ ∈]0, 1[ qui ne dépend que deΩ, Λ et de la régularité des maillages.

2.3. UNE ACCÉLÉRATION : PATCHS HARMONIQUES 19

Démonstration. Par le lemme 2.3, la vitesse de convergence asymptotique de l’algorithme 1.3 estdonnée parγ(VH , V0h)

2. Pour estimer cette quantité, on prend n’importe quelle fonction vH ∈VH , vH 6= 0 et on examine l’extension harmoniquev de vH |∂Λ à l’intérieur deΛ, c’est-à-dire lafonctionv ∈ H1(Λ) satisfaisant

∆v = 0 dans Λ, v|Ω\Λ = vH |Ω\Λ.

SoituH ∈ V 0H l’interpolant de Scott-Zhang [63] dev surTH à l’intérieur deΛ. On a alors

uH |Ω\Λ = vH |Ω\Λ et |uH − v|1,λ ≤ CSZ|v|1,Λ

avec une constanteCSZ qui ne dépend que deΛ et de la régularité deTH . Puisque

a(vH − v, uH − vH) = 0,

on voit que

|vH − v|21,λ = |uH − v|21,λ − |uH − vH |21,λ ≤ |uH − v|21,λ ≤ C2SZ|v|21,Λ.

De même, puisquea(vH − v, v) = 0,

on voit que|vH |21,Λ = |vH − v|21,Λ + |v|21,Λ ≥ (1 + C−2

SZ)|vH − v|21,Λ.

Donc, on a pour tousvH ∈ VH et vh ∈ Vh

a(vH , vh) = a(vH − v, vh) ≤ 1√1 + C−2

SZ

|vH |H1(Λ)|vh|H1(Λ) ≤ 1√1 + C−2

SZ

|vH |H1(Λ)|vh|H1(Λ)

ce qui implique

γ(VH , V0h) ≤ 1√1 + C−2

SZ

.

3 Sur la méthode Chimère

3.1 Convergence de l’algorithme de Schwarz sur des maillagesarbitraires non-conformes


Bien que la convergence de l’algorithme de Schwarz soit bienétablie au niveau continu depuisle début de l’histoire des méthodes de décomposition de domaine, une théorie complète manquetoujours au niveau discret surtout dans le cas de superposition arbitraire de maillages. En effet, laconvergence de cet algorithme sur des maillages uniformes aété démontrée seulement dans le cadrede différences finies dans [41]. La preuve est basée sur le principe du maximum et sur la décroissanceexponentielle des solutions des EDP elliptiques loin de la frontière. On a utilisé des idées similairesdans [32] dans le cas des éléments finisP1 sur des maillages triangulaires arbitraires.

On rappelle les conventions introduites dans la description de l’algorithme Chimère 1 :

Ω = ΩH ∪ Λ, ∂Λ ⊂ ΩH , ∂Λ ∩ Γ = ∅, ∂ΩH\Γ ⊂ Λ. (3.1)

On utilise les notationsSh = ∂Λ et SH = ∂ΩH\Γ. Etant données les triangulationsTH sur ΩH

et Th sur Λ, on introduit les espaces d’éléments finisP1 sur TH et Th qu’on désigne parVH et Vh

respectivement,V0H = VH∩H10 (ΩH) etV0h = Vh∩H1

0 (Λ). Pour simplifier les notations, on considèreici seulement le cas du problème de Laplace-Poisson (1.1) avecK = 1. On note donca(u, v) =∫Ω

∇u · ∇v,

ah(u, v) =

∫

Λ

∇u · ∇v et aH(u, v) =

∫

ΩH

∇u · ∇v. (3.2)

Hypothèse 3.1 (principe du maximum discret)On suppose que tous les coefficients non-diagonauxdes matrices de rigidité deah surVh et deaH surVH sont non-positifs. Par conséquent, siuh ∈ Vh

20

3.1. CONVERGENCE DE L’ALGORITHME DE SCHWARZ SUR DES MAILLAGESARBITRAIRES NON-CONFORMES 21

(respuH ∈ VH) est approximativement harmonique, c’est-à-dire

ah(uh, φh) = 0, ∀φh ∈ V0h,

aH(uH , φH) = 0, ∀φH ∈ V0H ,

alors le maximum deuh (respuH) est atteint sur la frontière deΛ (respΩH ).

Hypothèse 3.2 (une variante de principe du maximum fort surΛ) On suppose que la solutionνH ∈VH de

aH(νH , φH) = 0, ∀φH ∈ V0H , νH |SH= 1, νH |ΓH

= 0,

est telle que|νH |L∞(Sh) := λ < 1.

On peut montrer que le principe de maximum discret est validé(pour les éléments finisP1) surles triangulations dont tous les angles sont aigus [42]. Leshypothèses 3.1 et 3.2 impliquent

‖vH‖L∞(Sh) ≤ λ‖vH‖L∞(ΩH ) (3.3)

pour toute fonctionvH ∈ VH telle queaH(vH , φH) = 0, ∀φH ∈ V0H et vH |Γ = 0.

Proposition 3.3 Sous les hypothèses 3.1 et 3.2, l’algorithme de Schwarz converge vers la solutionunique(u∗

h, u∗H) ∈ Vh × VH de

aH(u∗H , wH) = (f, wH), ∀wH ∈ V0H , u

∗H|SH

= γHu∗h, u

∗H |ΓH

= gH

ah(u∗h, wh) = (f, wh), ∀wh ∈ V0h, u

∗h|Sh

= γhu∗H . (3.4)

Démonstration.Considérons le problème couplé : étant donnéuh ∈ Vh, trouvervH ∈ VH , vh ∈ Vh

tels que

aH(vH , wH) = 0, ∀wH ∈ V0H , vH |SH= rHuh, vH |ΓH

= 0

ah(vh, wh) = 0, ∀wh ∈ V0h, vm+1h |Sh

= rhvH .

SoitT : Vh → Vh l’application affine définie parT (uh) = vh. Par le principe du maximum et du faitque les interpolationsrH et rh n’augmentent pas la normeL∞, on voit que

‖vH‖L∞(ΩH ) ≤ ‖uh‖L∞(SH ), ‖vh‖L∞(Λ) ≤ ‖vH‖L∞(Sh).

et donc, par l’hypothèse 3.2,

‖vh‖L∞(Λ) ≤ ‖vH‖L∞(Sh) ≤ λ‖vH‖L∞(ΩH ) ≤ λ‖uh‖L∞(Λ).

Par conséquent,T est une contraction dans la normeL∞(Λ). Par le théorème de contraction deBanach, le processus itératifum

h = Tum−1h , qui est l’algorithme de Schwarz, converge versu∗

h lepoint fixe unique deT .

Afin d’obtenir une estimation d’erreur pour l’algorithme deSchwarz, on aura besoin des estima-tions d’erreur pour les éléments finis standards [61] : soitΠhu ∈ Vh la solution de

ah(Πhu, wh) = ah(u, wh) ∀wh ∈ V0h, Πhu = rhu surSh.

Alors,

||Πhu− u||L∞(Λ) ≤ h2 log1

h||u||H2,∞(Λ),

On a une estimation analogue pourΠHu définie dansVH au lieu deVh.

22 CHAPITRE 3. SUR LA MÉTHODE CHIMÈRE

Proposition 3.4 Sous les hypothèses 3.1 et 3.2, la solution de Schwarz(u∗h, u

∗H) dans (3.4) satisfait

max(||u∗H − u||L∞(ΩH ), ||u∗

h − u||L∞(ΩH ))

≤ C(H2 log1

H||u||H2,∞(ΩH ) + h2 log

1

h||u||H2,∞(Ωh)), (3.5)

avec une constanteC qui ne dépend que des domainesΩH etΩh.

Démonstration.La solution exacteu du problème de Poisson satisfaitu|Γ = 0 et

aH(u, w) = (f, w) ∀w ∈ H10 (Λ), u = rHu+ (u− rHu) surSH ,

ah(u, w) = (f, w) ∀w ∈ H10 (Λ), u = rhu+ (u− rhu) surSh.

Soit e = u∗H − u et ε = u∗

h − u. On choisitw = wH dans la première équation etw = wh dans laseconde :

aH(e, wH) = 0 ∀wH ∈ V0H , e = rHε− (u− rHu) surSH , e|Γ = 0

ah(ε, wh) = 0 ∀wh ∈ V0h, ε = rhe− (u− rhu) surSh.

SoientεH = uH − ΠHu = e+ u− ΠHu, εh = uh − Πhu = ε+ u− Πhu.

Alors εH ∈ VH , εh ∈ Vh et

aH(εH , wH) = 0 ∀wH ∈ V0H , εH = rH(εh + Πhu− u) surSH , εH |Γ = 0

ah(εh, wh) = 0 ∀wh ∈ V0h, εh = rh(εH + ΠHu− u) surSh.

Le principe de maximum donne

‖εH‖L∞(ΩH ) ≤ ‖Πhu− u‖L∞(SH ) + ‖εh‖L∞(SH ),

‖εh‖L∞(Λ) ≤ ‖ΠHu− u‖L∞(Sh) + ‖εH‖L∞(Sh),

et par (3.3)‖εH‖L∞(Sh) ≤ λ‖εH‖L∞(ΩH ).

Donc

max(‖εh‖L∞(Λ), ‖εH‖L∞(ΩH )) ≤ 1

1 − λ(‖ΠHu− u‖L∞(ΩH ) + ‖Πhu− u‖L∞(Λ)).

D’où le résultat.

3.2 Une extension au cas des problèmes non stationnaires

D’après l’article [26] inclus en annexe.

Les résultats sur la convergence de l’algorithme de Schwarzpeuvent être réutilisés dans le cadredes discrétisation en temps et en espace de l’équation de convection-diffusion du type

∂u

∂t+ b · ∇u− ∆u = f dansΩ avecu = g surΓ,

u|t=0 = u0. (3.6)

Bien que cela n’entre pas dans le cadre du présent manuscrit (équations multi-échelles elliptiques),on va indiquer brièvement les résultats de [26] pour donner un autre perspective possible.

3.2. UNE EXTENSION AU CAS DES PROBLÈMES NON STATIONNAIRES 23

La méthode de Schwarz pour l’équation de la chaleur (b = 0)

Comme dans la section précédente, on choisit deux sous-domaines deΩ, à savoirΩH et Λ satis-faisant (3.1), ainsi que les triangulationsTH deΩH et Th deΛ. On garde les notations de Fig. 1.1 eton travaille avec les éléments finisP1 qui forment les espacesVH surTH et etVh surTh.

L’algorithme d’Euler-Schwarz est initialisé paru0H = rHu0, u0

h = rhu0 et on procède en cher-chantun

H ∈ VH , unh ∈ Vh, n = 1, . . . , N tels que

(un

H − un−1H

∆t, wH)H + aH(un

H , wH) = (fn, wH), unH |SH

= rHun−1h , , un

H |Γ = rHgn,

(un

h − un−1h

∆t, wh)h + ah(u

nh, wh) = (fn, wh), u

nh|Sh

= rhun−1H , (3.7)

pour toutwH ∈ V(0H) et pour toutwh ∈ V(0h). Ici fn = f(tn, ·), gn = g(tn, ·), aH,h(u, v) sont défi-nis par (3.2) et les parenthèses(·, ·) désignent le produit scalaireL2 et (·, ·)H (resp(·, ·)h) désignentson approximation “mass lumping” surTH (respTh).

Théorème 3.5Sous les hypothèses 3.1, 3.2 et à condition queu, solution de (3.6) avecb = 0, soitsuffisamment régulière,(un

h, unH) dans (3.7) est une solution approchée pour (3.6) avec les estimations

d’erreurL∞ suivantes

max1≤n≤N

(||unH − u(tn, ·)||L∞(ΩH ), ||un

h − u(tn, ·)||L∞(Λ))

≤ CT

(∆t +H2 log1

H)||u||C2([0,T ],C(ΩH))∩C1([0,T ],H2,∞(ΩH ))

+(∆t+ h2 log1

h)||u||C2([0,T ],C(Λ))∩C1([0,T ],H2,∞(Λ))

, (3.8)

avecT = N∆t le temps final etC une constante qui ne dépend que des domainesΩH etΛ.

Remarque 3.6 On peut associer deux pas de temps différents pour les deux sous-domaines, parexemple∆t pour les problèmes avecun

H et δt pour les problèmes avecunh. Une variante de l’algo-

rithme de Schwarz peut alors être construite si∆t = mδt avec un entierm. On avance en temps lasolution dans le zoomm fois sur chaque pas de temps du problème grossier surΩH . La preuve deconvergence s’étend aisément à ce cas.

Remarque 3.7 L’extension de nos résultats à l’opérateur∇ · (K∇) au lieu de∆ est immédiate.

Convection Diffusion (b 6= 0)

Le terme de convection dans (3.6) peut être discrétisé par leschéma des caractéristiques. Lacombinaison avec la méthode de Schwarz donne l’algorithme suivant : trouverun

H ∈ VH , unh ∈ Vh,

tels que pour touswH ∈ V0H , wh ∈ V0h :

1

∆t(un

H − un−1H oXn−1, wH)H + aH(un

H, wH) = (f, wH),

unH |SH

= rHun−1h , un

H|Γ = gnH ,

1

∆t(un

h − un−1h oXn−1, wh)h + ah(u

nh, wh) = (f, wh),

unh|Sh

= rhun−1H , (3.9)

24 CHAPITRE 3. SUR LA MÉTHODE CHIMÈRE

oùXn−1(x) = X(tn−1) etX est la solution deX = b(X(t), t), X(tn) = x. On considère le cas oùXn−1 peut être calculé exactement (b constante par morceaux, par exemple).

Comme le montre [60], la précision de la méthode caractéristiques-Galerkin se détériore lorsquele pas de temps∆t devient trop petit devant le pas du maillage en espace. On ne peut s’attendre doncà généraliser complètement le théorème 3.5 au casb 6= 0. On a, cependant, un résultat plus faible :

Théorème 3.8Sous les hypothèses 3.1, 3.2 et à condition queu, solution de (3.6) avecb = 0, soitsuffisamment régulière,(un

h, unH) dans (3.9) est une solution approchée pour (3.6) avec les estimations

d’erreurL∞ suivantes

max1≤n≤N

(||unH − u(tn)||L∞(ΩH ), ||un

h − u(tn)||L∞(Λ))

≤ CT

(∆t+H2 log 1

H

∆t)||u||H2,∞(ΩH×[0,T ])

+(∆t +h2 log 1

h

∆t)||u||H2,∞(Λ×[0,T ])

, (3.10)

avecT = N∆t le temps final etC une constante qui ne dépend que des domainesΩH etΛ.

4 Une comparaison numérique desalgorithmes précédents

Nous illustrons maintenant les cinq algorithmes présentésdans chapitre 1 sur l’exemple du pro-blème de Poisson-Dirichlet (2.1) avecK = 1, g = 0 dansΩ = (−1, 1)2 dont la solution exacteest

u = cos(1

2πx1) cos(

1

2πx2) + ηχ(r) exp

(1

ε2− 1

|ε2 − r2|

)

où r =√x2

1 + x22 etχ(r) est la fonction indicatrice du disquer ≤ ε. On prend les valeurs suivantes

pour les paramètres :η = 20 et ε = 0.3. Dans une région proche de(0, 0) où la solution est à sonapogée, nous appliquons un patch avec une grille fine. On choisit les éléments finisP1 pour les deuxespacesVH et Vh. Le logicielFreefem [47] est utilisé pour générer les grilles et pour la mise enœuvre des algorithmes.

On considère trois situations en ce qui concerne le positionnement des maillages dans les algo-rithmes 1.2–1.5 :

I. Le cas des maillages emboîtés comme en Fig. 4.1. On choisitici les deux maillages structuréscartésiens :TH de21 × 21 nœuds surΩ etTh de41 × 41 nœuds surΛ = (−0.2, 0.2)2. On voitdonc que chaque triangle deTH à l’intérieur deΛ est décomposé en 100 petits triangles deTh.

II. Le cas des maillages non emboîtés mais construits de telle façon queΛ est composé dequelques triangles deTH , voir Fig. 4.2. On appellera de tels maillages “conformes aubord"parce que le bord du patchΛ est constitué d’arêtes de triangles grossiers deTH , ainsi qued’arêtes de triangles fins deTh. On obtient ce cas test en utilisant un maillage non structuréTH

et en imposant des arêtes le long du bord du zoomΛ = (−0.2, 0.2)2. Noter que les pas desmaillages sont comparables à ceux du cas présenté en Fig. 4.1. En effet, on construit le maillageTH avec 20 triangles adjacents à chaque côté de∂Ω, le domaine de zoomΛ est le même dansles deux cas et les deux maillagesTh sont presque identiques sauf que ici on a44 × 44 nœudsau lieu de41 × 41.

III. Le cas le plus général des maillages non emboîtés et non conformes au bord comme en Fig.4.3. Maintenant, même le bord du patch∂Λ coupe arbitrairement les triangles deTH . Pourque les maillages de ce cas soient semblable à ceux du cas précédent, on a choisi la mêmetriangulation grossièreTH qu’avant, la triangulation fine est toujours structurée cartésienne

25

26 CHAPITRE 4. UNE COMPARAISON NUMÉRIQUE DES ALGORITHMES PRÉCÉDENTS

avec44 × 44 nœuds, mais elle couvre le patchΛ = (−0.22, 0.22)2 qui est donc un peu pluslarge que le patch du cas II.

FIGURE 4.1: Les maillages emboîtés. A gauche : les deux maillages sur Ω ; à droite : un agrandisse-ment autour deΛ.

FIGURE 4.2: Les maillages non emboîtés, mais conformes au bord. A gauche : les deux maillagessurΩ ; à droite : un agrandissement autour deΛ.

La méthode Chimère 1 est testée sur des maillages comparables à ceux décrits précédemment.Comme il n’y a pas de correspondance directe entre les maillages utilisés par la méthode Chimèreet ceux utilisés par les autres algorithmes 1.3–1.5, on ne présente qu’un exemple de maillages pourla méthode Chimère. On prend notammentΩH comme étantΩ privé du trou(−0.2, 0.2)2 et Λ =(−0.2, 0.2)2. Les maillageTH , Th sont structurés cartésiens,TH contient21 × 21 nœuds (mis à partle trou) etTh contient41 × 41 nœuds.

Le point le plus délicat dans la mise en œuvre des algorithmesde patchs 1.2 et 1.3 est le calculdes termes “mixtes” du type

∫Λ

∇uH · ∇vh avecuH ∈ VH et uh ∈ Vh. Ce calcul ne pose pas deproblème dans le cas I des maillages emboîtés. Il suffit, dansce cas, d’employer une formule de

27

FIGURE 4.3: Les maillages non emboîtés, non conformes au bord. A gauche : les deux maillages surΩ ; à droite : un agrandissement autour deΛ.

quadrature assez précise sur le maillage finTh. Pour calculer les intégrales mixtes exactement surdes triangulations générales, on construit un maillage d’intersection entreTH et Th, c’est-à -direun maillageTHh qui est un raffinement et deTH et deTh. Il suffit alors d’employer une formule dequadrature assez précise surTHh. En pratique, nous construisonsTHh à l’aide du logicielTriangle.Notons qu’un algorithme plus efficace pour la construction d’un maillage d’intersection a été proposédans [45].

On retrouve des difficultés du même genre dans la mise en œuvredes algorithmes 1.4 et 1.5. Lestermes qui posent des problèmes ici sont les intégrales du type

∫ΛK∇uH · ∇vH ou

∫ΛK∇wh · ∇vH

avecuH , vH ∈ VH , wh ∈ Vh. Encore une fois, ces intégrales peuvent se calculer exactement à l’aidedu maillage d’intersectionTHh.

Bien que possible, le calcul exact des intégrales mixtes reste un tâche difficile du point de vuealgorithmique. Il est beaucoup plus facile, en pratique, d’approcher ces intégrales par des formulesde quadrature sur le maillage finTh. De cette façon, on introduit des erreurs d’interpolation dans lesalgorithmes et on ne sait pas maîtriser ces erreurs au niveauthéorique pour l’instant. Néanmoins,des expériences numériques montrent que l’approximation par quadrature ne gêne pas beaucoup laconvergence des algorithmes et l’erreur ainsi introduite est négligeable dans la plupart des situations.

Quelques résultats numériques sont présentés en Figs. 4.4–4.5. Dans la description de ces images,on sous-entend que les intégrales mixtes sont calculées exactement si rien n’est mentionné à cepropos. Sinon, on sous-entend par calcul inexact l’utilisation de quadratures surTh. Au vu de résultatsen Fig. 4.4, on peut faire les observation suivantes :

– La méthode originale des patchs 1.2 converge en général lentement, sauf dans les situationsdes maillages emboîtés. Sinon, cette méthode n’est pas beaucoup plus efficace que la méthodeChimère.

– La version harmonique 1.3 permet de retrouver le taux de convergence élevé du cas “emboîté”sur des maillages quelconques, même si ils ne sont pas conformes au bord.

– Les deux méthodes “semi-Schwarz” convergent au moins aussi vite que la méthode harmo-nique. Par contre, elles sont plus sensibles aux erreurs de quadratures dans les intégrales mixtes.Heureusement, ce petit défaut n’est pas trop important en pratique parce qu’il suffit de fairequelques itérations pour trouver la solution très proche del’optimalité.

28 CHAPITRE 4. UNE COMPARAISON NUMÉRIQUE DES ALGORITHMES PRÉCÉDENTS

– La convergence très lente de la méthode Chimère est conditionnée par la minceur de la zone derecouvrement entreΛ etΩH . Noter néanmoins que en faisant cette zone plus large, c’est-à-direen agrandissant le sous-domaineΛ, on aurait obtenu des problèmes surΛ plus coûteux.

Pour étudier le comportement des erreurs par rapport au raffinement des maillages, on choisit enFig. 4.5 la méthode des patchs harmonique 1.3, la plus robuste. On observe que l’ordre de conver-gence est optimal et que les erreurs ne dépendent pas beaucoup ni du positionnement des maillages(emboîtés – pas emboîtés) ni de la façon de calculer les intégrales mixtes.

Iteration

H1

Erro

r

0 25 50 75

10-1

100

101

chimèrepatchharmoniquesemi-Schwarzsemi-Schwarz-Lagrange

Iteration

H1

Erro

r

0 25 50 75

10-1

100

101


(a) (b)

Iteration

H1

Erro

r

0 25 50 75

10-1

100

101


Iteration

H1

Erro

r

0 25 50 75

10-1

100

101


(c) (d)

FIGURE 4.4: L’évolution de la normeH1(Ω) de l’erreur sur les itérations. (a) – les maillages emboîtésde Fig. 4.1 ; (b) – les maillages conformes au bord de Fig. 4.2 ;(c) – les maillages non conformes aubord de Fig. 4.3 ; (d) – les mêmes maillages que dans le cas (c),mais en utilisant le calcul non exactdes intégrales mixtes.

29

h0.02 0.04 0.06 0.08 0.1

10-4

10-3

10-2

10-1

erreur H 1

erreur L 2

HH2

h0.02 0.04 0.06 0.08 0.1

10-4

10-3

10-2

10-1

erreur H 1

erreur L 2

HH2

FIGURE 4.5: L’erreur par rapport au raffinement des maillages. A gauche : solutions limites obtenuespar l’algorithme des patchs harmoniques sur des maillages emboîtés (situation I de la page 25 et lesraffinements) ; à droite : solutions limites obtenues par le même algorithme avec le calcul non exactdes intégrales mixtes sur des maillages non emboîtés, non conformes au bord (situation III de la page25 et les raffinements).

5 Une variante de l’approche "ZoomNumérique" adaptée aux situations"multi-modèles"

Le travail en cours en collaborations avec Patrick Laborde

Toutes les méthodes décrites jusqu’à présent étaient principalement destinées aux situations qu’onpeut qualifier de problèmes avec des données multi-échelles. On sous-entend par cela que le caractèremulti-échelles de la solution exacte d’une EDP du type (1.1)est dû à la présence d’une forte varia-tion du second membref tandis que les coefficients à l’intérieur des opérateurs différentiels dans leséquations sont assez régulières, ne présentant pas de variations brusques localisées. Une seule dévia-tion de ce cadre jusqu’à ce point était l’application de la méthode des patchs aux problèmes avec dessolutions singulières dues aux particularités de la frontière du domaine du calcul.

Cependant, on trouve très souvent en pratique des situations où le caractère multi-échelles ap-paraît dans les opérateurs différentiels eux-mêmes. L’exemple le plus simple est celui de l’équation(1.1) avec le coefficientK qui varie brusquement à l’intérieur deΛ, sous-domaine deΩ, en étant plusou moins constant en dehors deΛ. On aimerait alors avoir une approche de Zoom Numérique quirésout au niveau grossier un problème qui ne voit pas l’échelle fine, c’est-à-dire un problème avecKplus ou moins constant partout. On peut aussi penser aux multi-échelles dans la géométrie deΩ. Parexemple, siΩ contient un trou ou une fissure d’une taille trop petite par rapport au pas du maillagequ’on envisage d’utiliser, on peut essayer de boucher le trou (ce qui donne le domaine sainΩ ⊃ Ω)et de mailler le domaine sain. La présence du trou sera alors prise en compte par le zoomΛ ⊂ Ω quientoure le trou ou la fissure et où on construit un maillage fin et conforme à la géométrie du trou. Onpeut qualifier ces problèmes comme "multi-modèles" en prévoyant que leur étude est un premier pasvers des situations encore plus complexes où même les équations peuvent changer à l’intérieur dupatchΛ. il pourra s’agir, par exemple, d’un couplage d’une description atomistique dansΛ avec unedescription macroscopique en dehors deΛ.

On ce concentre ici sur la situation la plus simple dans la classe des problèmes "multi-modèles" :l’équation (1.1) dans laquelle le coefficientK varie brusquement à l’intérieur deΛ. L’approche Chi-mère est applicable dans ce cas sans aucune modification. En effet, dans cette approche, on doitcreuser un trouD dansΩ qui peut couvrir la zone de grandes variations deK. Le problème grossier

30

31




∫

Ω

K∇unH · ∇vH =

∫

Ω\Λ

fvH −∫

∂Λ

λn−1h vH +

∫

Λ

K∇un−1H · ∇vH , ∀vH ∈ V0H . (5.1)

Trouverwnh ∈ Vh etλn

h ∈ Mh tels que∫

Λ

K∇wnh · ∇vh −

∫

∂Λ

λnhvh =

∫

Λ

fvh, ∀vh ∈ V0h, (5.2)∫

∂Λ

wnhµh =

∫

∂Λ

unHµh, ∀µh ∈ Mh.

finAlgorithme 5.1: Semi-Schwarz-Lagrange, version pour “multi-modèles”




∫

Ω

K∇unH · ∇vH = ρ

∫

Ω\Λ

fvH − ρ

∫

∂Λ

λn−1h vH + ρ

∫

Λ

K∇un−1H · ∇vH

+ (1 − ρ)

∫

Ω

K∇un−1H · ∇vH ∀vH ∈ V0H .

Trouver (wnh ∈ Vh, λn

h ∈ Mh), solution de (5.2).fin

Algorithme 5.2: Semi-Schwarz-Lagrange, version relaxée pour “multi-modèles”

est alors posé surΩH = Ω \ D et peut être discrétisé sur un maillage de pas beaucoup plus grandque la taille caractéristique des variations deK dansD. Par contre, les algorithmes 1.2–1.5 semblentinapplicables au premier regard parce qu’ils maillent toutle domaineΩ de manière grossière. Il estintéressant de contourner ce problème parce que les algorithmes 1.2–1.5 ont montré leur efficacité parrapport à l’approche Chimère dans les situations "données multi-échelles" décrites précédemment.On peut heureusement adapter notre dernier algorithme 1.5 "Semi-Schwarz Lagrange" pour notreproblème de classe "multi-modèles". L’idée de cette adaptation provient de l’observation que la seuleoccurrence de la partie multi-échelles deK au niveau grossier est dans les termes

∫ΩK∇un

H · ∇vH

et∫ΛK∇un−1

H · ∇vH . On peut alors remplacer dans ces termes le vrai coefficientK par une autrefonction K telle queK = K en dehors deΛ et qui est plus régulière queK dansΛ. Cela donnel’algorithme 5.1.

32 CHAPITRE 5. "ZOOM NUMÉRIQUE" POUR "MULTI-MODÈLES"

Etude de convergence des méthode itératives “multi-modèles” au niveau continu

Il convient d’étudier d’abord la convergence de l’analogueEDP de l’algorithme itératif 5.1 auniveau continu. Notons, qu’une telle étude était sans intérêt dans le cas des algorithmes 1.2–1.5 parcequ’ils convergent en une itération au niveau continu. Ce ne sera pas le cas pour l’algorithme 5.1. Onverra même qu’il diverge dans certaines situations.

On rappelle les notations :Λ est un sous-domaine deΩ. Sans perte de généralité, on peut neconsidérer que le casg = 0. On introduitUn comme l’analogue au niveau continu deun

H et wn

comme celui dewnh . L’analogue continue de l’algorithme 5.1 est alors : à partir desU0 ∈ H1

0 (Ω)et λ0 ∈ H−1/2(Λ) trouver successivementUn ∈ H1

0(Ω), wn ∈ H1(Λ) et λn ∈ H−1/2(Λ) pourn = 1, 2, . . .

∫

Ω

K∇Un · ∇V =

∫

Ω\Λ

fV −∫

∂Λ

λn−1V +

∫

Λ

K∇Un−1 · ∇V, ∀V ∈ H10 (Ω) (5.3)

∫

Λ

K∇wn · ∇v −∫

∂Λ

λnv =

∫

Λ

fv, ∀v ∈ H1(Λ) (5.4)∫

∂Λ

wnµ =

∫

∂Λ

Unµ, ∀µ ∈ H−1/2(∂Λ) (5.5)

En posant

Un =

Un dansΩ \ Λwn dans Λ

(5.6)

on voit tout de suite que si la suiteUn converge dansH10 (Ω), alors la limite est égale àu, c’est-à-dire

la solution de (1.1).Dans le casK = K, la méthode (5.3)–(5.5) est équivalente à la méthode des patchs sans relaxa-

tion (en posantun = wn − Un|Λ, un ∈ H10 (Ω)). Elle converge ainsi en une itération, c’est-à-dire

U1 = u. Dans le cas général :K 6= K la convergence de (5.3)-(5.5) n’est pas malheureusementgarantie. On peut voir cela sur l’exemple

K = 1, K =

α, x ∈ Λ1, x ∈ Ω\Λ

avec une constanteα assez grande. En effet, en posant pour simplifierf = 0, U0 = 0, λ0 6= 0, onmontre aisément par récurrence∆Un = ∆wn = 0 dansΛ pour toutn. Par conséquent,wn = Un|Λpour toutn et l’algorithme peut être réécrit en termes deλn seulement :

λ1 = −αSΛλ0 et λn = (1 − α)SΛλ

n−1 pourn ≥ 2,

oùSΛ : H−1/2(∂Λ) → H−1/2(∂Λ) est l’opérateur défini pour toute fonctionλ sur∂Λ par la relationSΛλ = ∂u

∂n

∣∣∂Λ

oùu ∈ H10 (Ω) satisfait

∫

Ω

∇u · ∇v =

∫

∂Λ

λv, ∀v ∈ H10(Ω).

Cela impliqueλn → ∞ lorsquen → ∞ si la valeur deα est assez grande et siλ0 est choisi, parexemple, comme une fonction propre deSΛ.

Pour remédier au manque de convergence, on peut relaxer la méthode. Cela veut dire dans notrecas qu’on remplaceUn dans (5.3) par une combinaison convexe de l’expression non-relaxée et de

33

l’itération précédente. Soitρ le paramètre de relaxation, alors la version relaxée de l’algorithmes’écrit∫

Ω

K∇Un · ∇V = ρ

∫

Ω\Λ

fV − ρ

∫

∂Λ

λn−1V + ρ

∫

Λ

K∇Un−1 · ∇V (5.7)

+ (1 − ρ)

∫

Ω

K∇Un−1 · ∇V, ∀V ∈ H10 (Ω)

∫

Λ

K∇wn · ∇v −∫

∂Λ

λnv =

∫

Λ

fv, ∀v ∈ H1(Λ) (5.8)∫

∂Λ

wnµ =

∫

∂Λ

Unµ, ∀µ ∈ H−1/2(∂Λ) (5.9)

L’algorithme relaxé converge si le paramètreρ est assez petit.

Proposition 5.1 Il existe un nombreρ0 > 0 (dépendant des domainesΩ, Λ et des coefficientsK, K)tel que l’algorithme (5.7)–(5.9) converge vers la solutionexacte de (1.1) pour toutρ ∈]0, ρ0[. Celaveut dire queUn → u dansH1(Ω) lorsquen → ∞, si on définitUn par (5.6) à partir deUn, wn

donnés par (5.7)–(5.9).

Démonstration.On introduit l’opérateurT : H10 (Ω) → H1

0 (Ω) comme suit : pour toutU ∈ H10 (Ω)

on trouvew ∈ H1(Λ) telle que

−∇ · (K∇w) = 0 dansΛ, w|∂Λ = U |∂Λ (5.10)

et on définitTU ∈ H10 (Ω) par

∫

Ω

K∇(TU) · ∇V =

∫

Ω\Λ

K∇U · ∇V +

∫

Λ

K∇w · ∇V, ∀V ∈ H10 (Ω). (5.11)

L’opérateurT est symétrique dans le produit scalairea(U, V ) =∫ΩK∇U · ∇V . En effet, on peut

réécrire sa définition (5.11) de manière symétrique∫

Ω

K∇(TU) · ∇V =

∫

Ω\Λ

K∇U · ∇V +

∫

Λ

K∇w · ∇w, ∀V ∈ H10 (Ω),

où w ∈ H1(Λ) est la même chose quew mais défini pourV au lieu deU , c’est-à-dire

−∇ · (K∇w) = 0 dansΛ, w|∂Λ = V |∂Λ.

On décomposeH10 (Ω) en somme directeH1

0 (Ω) = V0 ⊕⊥ V1 oùV0 = H10 (Λ) etV1 est le complé-

ment orthogonal deV0 au sens du produit scalairea. Cela veut dire que les fonctions deV1 satisfont−∇ · (K∇U) = 0 dansΛ. En étudiant l’action deT surV0 etV1 séparément, on observe facilementles faits suivants :

– TU = 0, ∀U ∈ V0.– TU ⊂ V1, ∀U ∈ V1, c’est-à-direT est bien défini comme un opérateurV1 → V1.– T est coercive et continu surV1. En effet, la continuité deT est évidente. Pour montrer sa

coercivité on prend n’importe quelleU ∈ V1 et on considèrew ∈ H1(Λ) donné par (5.10). Ona par intégration par parties ∫

Λ

K∇w · ∇(U − w) = 0


Cela implique

∫

Λ

K∇w · ∇U =

∫

Λ

K|∇w|2 ≥ α

∫

Λ

K|∇w|2

= α

(∫

Λ

K|∇U |2 +

∫

Λ

K|∇(U − w)|2)

≥ α

∫

Λ

K|∇U |2

avecα = infx∈ΛK(x)

K(x)> 0.

En posantV = U dans la définition deT on voit maintenant

a(TU, U) = a(T1U, U) ≥ min(1, α)a(U, U)

qui donne la coercivité deT surV1.En notanten = Un − u oùUn est calculé par (5.7)–(5.9), on voit queen et en−1 sont reliés par

en = en−1 − ρTen−1.

On décompose sur chaque itérationen enen = en0 + en

1 avecen0 ∈ V0 et en

1 ∈ V1. Les propriétés deTci-dessus impliquent

en0 = en−1

0

en1 = en−1

1 − ρTen−11

Comme l’opérateurT : V1 → V1 est auto-adjoint et coercif, on aen1 → 0 lorsquen → ∞ à condition

que0 < ρ < 2/||T ||. Cela impliqueUn → u dansH1(Ω \ Λ) et, en particulier,Un → u sur∂Λ dansH1/2(∂Λ). En combinant ce résultat avec (5.8)-(5.9), on voit quewn → u dansH1(Λ).

La discrétisation par éléments finis de l’algorithme relaxé(5.7)–(5.9) est détaillée dans l’algo-rithme 5.2.

Une estimation d’erreur pour les algorithmes 5.1 et 5.2

On rappelle que les approximations de la solution exacteu sont obtenues à chaque itération del’algorithme 5.1 ou 5.2 de la manière suivante :

unHh =

wn

h , dansΛun

H , en dehors deΛ

La fonctionunHh n’est pas continue en général. Par contre, elle appartient àl’espace combiné

VHh =v : Ω → R, tel que∃vH ∈ V 0

H , ∃vh ∈ Vh∫

∂Λ

vHµ =

∫

∂Λ

vhµ ∀µ ∈ Mh

v = vH surΩ \ Λ et v = vh surΛ

dans lequel la continuité à travers∂Λ est imposée de manière faible.

35

Supposons que l’algorithme 5.1 (ou 5.2) converge, c’est-à-dire unH → uH , un

h → uh, λnh → λh

lorsquen → ∞. AlorsuH ∈ VH , uh ∈ Vh, λh ∈ Mh satisfont∫

Ω\Λ

K∇uH · ∇vH =

∫

Ω\Λ

fvH −∫

∂Λ

λhvH , ∀vH ∈ V0H

∫

Λ

K∇wh · ∇vh −∫

∂Λ

λhvh =

∫

Λ

fvh, ∀vh ∈ Vh

∫

∂Λ

whµh =

∫

∂Λ

uHµh, ∀µh ∈ Mh

On peut combineruH etuh dans

uHh =

wh, dansΛuH , en dehors deΛ

qui est l’approximation de Galerkin sur l’espaceVHh, c’est-à-direuHh ∈ VHh satisfait∫

Ω\∂Λ

K∇uHh · ∇vHh =

∫

Ω

fvHh, ∀vHh ∈ VHh (5.12)

Cette formulation ressemble à la méthode “Mortar” sur des grilles non conformes, étudiée dans[33]. On y emprunte l’estimationa priori de cette méthode :

Théorème 5.2Soitr = s = 1 et supposons que la solutionu de (1.1) appartient àH2(Ω). Alors, ona l’estimation d’erreura priorisuivante pour la solutionuHh de (5.12) :

|u− uHh|H1(Ω) ≤ C(H||u||Hq(Ω\Λ) + h||u||Hq(Λ)

), (5.13)

avec une constanteC qui ne dépend pas de pas des maillagesTH etTh.

Il serait préférable de préciser la dépendance de la constanteC sur les coefficientK et K. Cela faitl’objet d’un travail en cours.

Un test numérique dans un domaine “fissuré”

On finit ce chapitre par une application de l’algorithme 5.1 dans une situation où le domaine decalculΩ contient une “fissure ouverte”, voir Fig. 5.1. On prend notammentΩ comme le disque unitéetΩ = Ω \ F avec

F = (x, y) ∈ R2 : x ≤ −0.5 et |y| ≤ |0.5 + x| tan(0.01).Le but est de résoudre numériquement le problème

−∆u = 1 dansΩ, u = 0 sur∂Ω,

en utilisant le maillage grossier qui ne voit pas la fissureF . On construit doncTH sur le disque entierΩ, on prend le zoomΛ dont la géométrie est conforme avec la fissure (voir Fig. 5.1)et on y construitla triangulationTh. L’algorithme 5.1 est modifié comme suit :

– On metK = K = 1 dans (5.1) et (5.2).– On remplaceΩ parΩ dans (5.1).

La convergence de cet algorithme est illustrée en Fig. 5.2. On y compare les itérés de l’algorithmeavec la solution “exacte” de référence obtenue sur une maillage très fin conforme à la géométrie deΩ. On voit que l’algorithme donne une très bonne approximation dès la 4-ième itération.


FIGURE 5.1: Le domaineΩ, la triangulation grossièreTH , le patchΛ et la triangulation fineTh.

IsoValue0.005475320.01642590.02737660.03832720.04927780.06022850.07117910.08212970.09308040.1040310.1149820.1259320.1368830.1478340.1587840.1697350.1806850.1916360.2025870.213537

IsoValue0.005485290.01645590.02742640.0383970.04936760.06033820.07130870.08227930.09324990.104220.1151910.1261620.1371320.1481030.1590730.1700440.1810140.1919850.2029560.213926

la solution exacte la solution grossière

IsoValue0.005480040.01644010.02740020.03836030.04932040.06028040.07124050.08220060.09316070.1041210.1150810.1260410.1370010.1479610.1589210.1698810.1808410.1918010.2027610.213722

IsoValue0.005452510.01635750.02726250.03816760.04907260.05997760.07088260.08178760.09269260.1035980.1145030.1254080.1363130.1472180.1581230.1690280.1799330.1908380.2017430.212648

l’itération 1 l’itération 4

FIGURE 5.2: La solution exacte du problème dans le domaineΩ dessiné en Fig. 5.1 ; la solutiongrossière qui sert à initialiser l’algorithme ; les solutions approchées sur les itérations 1 et 4.

6 Les méthodes de type "éléments finismulti-échelles - MsFEM "

Le travail en cours en collaboration avec plusieurs collègues de l’IMT

Beaucoup de problèmes fondamentaux et importants en pratique possèdent des solutions à échellesmultiples. Les matériaux composites, les milieux poreux, le transport de polluant en milieu urbainsont des exemples de ce type. A la différence des problèmes étudiés plus haut, le comportementmulti-échelles n’est plus localisé dans une petite région,mais il est présent partout dans le domainedu calcul. L’utilisation des idées de “Zoom Numérique” aboutirait donc aux calculs très lourds. Ce-pendant, il peut être nécessaire de réaliser des simulations de tels phénomènes en temps réel. Parexemple, une prédiction du transport de polluant en milieu urbain est un outil d’aide à la décisionpour les autorités lors de situations extrêmes. Il serait donc souhaitable d’avoir une méthode quiprenne en compte les effets des petites échelles sans nécessiter pour autant la résolution de tous lesdétails des petites échelles.

Dans ce but, T.Hou a développé la méthode des éléments finis multi-échelles (MsFEM), voir[44]. La particularité de cette technique est de travaillersur des mailles sous-résolues, c’est-à-direbeaucoup plus grandes que les tailles des échelles les plus petites. Les structures à l’échelle micro-scopique sont prises en compte par le biais de solutions locales représentatives (les fonctions de basemulti-échelles), dans un travail préparatoire (coûteux entemps de calcul) qui peut être réalisé une foispour toutes. Typiquement, ces fonctions sont calculées comme les solutions locales des problèmeshomogènes. Toute l’enjeu de la méthode est alors de bien choisir les conditions aux limites pour cesproblèmes.

Dans ce chapitre on va présenter quelques variantes de cetteméthode sur l’exemple de l’équationde diffusion-réaction. On considéra d’abord le cas 1D où on démontrera une estimation d’erreuroptimale. On tournera après au cas 2D où la théorie reste incomplète. On proposera une nouvellevariante de la méthode MsFEM qui s’avère en pratique plus robuste que les autres variantes proposéesdans la littérature.

37

38 CHAPITRE 6. MSFEM

Le cas 1D

Considérons d’abord le cas 1D : on poseΩ = (0, L) et on chercheu : Ω → R telle queLu = −(νu′)′ + σu = fu(0) = u(L) = 0

(6.1)

avecf ∈ L2(Ω) donnée. On suppose que les coefficientsν et σ sont des fonctionsL∞ satisfaisantν(x) ≥ νmin > 0 etσ(x) ≥ σmin ≥ 0 pour toutx ∈ Ω. En introduisant la forme bilinéaire surH1

0 (Ω)

a(u, v) =

∫ L

0

νu′v′dx+

∫ L

0

σuvdx, (6.2)

on écrit la formulation faible de (6.1) suivante : trouveru ∈ H10 (Ω) tel que

a(u, v) =

∫ L

0

fvdx, ∀v ∈ H10 (Ω). (6.3)

On introduit aussi un maillage surΩ en divisantΩ enN segmentsKi = [xi−1, xi], i = 1, . . . , Npar les nœuds0 = x0 < x1 < · · · < xN = L. On définit le pas de maillageH parmax |xi − xi−1|.Considérons l’espace d’éléments finis multi-échelles adapté à l’opérateurL

VH = uH ∈ C0(Ω) telle queLuH = 0 surKi, i = 1, . . . , N. (6.4)

L’approximation standard de GalerkinuH ∈ VH de la solution de (6.1) peut alors être introduitecomme

a(uH, vH) =

∫ L

0

fvHdx, ∀vH ∈ VH . (6.5)

Proposition 6.1 La solution de GalerkinuH de (6.5) satisfait

||u− uH ||E ≤ H

π√νmin

||f ||L2(Ω)

où || · ||E est la norme d’énergie surH10 (Ω) : || · ||2E = a(·, ·).

Démonstration.D’après le lemme de Céa

||u− uH ||E = infvH∈VH

||u− vH ||E (6.6)

On prendvH ∈ VH comme l’interpolant éléments finis deu, c’est-à-dire

vH(xi) = u(xi), i = 0, 1, . . . , N

et on considère l’erreur d’interpolatione = u− vH . On a sur chaque maille

−(νe′)′ + σe = f on (xi−1, xi)e(xi−1) = e(xi) = 0

Multiplication pare et intégration sur(xi−1, xi) donnent∫ xi

xi−1

ν|e′|2dx+

∫ xi

xi−1

σe2dx =

∫ xi

xi−1

fedx ≤ ||f ||L2(Ki)||e||L2(Ki) (6.7)

39

Comme la fonctione s’annule sur la frontière deKi, on peut appliquer l’inégalité de Poincaré avecla meilleure constante possiblehi/π. Ainsi

||e||L2(Ki) ≤ hi

π

(∫ xi

xi−1

|e′|2dx) 1

2

≤ hi

π√νmin

(∫ xi

xi−1

ν|e′|2dx+

∫ xi

xi−1

σe2dx

) 12

(6.8)

En substituant dans (6.7) on voit que∫ xi

xi−1

ν|e′|2dx+

∫ xi

xi−1

σe2dx ≤ h2i

π2νmin||f ||L2(Ki)

La sommation sur tous les segmentsKi donne le résultat désiré.

Remarque 6.2 L’intérêt de la proposition 6.1 réside dans la constante devantH dans l’estimationd’erreur. En effet, pour la méthode d’éléments finisP1 standards, l’erreur est aussi proportionnelleà H, mais la constante de la proportionnalité dépend de la normeH2 de la solution exacte. Parconséquent, elle peut être énorme si les coefficientsν etσ de l’opérateur différentiel oscillent à unepetite échelle. La méthode MsFEM permet de retrouver l’erreur, toujours proportionnelle àH, maisavec la constante qui ne dépend pas des oscillation des coefficients dans l’équation qu’on résout.

Remarque 6.3 Siσmin > 0 l’estimation d’erreur peut être améliorée :

||u− uH||E ≤ H√π2νmin +H2σmin

||f ||L2(Ω)

en utilisant une majoration plus précise dans (6.8).

Remarque 6.4 Le résultat de la proposition 6.1 peut être étendu aussi au cas d’éléments finis d’ordreplus élevé. Soitk ≥ 2 et

V kH = uH ∈ C0(Ω) tel queLuH ∈ Pk−2 surKi, i = 1, . . . , N.

On considère l’approximation de Galerkin (6.5) avecVH remplacé parV (k)H . On a alors la majoration

d’erreur suivante pour la solutionuH ∈ V(k)H

||u− uH ||E ≤ ckHk

π√νmin

|f |Hk−1(Ω) (6.9)

avecck = 1/(2k−1(k − 2)!√

(2k − 3)(2k − 2)).

En effet, il suffit de construire un interpolantvH ∈ V(k)H qui satisfait (6.9). Pour cela, on prend

fH comme une approximation def qui est un polynôme d’ordrek − 2 sur chaque segmentKi =(xi−1, xi). On peut construire une tellefh satisfaisant

||f − fH ||L2(Ki) ≤ ckhk−1i |f |Hk−1(Ki). (6.10)

On prendvH ∈ V(k)H comme un interpolant deu tel quevH(xi) = u(xi), i = 0, 1, . . . , N etLvH = fH

surKi, i = 1, . . . , N . On considère l’erreur d’interpolatione = u− vH . On a sur chaque maille

−(νe′)′ + σe = f − fH on (xi−1, xi)e(xi−1) = e(xi) = 0

Comme dans la preuve de la proposition 6.1, on voit que∫ xi

xi−1

ν|e′|2dx+

∫ xi

xi−1

σe2dx =

∫ xi

xi−1

(f − fH)edx ≤ ||f − fH ||L2(Ki)||e||L2(Ki)

Cela implique (6.9) en vue de (6.10) et (6.8).


L’implémentation de la méthode multi-échelles dans le cas 1D

La méthode présentée ci-dessus n’offre qu’un intérêt théorique puisque la construction d’espaceVH s’appuie sur des solutions exactes des EDPs sur chaque intervalleKi. Pour mettre cette méthodeen marche en pratique, on doit encore remplacer ces solutions exactes par leurs approximations élé-ments finis sur une grille fine de pash << H. Pour simplifier, on suppose quexi = iH, i = 0, . . . , n.On introduit ainsi les nœudsξi = ih, i = 0, . . . , N oùh = L/N etN >> n est divisible parn, desorte queH/h = N/n soit le nombre d’éléments fins dans chaque élément grossier.On définit

Vh = uh ∈ C0(Ω) tel queuh ∈ P1(ξi−1, ξi), i = 1, . . . , N.

L’espace d’éléments finis multi-échelles sera un sous-espace deVh introduit comme

VHh = uh ∈ Vh tel quea(uh, φh) = 0 ∀φh ∈ Vh tel que suppφh ⊂ Ki pour certaini ∈ [1, . . . , n].

On considère toujours l’approximation standard de Galerkin : trouveruHh ∈ X0 tel que

a(uHh, φHh) =

∫ L

0

fφHhdx, ∀φHh ∈ VHh. (6.11)

Théorème 6.5La solution de GalerkinuHh dans (6.11) satisfait

||u− uHh||E ≤ H

π√νmin

||f ||L2(Ω) + infvh∈Vh

||u− vh||E.

Démonstration. Soit V l’espace fonctionnelH10 muni du produit scalairea(·, ·). On décomposeV

en une somme directeV = V 0 ⊕⊥ VH où V 0 contient toutes les fonctions deV qui s’annulent entous les pointsxi, i = 1, . . . , n et oùVH est son complément orthogonal. On voit immédiatementqueVH est l’espace d’éléments finis multi-échelles "théorique" introduit plus haut dans (6.4). Onreproduit la même décomposition au niveau discret :Vh = V 0

h ⊕⊥ VHh oùV 0h = Vh ∩ V 0 etVHh est

le complément orthogonal deV 0h dansVh. Soituh ∈ Vh la solution éléments finis sur le maillage fin

a(uh, φh) =

∫ L

0

fφhdx, ∀φh ∈ Vh.

Considérons les décompositions uniques deu et deuh :

u = u0 + uH avec u0 ∈ V 0 etuH ∈ VH ,

uh = u0h + uHh avec u0

h ∈ V 0h etuHh ∈ VHh,

oùuH etuHh sont définis par (6.5) et (6.11) respectivement. On a

||u0h||2E = a(uh, u

0h) =

∫ L

0

fu0hdx = a(u, u0

h) = a(u0, u0h) ≤ ||u0||E|| u0

h||E,

puisqueV 0h est orthogonal àVH et àVHh. Par conséquent,||u0

h||E ≤ ||u0||E et

||u− uHh||E ≤ ||u− uh||E + ||u0h||E ≤ ||u− uh||E + ||u0||E = ||u− uh||E + ||u− uH||E.

On conclut en appliquant le lemme de Céa et la proposition 6.1.

41

MsFEM en 2D : des variantes conformes et non conformes

Soit Ω ⊂ R2 un domaine borné polygonal. On considère le problème aux limites : trouveru :Ω → R tel que

Lu = −∇ · (ν∇u) + σu = f dansΩu = 0 sur∂Ω

(6.12)

avecf ∈ L2(Ω) donné. On suppose que les coefficientsν et σ sont des fonctionsL∞ satisfaisantν(x) ≥ νmin > 0 etσ(x) ≥ σmin ≥ 0 pour toutx ∈ Ω. En introduisant la forme bilinéaire surH1

0 (Ω)

a(u, v) =

∫

Ω

ν∇u · ∇vdx+

∫

Ω

σuvdx, (6.13)

la formulation faible du (6.12) s’écrit : trouveru ∈ H10 (Ω) tel que

a(u, v) =

∫

Ω

fvdx, ∀v ∈ H10 (Ω). (6.14)

On notera|| · ||E la norme d’énergie surH10 (Ω) définie comme|| · ||2E = a(·, ·).

Soit TH un maillage surΩ. Pour la simplicité, on se restreint ici au cas oùΩ est le carré unitéet TH est une grille cartésienne rectangulaire surΩ. On noteraEH l’ensemble de tous les arêtes derectangles deTH . Le pas du maillage estH = maxK∈TH

diam(K). Comme on a vu dans le cas 1D,l’idée de la méthode MsFEM consiste à introduire un espace des éléments finis multi-échelles adaptéà l’opérateurL. Au niveau théorique, les fonctions de cet espaces satisferont toujours l’équationLuH = 0 dans chaque mailleK du maillageTH . Au niveau pratique, il faut discrétiser ces EDPssur un maillage finTh. On va ici supposer queTh est une subdivision deTH . On noteraWh l’espacestandard d’éléments finiQ1 surTh et on poseraVh = Wh ∩H1

0 (Ω).Tout l’enjeu de MsFEM, dans le cas 2D, consiste en un choix convenable des conditions aux

limites sur les arêtes du maillageTH pour les fonctions dans l’espace adaptéVH . Dans ce qui suit, onva présenter quelques options possibles. Certains de nos choix amènent à des espaces non conformes,c’est-à-dire,VH 6⊂ H1(Ω) généralement et, par conséquent,VH 6⊂ Vh. Concrètement, les fonctionsde VH seront parfois discontinues à travers les arêtes deTH . On est obligé donc d’introduire uneforme bilinéaire brisée

aH(u, v) =∑

K∈TH

[∫

K

ν∇u · ∇vdx+

∫

K

σuvdx

](6.15)

L’approximation GalerkinuH ∈ VH de la solution du problème (6.12) sera définie comme

aH(uH , vH) =

∫

Ω

fvHdx, ∀vH ∈ VH . (6.16)

1. MsFEM classique [44] On construitVH de manière suivante

VH = uh ∈ Vh : a(uh, vh) = 0 pour toutvh ∈ Vh tel que supp(vh) ⊂ K avecK ∈ TH

etuh est linéaire sur tous les arêtes deTH .

Cela donne un espaceH1-conforme :VH ⊂ Vh ⊂ H10 (Ω). L’inconvénient de cet approche est que

l’approximation linéaire sur les arêtes du maillage n’est pas vraiment adaptée à l’opérateurL, ce quidégrade la qualité de la solution MsFEM.


2. MsFEM avec l’oversampling [44] On agrandit d’abord les mailles : pour toutK ∈ TH , onchoisit un rectangle plus grandO(K) ⊃ K en prenantO(K) comme un ensemble de mailles deTh.L’espace d’éléments finisVH est défini par

VH = u ∈ L2(Ω) : u est continue dans les nœuds deTh,

pour toutK ∈ Th, il existeuh ∈ Vh tel que

uh est linéaire sur toutes les arêtes de∂O(K),

a(uh, vh) = 0 pour toutvh ∈ Vh tel que supp(vh) ⊂ O(K),

etu|K = uh|K

PuisqueuH ∈ VH est continue seulement aux nœuds du maillage, on observe queVH 6⊂ H1(Ω) etpar conséquentVH 6⊂ Vh.

3. MsFEM non conforme Bien que la qualité de la solution MsFEM avec l’oversamplingsoit engénéral meilleur que celle sans oversampling, la deuxième méthode n’est pas complètement satisfai-sant, au moins du point de vue théorique. En effet, la notion de continuité aux nœuds n’a même pasde sens pour les fonctions deH2(Ω) qui est l’espace fonctionnel naturel de notre problème. C’estpour cette raison qu’il est intéressant d’introduire encore un autre approche pour la construction del’espace des éléments finis non conformes (à La Crouzeix-Raviart) dans lequel la continuité est pré-servée au sens des valeurs moyennes sur les arêtes du maillage. Ainsi, on fera de sorte que siE estune arête interne du maillage partagée par deux maillesK1 etK2, alors toutes les fonctionsuH dansle nouvel espace satisferont ∫

E

uH |K1 =

∫

E

uH |K2.

A cet effet, on introduit d’abord un espace auxiliaire où on autorise les sauts sur les arêtes internesdu maillage :

WhH = u ∈ L2(Ω) :

∫

E

[u] = 0 pour toutE ∈ EH ,

pour toutK ∈ TH il existeuh ∈ Wh avecu|K = uh|K.

Ici, [u] désigne le saut sur une arête si cette arête est interne, sinon [u] = u sur une arête du bord. Soit

W 0hH = u ∈ WhH :

∫

E

u = 0 pour toutE ∈ EH.

On introduit l’espace MsFEM non conforme

VH = uh ∈ WhH : aH(uh, vh) = 0 pour toutvh ∈ W 0hH.

En d’autres termes, toute fonctionuh ∈ VH doit satisfaire sur n’importe quelle mailleK ∈ TH

∫

K

ν∇uh · ∇vhdx+

∫

K

σuhvhdx = 0, ∀vh ∈ Wh tel que∫

∂Λi

vh = 0, i = 1, . . . , 4,

43

où ∂Λi, i = 1, . . . , 4 sont les quatre arêtes qui forment le bord deK. Donc il existent des nombresλ1, . . . , λ4 tels que

∫

K

ν∇uh · ∇vhdx+

∫

K

σuhvhdx =4∑

i=1

λi

∫

∂Λi

vh, ∀vh ∈ Wh.

On voit que la restriction deuh àK soit une approximation éléments finis surTh de la solution duproblème aux valeurs limites

Lu = 0 onK, ν∂u

∂n

∣∣∣∣∂Λi

= λi, i = 1, . . . , 4.

Dans le cas le plus simple du laplacien (L = −∆), on retrouve les éléments finis non conformesclassiques de Rannacher-Turek sur des rectangles :uh|K ∈ span 1, x, y, x2 − y2 sur chaque mailleK ∈ TH . Si on travaillait sur des maillages triangulaires, on retrouverait les éléments finis finis deCrouzeix-Raviart :uh|K ∈ span 1, x, y.

Décrivons la construction d’une base pourVH . Chaque fonction de baseφE peut être associée àune arête interneE ∈ EH . La fonctionφE est définie de sorte que son intégrale surE soit égale à 1,tandis que les intégrales sur toutes les autres arêtes sont 0. Le support d’une telle fonction est dansK1 ∩K2 oùK1 et K2 sont deux mailles partageant l’arêteE. NotonsΓ1

i , i = 1, . . . , 4 les arêtesqui composent le bord∂K1 et Γ2

i , i = 1, . . . , 4 celles pour∂K2, en numérotant de telle sorte queΓ1

1 = Γ21 = E. SoitW 1

h (respW 2h ) l’espace de restrictions de fonctions deWh surK1 (resp surK2).

Soitu1h ∈ W 1

h , u2h ∈ W 2

h les solutions des problèmes suivants

∫

K

ν∇u1h · ∇v1

hdx+

∫

K

σu1hv

1hdx =

4∑

i=1

λ1i

∫

Γ1i

v1h, ∀v1

h ∈ W 1h

∫

Γ1i

u1h = δi1, i = 1, . . . , 4

∫

K

ν∇u2h · ∇v2

hdx+

∫

K

σu2hv

2hdx =

4∑

i=1

λ2i

∫

Γ2i

v2h, ∀v2

h ∈ W 2h

∫

Γ2i

u2h = δi1, i = 1, . . . , 4

On voit que la fonction de baseφE peut être construite comme

φE =

u1h surK1,u2

h surK2,0 ailleurs.

Les fonctions de base de l’espace MsFEM non conforme ont doncles mêmes supports localisésque les fonctions éléments finis non conformes habituelles.En pratique, la construction de chaquefonction de base requiert des calculs lourds sur les maillesautour de chaque arête.

Remarque 6.6 Les fonctions de base des espaces MsFEM dans les premières deux variantes sontcaractérisés aussi par leurs supports localisés. notamment, on associe les fonctions de base auxnœuds du maillageTH comme pour les éléments finis conformes habituels. Les détails de constructiondes fonctions de base dans MsFEM classique et MsFEM-oversamling peuvent être trouvés dans [44].


Des résultats numériques

On considère le cas test suivant : le problème (6.12) surΩ = (0, 1)2 avec les coefficients

ν = 1 + 1001cos(50x)>01sin(50y)>0

σ = 0,

où 1 désigne la fonction indicatrice. Le maillageTH est rectangulaire cartésien composé den × ncarrés de tailleH × H, H = 1/n. Le maillageTh est une partition deΩ en triangles obtenue endivisantΩ enN ×N carrés de tailleh×h, h = 1/N et en coupant chaque carré ainsi obtenu en deuxtriangles. Dans MsFEM avec l’oversampling, on choisit les mailles élargiesO(K) 1.5 fois plus largequeK et on les construit par homothétie autour du centre deK.

On présente en Fig. 6.1 le comportement de l’erreur par rapport au raffinement. Les erreurs sontcalculées en prenant la solution standard éléments finisP1 sur le maillage512 × 512 comme solu-tion de référence. On observe que les variantes oversampling et non conforme sont toutes les deuxbeaucoup plus précises que la variante classique. L’avantage de l’approche non conforme sur l’over-sampling est illustré par le fait que ses courbe de convergences ont les mêmes pentes que celles d’uneméthode éléments finisQ1 dans le cas bien résolu. Ce n’est pas le cas pour la MsFEM-oversampling.On voit aussi que la qualité de cette dernière méthode se dégrade quand le pas de maillage devientproche de la taille caractéristiqueε de la petite échelle, c’est-à-dire la période des oscillations deν.Cette observation est consistante avec l’analyse théorique de l’erreur dans [44, Theorem 6.12]. Eneffet, la majoration d’erreur contient un terme proportionnel àε/H. Cette particularité de la varianteavec l’oversampling (visiblement absente de la méthode nonconforme) peut avoir des conséquencesnégatives dans les applications pratiques où la séparationdes échelles n’est pas toujours bien établie.On pense donc que la variante non conforme est un bon point de départ pour le développement desméthodes robustes pour des problèmes du type propagation des polluants dans un milieu urbain.

45

H

erre

urL

2

0.05 0.1 0.1510-6

10-5

10-4

10-3

10-2

10-1

classiqueoversamplingnon conformeréférence H 2

H

erre

urH

1

0.05 0.1 0.15 0.2

10-2

10-1

100

classiqueoversamplingnon conformeréférence H

H

erre

urno

rme

d’en

ergi

e

0.05 0.1 0.15 0.2

10-2

10-1

100


H

erre

urno

rme

max

0.05 0.1 0.15

10-2

10-1


FIGURE 6.1: L’errer de MsFEM. La ligne pointillée montre la puissance appropriée deH, c’est-à-direH2 pour le grapheL2 etH pour tous les autres graphes.

7 Perspectives

Les travaux présentés dans cette parties pourront être poursuivis dans plusieurs directions :– Application des méthodes “Zoom Numérique” à des problèmesindustrielles. L’algorithme 5.1

est particulièrement prometteur pour des problèmes de la mécanique de structures avec depetites inclusions.

– Extension des méthodes “Zoom Numérique” aux problèmes nonsymétriques. Une idée seraitd’utiliser ces algorithmes comme des préconditioneurs dans GMRES.

– L’étude théorique de la nouvelle variante de la méthode MsFEM non conforme.– Une extension de MsFEM aux problèmes du type Stokes (un travail déjà bien entamé).

46

Deuxième partie

Sur la modélisation des fluidesviscoélastiques

47

48 CHAPITRE 7. PERSPECTIVES

8 Une construction du modèle dechaînes bille-ressort pour lessolutions de polymère


Cette partie est consacrée à quelques questions liées à la modélisation des fluides non newtoniens.Rappelons que la modélisation des écoulements des fluides est basée sur une relation entre le champde vitesse et le tenseur des contraintes de Cauchyτ . L’exemple le plus simple d’une telle relation estla loi de Newton,τ = η(∇u + (∇u)T ) − pI, oùη, u etp sont respectivement la viscosité, la vitesseet la pression du fluide. Cette loi mène aux équations de Navier-Stokes qui décrivent convenablementle comportement des fluides les plus répandus composés de petites molécules. Cependant, certainsfluides de structure interne complexe (des fluides composés de macro-molécules, suspensions departicules élastiques, etc.) peuvent exiger des relationsplus compliquées. On parle alors des fluidesnon newtoniens ou, plus spécifiquement, des fluides viscoélastiques lorsque le tenseur des contraintesdépend de l’histoire des déformations subies par un élémentdu fluide.

Nous considérerons ici un des exemples les plus simples de fluides viscoélastiquesé : une so-lution diluée de polymère. On appliquera l’approche de la théorie cinétique, aussi connue commel’approche micro-macro. On représente donc les molécules du polymère par des objets relativementsimples comme des chaînes bille-ressort (Fig. 8, à gauche) ou des dumbbells (“haltères” en français)– deux billes reliés par un ressort, voir Fig. 8, à droite. Bien que cette description ne descende pasvraiment au niveau atomistique, chaque ressort correspondant à un long segment d’une molécule, ellepermet de donner de manière assez précise la distribution statistique des configurations moléculaires.Le couplage micro-macro consiste à étudier l’influence de l’écoulement du solvant (vue au niveaumacroscopique) sur l’évolution de la distribution statistique de chaînes ou de dumbbells et la rétroac-tion des polymères sur le solvant. Bien que les expressions mathématiques qui décrivent l’interactionentre les polymères et le solvant soient bien connues et disponibles dans plusieurs manuels, dont [37]et [57], nous en présentons ici une nouvelle construction qui diffère des approches traditionnelles enplusieurs aspects :

– Notre point du départ est une modélisation complète qui prend en compte les effets d’inertiedes billes dans les chaînes représentant les molécules du polymère. Le cas sans inertie est

49

50 CHAPITRE 8. CONSTRUCTION DU MODÈLE DE CHAÎNES BILLE-RESSORT

(q )F

O

r

r r

r1

23

4

q

q

q1 2

3

x0

−

1

1

F (q ) r

r

1

2O

m

m

x

Q

FIGURE 8.1: Une chaîne bille-ressort (à gauche) et une dumbbell (à droite).

obtenu alors à la limite quand la masse de la bille tend vers zéro, en utilisant les techniques dedéveloppements asymptotiques. Par contre, dans les constructions que l’on peut trouver dans lalittérature, les termes d’inertie dans les équations de mouvement des billes sont négligés tout desuite avec des justifications plus ou moins obscures, voir par exemple paragraphe 13.1 du livre[37]. Bien que le modèle sans inertie soit bien approprié à l’étude de la plupart des phénomènesphysiques, le modèle complet avec l’inertie pourrait apporter de nouvelles informations dansle cas de brusque changement de vitesse, par exemple.

– Notre expression de la force exercée sur le solvant par le polymère est une conséquence directedes équations du mouvement des billes et du principe des actions réciproques. Nous retrouvonsl’expression traditionnelle – la formule de Kramers – dans la limite quand la masse de la billetend vers zéro, par le même procédé des développements asymptotiques que celui utilisé pourdéduire l’équation de Fokker-Planck pour la distributionsstatistiques de chaînes. Par contre,traditionnellement, l’expression de Kramers est obtenue par un raisonnement qui est logique-ment indépendant de la construction de l’équation de Fokker-Planck, voir paragraphe 13.3 de[37] par exemple.

– Notre construction est effectué dans l’hypothèse d’un écoulement non-homogènes de la so-lution de polymère, c’est-à-dire sans aucune restriction sur le champ de vitesse. Cela inclut,entre autre chose, une description du phénomène important associé à de tels écoulements quiest la migration moléculaire. Cela veut dire que la vitesse du centre de masse d’une moléculeest en général différente de celle du solvant ambiant. De tels phénomènes sont hors de portéede modélisation traditionnelle qui est effectuée à la base dans l’hypothèse d’un écoulement ho-mogène, c’est-à-dire en supposant que la vitesse du solvant(et la vitesse moyenne de chaînes)dépend linéairement de la position en espace.

On va présenter le modèle seulement dans le cas des dumbbells. Une extension au cas de chaînesavec un nombre de ressorts quelconque ne présent aucune difficulté.

8.1 Le modèle prenant en compte l’inertie des billes

On considère une solution diluée de polymères qui remplit undomaineΩ ⊂ Rd. La solutionest composée de deux phases, du polymère et du solvant, avec deux vitesses macroscopiques dis-tinctesvp etvs, respectivement. La fraction volumique du polymèreϕ (et, par conséquent, la fractionvolumique du solvant1 − ϕ) peut varier en espace et en temps. On suppose que le solvant est unfluide incompressible newtonien. Sa vitessevs est donc régie par les équation macroscopiques de

8.1. LE MODÈLE PRENANT EN COMPTE L’INERTIE DES BILLES 51

conservation du moment linéaire et de la masse (Navier-Stokes), auxquelles on ajoutera la forced’interaction entre les deux phases. Par contre, la phase dupolymère est modélisée au niveau méso-scopique, tandis que les quantités macroscopiquesvp etϕ sont définies comme des moyennes sur lesgrands ensembles de dumbbells.

On choisit représenter les molécules du polymère par les dumbbells comprenant deux billes (plusprécisément, des points matériels), de massem chacune, réliées par un ressort qui ne porte pas demasse, voir Fig. 8, à gauche. Pour écrire l’équation du mouvement d’une dumbbell, on note d’abordles positions de ses billes numéros1 et 2 par les vecteursr1(t) et r2(t) et leur vitesses patVi = ri,i = 1, 2. La vitesse du fluide ambiant, c’est-à-dire celle du solvant, au pointri est notée parvi =vs(ri). Les équations du mouvement de la billei s’écrivent alors ainsi :

mdVi = (−ζ(Vi − vi) + Fi) dt+√

2ζkBTdWi, (8.1)

dri = Vi dt. (8.2)

On a pris en compte ici les trois forces qui agissent sur une bille d’une dumbbell :

1. La force du ressortFi agissant sur la billei. On suppose queF1 = −F2 = F , où F estune fonction donnée du vecteur bout-à-boutQ = r2 − r1. La justification de cette force setrouve dans une étude de la distribution du vecteur aléatoireq à l’équilibre thermique. Elle doitassurer notamment que cette distribution est proche de la distribution du vecteur bout-à-boutd’une vraie molécule de polymère en équilibre.

2. La force de frottement qui est proportionnelle à la vitesse relative de la bille par rapport ausolvant d’après l’hydrodynamique classique (on peut supposer que le nombre de Reynolds est0 à l’échelle mésoscopique). Le coefficient de traînée est noté ζ .

3. La force brownienne, provenant des impacts aléatoires des petites molécules du solvant surla bille. W1 et W2 désignent donc deux mouvement browniens (processus de Wiener) mu-tuellement indépendants. Le coefficient

√2ζkBT avec la constante de BoltzmannkB et la

températureT provient du principe d’équipartition de l’énergie.

Remarque 8.1 La vitessevi = vs(ri) dans (8.1) doit être comprise comme celle du solvant moyen-née sur une boîte de taille plus grande que la taille de la bille mais plus petite que la distance entreles deux extrémités d’une dumbbell. Les fluctuations dans lavitesse du solvant autour devs sont doncnégligées à l’échelle de cette boîte. Cela implique que le mécanisme de l’interaction hydrodynamiquen’est pas inclus dans notre modélisation, c’est-à-dire lesfluctuations de la vitesse du solvant induitespar les vibrations d’une bille s’effacent sur une distance plus petite que la distance à l’autre bille.Dans une modélisations plus complète, on devrait tenir compte du bruit dans la vitesse du solvant.Le champvs satisferait alors une équation stochastique aux dérivées partielles. Une telle approcheest esquissée dans [56] sous le nom de Coupled Langevin Equations.

Les équations (8.1)–(8.2) sont des équations différentielles stochastiques (EDS) pour les pro-cessus aléatoiresri, Vi. Il est bien connu que la densité de probabilité d’un processus stochastiquesatisfaisant à une EDS, évolue selon l’équation de Fokker-Planck, dont les coefficients sont facilesà déterminer à partir de coefficients de l’EDS, voir [57] par exemple. Pour alléger les notations onécrirar = (rT

1 , rT2 )T , V = (V T

1 ,VT

2 )T , etc. On introduit la fonction de distribution des dumbbellsΨ(r,V , t) telle queΨ(r,V , t) dr dV donne l’espérance du nombre des dumbbells dont les billessont dans l’intervalle[r, r + dr] avec les vitesses dans l’intervalle[V ,V + dV ] au tempst. La fonc-tion de distributionΨ(r,V , t) diffère de la densité de probabilité de processus (r, V ) juste par une


constante de normalisation. Cette fonction évolue donc selon l’équation de Fokker-Planck suivante

∂Ψ

∂t+ ∇r · (V Ψ) +

1

m∇V · ((−ζ(V − v) + F )Ψ) =

ζkBT

m2∇2

V Ψ. (8.3)

On se tourne maintenant vers la modélisation de la phase de solvant. En utilisant les hypothèsesque le tenseur de contrainte est newtonien et que le fluide estincompressible, on peut décrire laconservation de moment linéaire et de la masse par

ρs(1 − ϕ)Dvs

Dt= ηs∇2

xvs + ηs∇x(∇x · vs) − ∇xps + f , (8.4)

∂(1 − ϕ)

∂t+ ∇x · ((1 − ϕ)vs) = 0. (8.5)

Ici, ps est la pression dans le solvant etf = f (x, t) est la densité de force exercée en pointx et entempst sur le solvant par les dumbbells. La dérivée matérielle est associée à la vitesse du solvant,c’est-à-direD

Dt= ∂

∂t+ vs · ∇x.

L’idée cruciale qui permet définirf consiste à observer que toutes les interactions entre le po-lymère et le solvant sont décrites par l’équation du mouvement des dumbbells (8.1). Mise à part laforce du ressort (qui est une force interne de la phase polymérique), l’action du solvant sur le poly-mère est représentée par la force de frottement−ζ(Vi − vi) et la force brownienne. Cette dernièrene doit pas entrer dans le calcul de la force moyennef . En effet, les force Browniennes représententdes impacts aléatoires qu’une molécule du polymère reçoit de la part du solvant ambiant. Maisf estune quantité macroscopique, et donc son calcul entraîne la moyennisation de grandes ensembles dechaînes. Les impacts aléatoires doivent donc s’annuler mutuellement dans cette procédure puisquela probabilité d’un tel impact est la même que celle d’un impact de la même intensité mais dans ladirection opposée. On voit donc que la formule pourf ne doit contenir que la force de frottement. Parle principe de l’action réciproque, cette force est de signeopposé dans l’expression def par rapportà l’équation du mouvement des billes (8.1) :

f (x, t) =

∫ ∫

Ω

ζ(V1 − v1) Ψ|r1=x dr2dV +

∫ ∫

Ω

ζ(V2 − v2) Ψ|r2=x dr1dV . (8.6)

La première intégrale ici prend en compte les dumbbells dontla bille numéro 1 est à la positionx etla deuxième intégrale prend en compte les billes numéro 2 enx.

Pour fermer le système, on doit encore fournir une équation pour la fraction volumique du poly-mèreϕ. On postule qu’elle est proportionnelle à la densité numériquen des dumbbells

ϕ = nVd, (8.7)

où Vd peut être interprétée comme le volume occupé par une dumbbell. La densité numérique estdonnée par

n(x, t) :=1

2

(∫ ∫

Ω

Ψ|r1=x dr2dV +

∫ ∫

Ω

Ψ|r2=x dr1dV

). (8.8)

Cela achève la modélisation de la solution des dumbbells. Lesystème des équations régissantl’écoulement est composé donc de l’équation de Fokker-Planck (8.3) pour la phase de polymère etdes équations du type Navier-Stokes (8.5)–(8.4) pour la phase de solvant qui sont couplées par lesexpressions de la densité numérique, de la fraction volumique et de la force inter-phasique, c’est-à-dire (8.8), (8.7) et (8.6), respectivement. On voit que ces équations ne font pas intervenir la vitesse

8.2. LA LIMITE SANS INERTIE DES BILLES 53

du polymèrevp ce qui est consistant avec notre choix de modéliser le polymère à l’échelle mésosco-pique. Néanmoins, la vitesse du polymèrevp ainsi que la vitesse moyenne de la solutionu, peuventêtre calculées dans un post-processing :

vp =1

2n

(∫ ∫

Ω

V1Ψ|r1=x dr2dV +

∫ ∫

Ω

V2Ψ|r2=x dr1dV

), (8.9)

u = ϕvp + (1 − ϕ)vs. (8.10)

En comparant (8.6) avec les définitions (8.8) et (8.9) et en observant quevi = vs(x) lorsqueri = x,on obtient encore une formule utile pour la forcef :

f = 2nζ(vp − vs). (8.11)

Notons encore que l’équation de Fokker-Planck (8.3) implique la loi de conservation pour la densiténumériquen

∂n

∂t+ ∇x · (vpn) = 0. (8.12)

En multipliant cette équation parVd et en ajoutant (8.5), on a

∇x · u = 0. (8.13)

La solution des dmbbells est donc un fluide incompressible.

8.2 La limite sans inertie des billes

L’équation de Fokker-Planck

On procède maintenant à la dérivation d’une variante sans inertie de notre modèle, c’est-à-dire dela limitem → 0. On introduit les scalings suivants :

m = ε2, p = εV . (8.14)

Ces scalings sont motivés par le théorème de l’équipartition d’énergie qui donne la relation

⟨mV 2

i

⟩= 3kBT

dans l’état d’équilibre thermique. En supposant que la distribution des vitesse n’est pas trop éloignéede celle en équilibre, on obtient donc la valeur caractéristiques de vitesse

√kBT/m, ce qui implique

queV se comporte comme1/ε lorsqueε → 0.

Remarque 8.2 On traiteraε comme un petit paramètre dans la suite, ce qui n’a pas vraiment de senspuisqueε est une quantité dimensionnelle. Cependant, avec les non-dimensionalisations standards,on peut voir facilement que la contre-partie adimensionnelle deε est

√λB/λH où λB = m/ζ est

l’échelle de temps caractéristique pour les fluctuations devitesse etλH = ζ/4H est la périodede relaxation caractéristique d’une dumbbell,H étant la rigidité typique du ressort. L’hypothéseλB << λH est assez bien vérifiée dans la plupart des expériences, cf. [62], ce qui justifie la limiteε → 0. Ici, nous gardons unε dimensionnel pour alléger les notations.


On introduit le développement asymptotique enε pourΨ

Ψ = Ψ0 + εΨ1 + ε2Ψ2 + . . . . (8.15)

On va établir une équation directement pour le terme principal Ψ0 et pour sa distribution marginaleenr : ψ(r, t) =

∫Ψ0 dp .

L’équation de Fokker-Planck (8.3) se réécrit dans les nouvelle variables comme

∂Ψ

∂t+

1

ε∇r · (pΨ) +

1

ε∇p · ((ζv + F )Ψ) =

1

ε2ζQ(Ψ), (8.16)

où

Q(Ψ) := ∇p · (pΨ) + kBT∇2pΨ. (8.17)

= kBT∇p ·(M∇p

(Ψ

M

))(8.18)

etM est la distribution de Maxwell

M(p) = C exp(−p2/2kBT ), (8.19)

avec la constante de normalisationC. En substituant le développement asymptotique (8.15) dans(8.16), on obtient

∂Ψk−1

∂t+ ∇r · (pΨk) + ∇p · ((ζv + F )Ψk) = ζQ(Ψk+1), (8.20)

pourk = −1, 0, 1, . . . avecΨ−1 = Ψ−2 = 0.Posons d’abordk = −1 dans (8.20). Cela donneQ(Ψ0) = 0, ce qui implique queΨ0(r,p, t) est

proportionnel àM(p) pour toutr, t et doncΨ0(r,p, t) = ψ(r, t)M(p).Posons maintenantk = 0 dans (8.20). En multipliant (8.20) parp/ζ et en intégrant par rapport à

p on arrive à

∫pQ(Ψ1) dp =

1

ζ

(∇r ·

∫ppΨ0 dp +

∫p∇p · ((ζv + F )Ψ0) dp

). (8.21)

En utilisant l’identitép + kBTM

∇pM = 0 et en faisant quelques intégrations par parties, on voit que

∫pQ(Ψ1) dp = kBT

∫p∇p ·

(M∇p

(Ψ1

M

))dp = −kBT

∫M∇p

(Ψ1

M

)dp,

= kBT

∫Ψ1

M∇pM dp = −

∫pΨ1 dp. (8.22)

et∫

ppΨ0 dp = ψ

∫ppM dp = −kBTψ

∫p (∇pM) dp

= kBTψδ

∫M dp = kBTψδ. (8.23)

8.2. LA LIMITE SANS INERTIE DES BILLES 55

En substituant ces identités dans (8.21), on obtient la relation entre le premier moment deΨ1 etψ∫

pΨ1 dp =1

ζ((ζv + F )ψ − kBT∇rψ) . (8.24)

D’autre part, en posantk = 1 dans (8.20) et en intégrant par rapport àp, on arrive à

∂ψ

∂t= −∇r ·

∫pΨ1 dp. (8.25)

L’équation (8.24) donne une fermeture pourψ(r, t) dans (8.25) :

∂ψ

∂t=

1

ζ∇r · (kBT∇rψ − (ζv + F )ψ) . (8.26)

=1

ζ

2∑

i=1

∇ri· (kBT∇ri

ψ − (ζvi + Fi)ψ) .

C’est l’équation bien connue de Fokker-Planck pour la solution diluée de dumbbells avec l’inertienégligée.

Expression de la force d’interaction entre le polymère et lesolvant

Il reste à déterminer, dans la même limite de l’inertie négligée, une expression pour la forcefqui apparaît dans (8.4). On écritVi = pi/

√m (= pi/ε) dans (8.6) et on garde seulement les termes

d’ordreO(1) enε. Cela donne la formule pour la force

f (x, t) = ζ

∫ ∫

Ω

p1Ψ1|r1=x dr2dp − ζ

∫

Ω

v1ψ|r1=x dr2

+ ζ

∫ ∫

Ω

p2Ψ1|r2=x dr1dp − ζ

∫

Ω

v2ψ|r2=x dr1. (8.27)

En utilisant (8.24) elle peut être réécrite ainsi

f (x, t) =

(∫

Ω

F1ψ|r1=x dr2 +

∫

Ω

F2ψ|r2=x dr1

)− kBT

(∫

Ω

∇r1ψ|r1=x dr2 +

∫

Ω

∇r2ψ|r2=x dr1

)

=: f s(x, t) + f t(x, t). (8.28)

On identifie icif s comme la partie de force due aux ressorts etf t comme la partie de force due auxfluctuations thermiques.

En passant aux coordonnées du centre de masse de la dumbbellx et du vecteur bout-à-boutQ

x :=1

2(r1 + r2), Q := r2 − r1, (8.29)

on peut représenterf s comme la divergence d’un tenseur

f s(x, t) =

∫F(ψ(x +

1

2Q,Q, t) − ψ(x − 1

2Q,Q, t))dq

= ∇x ·∫ 1/2

−1/2

∫ψ(x + sQ,Q, t)Q ⊗ F dQds.


Pour le deuxième terme de la forcef t, on obtient immédiatement

f t(x, t) = −kBT∇x

(∫

Ω

ψ|r1=x dr2 +

∫

Ω

ψ|r2=x dr1

)= −2kBT∇xn, (8.30)

oùn est toujours la densité numérique des dumbbells calculée à la limiteε → 0 par

n(x, t) =1

2

(∫

Ω

ψ|r1=x dr2 +

∫

Ω

ψ|r2=x dr1

). (8.31)

La forcef dans (8.4) peut être donc représentée comme la divergence d’un tenseur

f = ∇x · τp, (8.32)

oùτp est la contribution polymérique dans le tenseur de contraintes

τp(x, t) :=

∫ 1/2

−1/2

∫

Qs(x)

ψ(x + sQ,Q, t)Q ⊗ F dQds− 2n(x, t)kBTI (8.33)

On obtient donc l’expression de Kramers pour le tenseur des contraintes dans la forme généraliséepour les écoulements non homogènes, que l’on trouve par exemple dans [36].

8.3 Le cas des écoulements localement homogènes

On va retrouver maintenant les équations habituelles du modèle des dumbbells dans le cas desécoulements localement homogènes. Cela veut dire que la vitesse du fluide dépend linéairement deposition sur l’échelle d’une dumbbell. Autrement dit, c’est le cas où 0/L << 1 avec`0 la taillecaractéristique d’une dumbbell etL l’échelle macroscopique de l’écoulement.

On noteV la vitesse caractéristique du fluide,nav la valeur moyen de la densité numériquen.L’équation de Fokker-Planck (8.26) donne la longueur typique d’une dumbbell0 =

√kBT/H où

H est la rigidité typique du ressort. On introduit des variables adimensionnelles :

x∗ =x

L, v∗

s =vs

V, t∗ =

tV

L, Q∗ =

Q

`0, F ∗ =

F

H`0. (8.34)

On réécrit l’équation de Fokker-Planck (8.26) dans les coordonnées du centre de masse de la dumb-bell x et du vecteur bout-à-boutQ (8.29) et on passe aux variables adimensionelles. Cela donne

Dψ

Dt∗= ∇Q∗ ·

(1

2De∇Q∗ψ +

F ∗

2Deψ −

(Q∗ · ∇x∗)v∗

s +1

24

(`0L

)2

(Q∗ · ∇x∗)3v∗s + . . .

ψ

)

+ ∇x∗ ·(

1

8De

(`0L

)2

∇x∗ψ −

1

4

(`0L

)2

(Q∗ · ∇x∗)2v∗s + · · ·

ψ

), (8.35)

oùDe = λV/L est le nombre adimensionnel de Deborah qui mesure l’importance des effets visco-élastiques etλ = ζ/(4H) est le temps de relaxation du polymère. En négligeant les termes d’ordre(`0/L)2 ou plus petit et en revenant aux variables dimensionnellesx et t, mais en gardant la forme

8.3. LE CAS DES ÉCOULEMENTS LOCALEMENT HOMOGÈNES 57

adimensionnelle pourQ, on arrive à l’équation de Fokker-Planck classique pour lesécoulementslocalement homogènes :

Dψ

Dt= ∇Q ·

(1

2λ∇Qψ +

1

2λFψ − (∇xvs)Qψ

). (8.36)

Comme le laplacien par rapport àx a maintenant disparu, on voit que l’intégrale∫ψ(x,Q, t)dQ est

conservée le long des lignes du courant, ce qui implique la conservation de la densité numériquen.On peut donc supposer quen = nav = const. De même, la formule (8.11) implique que

vp − vs = O

(kBT

2ζL

)= O

(V

De

(`0L

)2)

et donc on peut supposervp = vs = u dans ce régime.Dans le cas des écoulements localement homogènes, il est plus commode de travailler avec la

densité de probabilité au lieu de densité de distribution numérique de dumbbells. On fait donc lechangement de variableψ → ψnav/`

d0 de manière queψ(t,x,Q)dQ désigne maintenant la probabi-

lité de trouver une dumbbell ayant le vecteur bout-à-bout dans l’intervalle[Q,Q + dQ] parmi toutesles dumbbells dont le centre de masse est enx au tempst. Ce changement deψ ne modifie pas (8.36)car cette équation est linéaire enψ, mais il va introduire un coefficient dans la formule pour le tenseurdes contraintes (8.33). En effectuant un développement limité enQ dans cette formule, comme on adéjà fait dans (8.35), et en négligeant des termes d’ordre(`0/L)2, on arrive finalement à l’expressionsuivante pour la contribution polymérique au tenseur des contraintes au cas localement homogène

τp(x, t) = nkBT

(∫ψ(x,Q, t)Q ⊗ F dQ − I

)(8.37)

qui estl’expression de Kramershabituelle. On a fait implicitement ici un petit changementdans ledernier terme de (8.37) en ajoutantnkBTI. Cela ne modifie en rien le modèle complet parce quece terme est récupéré comme pression dans l’équation du moment linéaire (8.4). L’avantage de ladéfinition (8.37), par rapport à (8.33), réside dans le fait queτp dans (8.37) s’annule à l’équilibre.

Pour donner le système complet des équations dans le cas localement homogène, on rappelle queles équations (8.36)–(8.37) doivent être couplées avec leséquations Navier-Stokes qui prennent laforme

ρDvs

Dt= ηs∇2

xvs − ∇xp+ ∇x · τp,

∇x · vs = 0

avecρ = ρs(1 − ϕ) etp = ps + nkBT .

9 Sur le choix de la force de ressort


9.1 Motivation physiques

On se pose dans ce chapitre la question du choix raisonnable de la fonctionF(Q) qui donne la loide force de ressort dans le modèle des dumbbells. On se restreint aux écoulements homogènes, c’est-à-dire on suppose que l’écoulement macroscopique est imposé et que le champ de vitesse est linéairepar rapport à la position dans le fluide. On considère toujours une solution diluée des dumbbells eton rappelle que la distribution du vecteur bout-à-boutQ satisfait l’équation de Fokker-Planck (8.36)avec le gradient de vitesse∇xvs = κ supposé être une matrice donnée. Comme on a une équivalenceentre les équations de Fokker-planck et des EDS, on voit que le processus stochastiqueQt évolueselon l’EDS

dQt =

(κQt − 1

2λF(Qt)

)dt+

√1

λdBt, (9.1)

où Bt est le mouvement brownien. La contribution polymériqueτ au tenseur des contraintes peutêtre calculé par (8.37), c’est-à-dire

τ (t) =ηp

λ(−I + E(Qt ⊗ F(Qt))) , (9.2)

où E(·) désigne l’espérance. Le paramètreηp = nkBTλ est connu sous le nom de la viscosité poly-mérique.

La fonctionF(·) devrait être choisie, en pratique, sur la base d’une description plus détaillée desforces à l’intérieur des molécules du polymère. A cette fin, on peut d’abord représenter le polymèrepar une chaîne composée d’un grand nombre de liens, chacun delongueur fixée, qui peuvent être soitindépendants l’un de l’autre (Freely Jointed Chain), soit avoir des directions aléatoires en préservant

58

9.1. MOTIVATION PHYSIQUES 59

les angles fixés entre les liens adjacents (Freely Rotating Chain). Dans les deux cas, le vecteur aléa-toireQt dans (9.1) représente approximativement le vecteur qui va d’un bout à l’autre de la chaîne.En considérant la distribution d’équilibre et en appliquant le théorème central limite, on peut mon-trer au la loi de la force est alors linéaire :F(Q) = Q pour des petite extensionsQ, c’est-à-direpour des petites déviations de l’équilibre. Toutefois, cette loi n’est certainement pas réaliste dans lalimite d’extension complète de la chaîne, parce qu’elle permet aux dumbbells d’être étirées infini-ment. Une alternative mathématiquement attrayante pour introduire l’extensibilité maximum dans lamodélisation est de définir la force comme le sous-differentiel du potentiel suivant

Π(X) =

12|X|2, if |X| < R

+∞, sinon(9.3)

avec unR fixé. Physiquement, cela veut dire que la forceF(.) reste linéaireF(Q) = Q quand lachaîne n’est pas étirée au maximum (|Q| < R). Mais, dès que|Q| atteint la valeur maximaleR, laforce prend momentanément l’amplitude nécessaire pour empêcher la dumbbell d’être étirée encoreplus. Les équations différentielles stochastiques avec unsous-différentiel d’un potentiel infini commele terme de dérive sont bien connues dans la littérature mathématique sous le nom de problème deSkorohod. L’écriture rigoureuse de (9.1) avec la force dérivée de (9.3) est établie de manière suivante :étant donnée la condition initialeq ∈ B(0, R), trouver les processusQt et µt à valeurs dansRd,continus et progressivement mesurables, tels qu’ils vérifient :

– l’équation

dQt =

(κQt − 1

2λQt

)dt− dµt +

√1

λdBt, (9.4)

– la condition initialeQ0 = q,– Qt ∈ B(0, R),– µt est de variation bornée sur[0, T ] etµ0 = 0,– pour tout processusZt continu et progressivement mesurable à valeurs dansB(0, R), on a

∫ T

0

(Qt − Zt)dµt ≥ 0, ∀T > 0.

oùµt est an processus stochastique qui force le processus à rester dans la bouleB(0, R).La formule pour le tenseur des contraintes (9.2) n’est pas facile à interpréter dans ce cas puisque

la forceF n’est plus définie comme une fonction univoque deQ. Pour contourner ce problème, onpropose d’approcher la force exacte∂Π par une expression pénalisée

F ε(Q) = Q +1

εβ(Q), (9.5)

oùβ(Q) = Q − π(Q) etπ(Q) est la projection deQ surB(0, R)

π(Q) =

Q, if |Q| ≤ RR

|Q|Q, sinon.

La solution de (9.4) est donc approchée par le processusQεt qui satisfait

dQεt =

(κQε

t − 1

2λQε

t − 1

2λεβ(Qε

t)

)dt+

√1

λdBt. (9.6)

60 CHAPITRE 9. SUR LE CHOIX DE LA FORCE DE RESSORT

La pénalisation des équations stochastiques réfléchies a été étudiée dans [55] dans le but de montrerl’existence de diffusions réfléchies dans le cas général d’un domaine convexe borné. On peut doncadmettre déjà que les processusQε

t etQt sont bien définis et, en plus,Qεt converge versQt lorsque

ε → 0 dans le sens précisé plus bas. Néanmoins, ces résultats connus ne permettent pas d’assurerl’existence de la limite du tenseur de contrainte pénalisé

τ ε(t) =ηp

λ

(E(Qε

t ⊗ Qεt + Qε

t ⊗ 1

εβ(Qε

t )

)− I

)(9.7)

lorsqueε tend vers 0. Le résultat principal de ce chapitre sera la preuve de l’existence de cette limiteτ(t) = limε→0 τ

ε(t). On donnera aussi une formule pour calculer cette limite à partir de la solutionde (9.4) elle-même. On finira par quelques simulations numériques illustrant notre approche.

Remarque 9.1 Il existe un autre choix usuel deF(·) – la loi FENE (Finitely Extensible Non-linearElastic) [65]

FFENE(Q) =Q

1 − |Q|2/R2. (9.8)

L’existence et unicité des solutionsQt à (9.1) avec la force FENE est montrée dans [49] sous l’hy-pothèseR ≥

√2. Le processusQt ne quitte jamais la bouleB(0, R) dans ce cas. L’objectif de

restreindre l’extension des dumbbells à une longueur maximum est ainsi atteint avec la force FENE.De plus, cette loi donne une bonne approximation de la fonction de répartition pour une “FreelyJointed Chain” en équilibre. Nous pensons néanmoins que la force dérivée du potentiel (9.3) mériteégalement une attention comme une alternative à la loi FENE qui pourrait être plus appropriée pourd’autres modélisations sous-jacentes des macro-molécules.

9.2 Des résultats théoriques sur le modèle avec réflexion

On énonce d’abord le résultat principal.

Théorème 9.2SoitR > 0, T > 0, λ > 0, ηp > 0, q ∈ B(0, R) la condition initiale, etκ une matriceRd×d donnée. Si(Qt, µt) est la solution de (9.4),Qε

t est la solution de (9.6) etτ ε est donné par (9.7),alors la limiteτ := limε→0 τε existe pour toutt ∈ [0, T ]. De plus,τ(t) est un fonction continue detqui peut être calculée par la formule suivante

τ = −ηp∂E(Qt ⊗ Qt)

∂t+ ηpE((Qt ⊗ Qt)κ

T + κ(Qt ⊗ Qt)). (9.9)

Pour simplifier les notations, on va écrireΓ = κ − 12λI etσ2 = 1/λ. Le problème pénalisé (9.6)

devient

dQεt =

(ΓQε

t − 1

εβ(Qε

t)

)dt+ σdBt, Qε

0 = q. (9.10)

Rappelons les résultats de [55] :– Pour chaque1 ≤ p < ∞, il existe une constantC indépendante deε telle que

E

(sup

t∈[0,T ]

|Qεt |p)

≤ C (9.11)

9.2. DES RÉSULTATS THÉORIQUES SUR LE MODÈLE AVEC RÉFLEXION 61

– Pour chaque1 ≤ p < ∞ et0 < T < ∞, on a

limε→0

E(

sup0≤t≤T

|Qt − Qεt |p)

= 0. (9.12)

On peut noter qu’une forme faible du théorème 9.2, c’est-à-dire la formule (9.9) intégrée en tempssur un intervalle(0, T ), est une conséquence immédiate de la formule d’Itô appliquée àQε

t ⊗ Qεt et

de (9.12). La question à laquelle notre théorème répond est de savoir si il est possible d’échangerl’ordre de la limite et de la somme dans cette formule.

La preuve du théorème 9.2 sera présentée comme une suite de lemmes, le moins évident entreeux étant celui d’equi-continuité ent de 1

εE(Qε

t ⊗ β(Qεt )) par rapport àε. On montre, en effet, dans

le lemme 9.6 (avecφ(Q) = Q · β(Q)) que

limh→0+

supε>0

sup0≤t≤T−h

E(Qεt+h ⊗ 1

εβ(Qε

t+h) − Qεt ⊗ 1

εβ(Qε

t)) = 0. (9.13)

Remarque 9.3 Il est intéressant d’observer que ce résultat serait faux sans le mouvement brownien.Pour voir un contre-exemple, considérons le cas uni-dimensionnel (d = 1) de (9.10) avecR = 1,Γ = γ ∈ R, γ > 0, σ = 0 etx ∈]0, 1[. La solution pourε < 1/γ prend la forme

Qεt =

xeγt, t < t,(1 − 1

1−εγ

)e(γ−1/ε)(t−t) + 1

1−εγ, t ≥ t,

où t = 1γ

ln(

1x

). Fixonsh > 0, alors

1

εQε

t+hβ(Qεt+h) − 1

εQε

tβ(Qεt ) =

1

ε(Qε

t+h − Qεt )(Q

εt+h + Qε

t − 1)

Puisquelimε→01ε(Qε

t+h− Qε

t) = γ, limε→0(Q

εt+h

+ Qεt) = 2, on voit que

limh→0+

supε>0

sup0≤t≤T−h

∣∣∣∣1

εQε

t+hβ(Qεt+h) − 1

εQε

tβ(Qεt)

∣∣∣∣ ≥ γ > 0.

Sans perdre de généralité, on va supposer dorénavantR = 1 etσ = 1.On établit d’abord un principe de comparaison pour les solutions de (9.10). On considère

dRεt = (γRε

t − 1

εβ(Rε

t))dt+ dBt, Rε0 = r, (9.14)

avecΓ = γI, γ ∈ R, y ∈ B(0, 1) etγ = sup

|Q|=1

Q · ΓQ.

Lemme 9.4 Si Qεt et Rε

t sont les solutions de (9.10) et (9.14) respectivement, avec|q| ≤ |r| etg : [0,∞[→ R est une fonction croissante, alors

E(g(|Qεt |)) ≤ Ey(g(|Rε

t |)), ∀0 ≤ t ≤ T. (9.15)


Démonstration. Comme le lemme porte sur le comportement des composantes radiales des solu-tions, l’idée de sa preuve est de bien choisir les réalisations des mouvements browniens dans leséquations stochastiques pour synchroniser les composantes du bruit dans les directions deQε

t etRεt .

On obtiendra ainsi des équation quasi unidimensionnelles pour |Qεt | et |Rε

t | et il sera facile de compa-rer leurs solutions. Pour concrétiser cette idée, introduisons pour toutQ ∈ Rd la matrice orthogonaleS(Q) telle queSQ = |Q|e1 etSX = X pour tous les vecteursX orthogonaux au plan(Q, e1).

Considérons les mouvements BrowniensBt =∫ t

0S(Qs)dBs et ˜Bt =

∫ t

0S(Rs)dBs. La substi-

tution dans (9.10) et (9.14) donne

dQεt = (ΓQε

t − 1

εβ(Qε

t))dt+ ST (Qt)dBt, (9.16)

dRεt = (γRε

t − 1

εβ(Rε

t))dt+ ST (Rt)d˜Bt. (9.17)

Comme l’énoncé du lemme ne parle que des espérances deg(|Qεt |) et g(|Rε

t |), on peut changerlibrement les réalisations des mouvements Browniens dans les équations ci-dessus. On choisit donc

Bt =˜Bt = Bt = (B1

t , . . . ,Bdt ). La formule d’Itô donne

d|Qεt |2 = 2(Qε

t · ΓQεt − Qε

t · 1

εβ(Qε

t ) +d

2)dt+ 2|Qε

t |dB1t , (9.18)

d|Rεt |2 = 2(γ|Rε

t |2 − Rεt · 1

εβ(Rε

t) +d

2)dt+ 2|Rε

t |dB1t . (9.19)

Il est facile de voir maintenant que chaque trajectoire de|Qεt |2 reste toujours à gauche de celle de

|Rεt |2, prise sur la même trajectoire du mouvement brownien, ce quiprouve (9.15). En effet, si

|Qεt |2 = |Rε

t |2 en un certain momentt = t0, le coefficient de dérive dans (9.19) est plus grandque celui dans (9.18) à ce moment-là, et donc on va avoir|Qε

t |2 < |Rεt |2 immédiatement aprèst0.

Pour une démonstration plus rigoureuse, on renvoie à [50, par. 5, proposition 2.18] et [24].Ce principe de comparaison permet à donner une majoration uniforme enε pour le tenseur péna-

lisé.

Lemme 9.5 Pour toutr ∈ [0, 1) il existe existe une constanteC(r) telle que siQεt est la solution de

(9.10) avec la condition initialeq ∈ B(0, r), alors pour toutt ∈ [0, T ]

Ex(Qεt · 1

εβ(Qε

t )) ≤ C(r),

La constanteC(r) ne dépend que der etγ.

Démonstration.Soitπε la distribution stationaire de (9.14) etπε ∩ I+ la distributionπε conditionnéesur l’ensembleI+ = z ∈ Rd : |z| > r. Le lemme 9.4 avecg(Q) = Q · 1

εβ(Q) implique

Ex(Qεt · 1

εβ(Qε

t)) ≤ Eπε∩I+(Rεt · 1

εβ(Rε

t)) ≤ Eπε(Rεt · 1

εβ(Rε

t))

πε(I+)(9.20)

La règle d’Itô pour|Rεt |2 donne

Eπε(Rεt · 1

εβ(Rε

t)) = γEπε(|Rεt |2)+

d

2. (9.21)

9.2. DES RÉSULTATS THÉORIQUES SUR LE MODÈLE AVEC RÉFLEXION 63

Il est facile de trouver une formule explicite pour la distribution stationnaireπε(R) = 1Cεφε(R) avec

φε(R) =

eγ|R|2, |R| < 1

eγ|R|2− (|R|−1)2

ε , sinon

etCε =∫

Rd φε(R)dR. En appliquant le théorème de la convergence dominée, on voit que les limiteslimε→0 Eπε(|Rε

t |2) et limε→0 πε(I+) existent et sont finis. Cela implique le résultat du lemme en

combinant les inégalités (9.20) et (9.21).On peut maintenant montrer le résultat d’équi-continuité (9.13) que l’on énonce de manière un

peu plus générale :

Lemme 9.6 SoitQεt la solution de (9.10) avecq ∈ B(0, 1) et soitφ : Rd → R+ telle queφ(z) = 0

for all z ∈ B(0, 1). On suppose que pou toutt ∈ [0, T ]

E∣∣∣∣1

εφ(Qε

t )

∣∣∣∣ ≤ C(|x|) (9.22)

avecC(|x|) indépendante deε. Alors,

limh→0+

supε>0

sup0≤t≤T−h

∣∣∣∣E(

1

ε(φ(Qε

t+h) − φ(Qεt))

)∣∣∣∣ = 0. (9.23)

Démonstration. Fixons un nombrer tel que|q| < r < 1 et α ∈ (0, 1). D’après la propriété deMarkov, on a pour touth > 0, t ≥ hα

E(

1

εφ(Qε

t )

)=

∫

B(0,r)

pQε

hα (z)Ez

(1

εφ(Qε

t−hα)

)dz

+

∫

Rd\B(0,r)

pQε

hα (z)Ez

(1

εφ(Qε

t−hα)

)dz (9.24)

où pZa désigne la densité de probabilité du processusZ au tempsa et Ez désigne l’espérance en

supposant que le processus concerné est parti ent = 0 du pointz. Pour estimer la seconde intégraleci-dessus, on considère le temps d’arrêtσ = infs > 0 : |Qε

s| = r. Cette intégrale peut alors êtremajorée en utilisant l’hypothèse (9.22) :

∫

Rd\B(0,r)

pQε

hα (z)Ez

(1

εφ(Qε

t−hα)

)dz = E

(1

εφ(Qε

t)1|Qεhα |≥r

)≤ E

(1

εφ(Qε

t )1σ≤hα

)

= Ex

(EQε

σ

(1

εφ(Qε

t−σ)

)1σ≤hα

)≤ C(r)P(σ≤hα). (9.25)

Puisque l’événementσ≤hα ne concerne que les partie initiales de trajectoires deQt avant qu’il sortede la boule unité, la probabilitéP (σ≤hα) ne dépend pas deε et, en plus, elle tend vers zéro lorsqueε → 0.

On écrirao(1) pour toutes les quantités qui peuvent être majorées par une fonction deh qui estindépendante deε et qui tend vers zéro lorsqueh → 0. On a montré donc

E(

1

εφ(Qε

t )

)=

∫

B(0,r)

pQε

hα (z)Ez

(1

εφ(Qε

t−hα)

)dz + o(1).


Par des raisonnements similaires, on peut aussi montrer que

E(

1

εφ(Qε

t )

)=

∫

B(0,r)

pQhα(z)Ez

(1

εφ(Qε

t−hα)

)dz + o(1), (9.26)

oùQt est le processus qui ne voit pas la boule :

dQt = ΓQtdt+ dBt : Q0 = x.

On peut maintenant écrire (9.26) en tempst et en tempst + h et prendre la différence entre lesdeux : ∣∣∣∣E

(1

εφ(Qε

t+h)

)− E

(1

εφ(Qε

t )

)∣∣∣∣

=

∣∣∣∣∫

B(0,r)

(pQhα+h(z) − pQhα(z))Ez

(1

εφ(Qε

t−hα)

)dz

∣∣∣∣ + o(1)

≤ C(r)

∫

B(0,r)

|pQhα+h(z) − pQhα(z)|dz + o(1).

La preuve du lemme est ainsi réduite à une estimation de|pQhα+h(z) − pQhα(z)| (noter que cette ex-pression ne dépend pas deε). Pour cela, on peut montrer par un calcul direct, détaillé en [24], que

|pQhα+h(z) − pQhα(z)| ≤ Chδ pourz ∈ B(0, 1), 0 ≤ h ≤ 1 (9.27)

à condition que0 < α < 2d+2

et δ = 1 − α(d+2)2

> 0. La constanteC ne dépend que deΓ et deα.

Démonstration du théorème 9.2Prenonsq ∈ B(0, 1) et t ∈ [0, T ]. On a les estimations sui-vantes pour la solutionQε

t de (9.10) pouri, j tels que1 ≤ i, j ≤ d

E(∣∣∣∣(Qε

t )i1

εβ(Qε

t )j

∣∣∣∣)

≤ C(|q|)

avecC(|q|) indépendante deε. Soit (εn)n≥1 ⊂ R une suite aveclimn→∞ εn = 0 et 1 ≤ i, j ≤ d. Ondésigne parfn(t) = 1

εnE ((Qεn

t )i(β(Qεnt ))j) la composanteij du tenseur1

εnE (Qεn

t ⊗ β(Qεnt )). La

formule d’Itô montre que∀t ∈ [0, T ] et∀n ≥ 1∫ t

0

fn(s)ds = −1

2

(E((Qεn

t )i(Qεnt )j

)− xixj

)+

1

2

(∫ t

0

E(Qεns ⊗ Qεn

s )ΓT + Γ(Qεns ⊗ Qεn

s ))ds

)

ij

+1

2tδij .

Soit

g(t) = −1

2

(E((Qt)i(Qt)j

)− xixj

)+

1

2

(∫ t

0

E(Qs ⊗ Qs)ΓT + Γ(Qs ⊗ Qs))ds

)

ij

+1

2tδij .

En utilisant (9.12), on obtient

limn→∞

∫ t

0

fn(s)ds = g(t). (9.28)

De l’autre côté, d’après les lemmes 9.5 et 9.6, la suite(fn(t))n≥1 est bornée et équi-continue. Par lethéorème d’Ascoli-Arzelà, il existe une sous-suite qui converge dansC0([0, T ]; R) vers la limitef ∈C0([0, T ]; R). La convergence uniforme implique

∫ t

0f(s)ds = g(t) et doncf(t) = g′(t) ∀t ∈ [0, T ].

Comme ce raisonnement ne dépend pas du choix de la suite(εn)n≥1, cela prouve (9.9) sur[0, T ].

9.3. SIMULATIONS NUMÉRIQUES 65

9.3 Simulations numériques

Dans cette section, nous présentons quelques schémas numériques pour le modèle (9.4) condui-sant à une approximation du tenseurτ donné par (9.9) ou comme la limitelimε→0 τ

ε. Comme aucuneanalyse théorique n’a été fait pour les approximations numériques stochastique, on va utiliser commesolution de référence celle de la méthode EDP basée sur l’équation de Fokker-Planck.

Les résultats présentés ci-dessous correspondent à la situation physiques suivante : on considèrel’écoulement de cisaillement en 2D avec

κ =

(0 γ0 0

).

La distribution initiale deQ est prise, dans toutes nos expériences, comme la distribution stationnairedu modèle (9.10) avecκ = 0, autrement dit la distribution normaleN (0, 1) restreinte à la bouleB(0, R).

La méthode Fokker-Planck

Soitψ(t,Q), Q ∈ B (B = B(0, R)) la densité de probabilité du processusQt. Il est bien connuqueψ vérifie l’équation de Fokker-Planck

∂ψ

∂t+ divQ

((κQ − 1

2λQ

)ψ

)=

1

2λ∆Qψ (9.29)

avec les condition aux limites de flux zero :

(−κQψ +

1

2λQψ +

1

2λ∇Qψ

)· n|∂B = 0. (9.30)

L’expression pour le tenseurτ (9.9) peut être réécrite en termes de densité de probabilitéet en utilisant(9.29)–(9.30)

τ = −ηp∂

∂t

∫

B

Q ⊗ QψdQ + ηp

∫

B

(κQ ⊗ Q + Q ⊗ QκT

)ψdQ

=ηp

λ

∫

B

Q ⊗ QψdQ +ηp

λR

∫

∂B

Q ⊗ Qψds− ηp

λI

On utilise une méthode numérique spectrale en passant aux coordonnées polairesr ∈ [0, R],θ ∈ [0, 2π) et en approchantψ par les polynômes enr (sur une grille des points de collocationde Gauss-Legendre-Radau) multipliés par des polynômes trigonométriques enθ. Le système d’équa-tions linéaires est obtenue par la méthode de Galerkin. Il est ensuite discrétisé en temps par le schémaimplicite d’Euler. Les résultats numériques sont présentés en Fig. 9.1. Ils montrent que la méthodeconverge par rapport au raffinement dans l’espace-temps. Onprend donc la solution numérique auniveau le plus raffiné comme référence pour les suite simulations suivantes.


NR NF ∆tLevel 3 8 4 0.1 × 2−3

Level 4 12 8 0.1 × 2−4

Level 5 20 12 0.1 × 2−5

Level 6 26 16 0.1 × 2−6

-5

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

τ xx

t

Level 3Level 4Level 5Level 6

FIGURE 9.1: La partie gauche de la figure décrit les paramètres numériques correspondant aux ni-veaux de discrétisation différents. La partie droite de la figure présente l’évolution desτxx (Top :t ∈ [0, 2], Bottom :t ∈ [1.4, 2]).

Simulations stochastiques

On considère deux méthodes de discrétisation de (9.4) en temps. Introduisons la grille uniformeen tempstn = n∆t et désignerons une approximation deQtn parQ∆t

n . Le nombre de réalisationsaléatoires utilisée pour calculer l’espérance est notéM . Pour toutes les simulations, nous imposonsla relationM ∝ (∆t)2 pour assurer que le bruit stochastique n’affecte pas l’ordre de convergence ence qui concerne le pas de temps.

La méthode de pénalisation

La première méthode est l’algorithme par pénalisation qui est une discrétisation directe de l’EDSpénalisée (9.6)

Q∆tn+1 − Q∆t

n =

(κ(tn)Q∆t

n − 1

2λQ∆t

n − 1

2λεβ(Q∆t

n )

)∆t +

√1

λ(Btn+1 −Btn). (9.31)

Le tenseurτ au tempstn est alors approché par

τ∆tn =

ηp

λE(Q∆t

n ⊗ Q∆tn +

1

εQ∆t

n ⊗ β(Q∆tn )

)− ηp

λI. (9.32)

Le choix du paramètre de pénalisationε est une question délicate. D’après [59], le schéma (9.31)converge faiblement à l’ordre

√∆t si ε ≥ ∆t. Ce résultat n’est pas suffisant pour assurer la conver-

gence de l’approximation du tenseur des contraintes (9.32). Toutefois, on a observé expérimentale-ment que la convergence deτ est d’ordre∆t en normeL2(0, T ) si on poseε = ∆t. On présente enFig. 9.2 l’évolution du tenseur des contraintes et de sa variance qui est estimée sur l’ensemble de 100simulations indépendantes. On observe clairement la convergence de la solution par pénalisation versla solution de référence en Fig. 9.2.

9.3. SIMULATIONS NUMÉRIQUES 67

-5 0 5

10 15 20 25 30 35 40 45 50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

τ xx

t

Fokker-PlanckLevel 0Level 3Level 6

25

30

1.4 1.5 1.6 1.7 1.8 1.9 2

τ xx

t

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Var

(τxx

)

t

Level 0Level 1Level 2Level 3Level 4Level 5Level 6

FIGURE 9.2: La méthode de pénalisation dans l’écoulement de cisaillement avecγ = 10. On consi-dère sept niveaux de discrétisationL ∈ 0, 1, 2, 3, 4, 5, 6. Les paramètres numériques à chaqueniveauL sont∆tL = 0.1 × 2−L etML = 250 × 4L. A gauche : l’évolution deτxQ ; en bas : un zoomsur l’intervallet ∈ [1.4, 2]) ; à droite : l’évolution de la variance.

-5

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

τ xx

t

Fokker-PlanckLevel 0Level 3Level 6

25

30

1.4 1.5 1.6 1.7 1.8 1.9 2

τ xx

t

10-3

10-2

10-1

100

101

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Var

(τxx

)

t

Level 0Level 1Level 2Level 3Level 4Level 5Level 6

FIGURE 9.3: La méthode de réflexion symétrique dans l’écoulement decisaillement avecγ = 10 etles mêmes paramètres de discrétisation qu’en Fig. 9.2.


La méthode de réflexion symétrique

La seconde méthode est basée directement sur l’EDS réfléchie(9.4). L’idée est de faire une ré-flexion par rapport à la frontière∂B lorsque le processus la touche [38]. Chaque pas de cet algorithmeest composé de deux sous-pas suivants :

– On fait avancer d’abord le processus stochastique sans prendre en compte la force de réflexionsur la frontière∂B(0, R) :

Q∆tn+1 − Q∆t

n =

(κ(tn)Q∆t

n − 1

2λQ∆t

n

)∆t+

√1

λ(Btn+1 − Btn) (9.33)

– S’il n’y a pas de réflexion, c’est-à-dire|Q∆tn+1| ≤ R, on poseQ∆t

n+1 = Q∆tn+1. Sinon, on prend

Q∆tn+1 comme l’image du miroir deQ∆t

n+1 par rapport à∂B(0, R), c’est-à-dire

Q∆tn+1 =

2R− |Q∆tn+1|

|Q∆tn+1|

Q∆tn+1

Ce schéma converge faiblement à l’ordre∆t [38]. Le tenseur des contraintes est définie par l’expres-sion (9.9). La discrétisation de la dérivée en temps par une différence finie donne l’approximation

τ∆tn = − ηp

∆tE(Q∆t

n+1 ⊗ Q∆tn+1 − Q∆t

n ⊗ Q∆tn

)

+ ηpE(κ(tn)Q∆tn ⊗ Q∆t

n + Q∆tn ⊗ Q∆t

n κT (tn)).(9.34)

Les résultats obtenus par cette méthode sont présentés en Fig. 9.3. On observe toujours la convergencevers la solution de référence.

En résumé, les deux méthodes stochastiques donnent des approximation deτ avec une erreurd’ordre∆t. Les résultats de la méthode de réflexion symétrique sont plus précis que ceux de la mé-thode de pénalisation. Par contre, la méthode de pénalisation donne une approximation deτ beaucoupmoins bruitée que l’autre méthode.

10 Perspectives

Les travaux que nous venons de présenter dans cette partie pourront être poursuivis dans plusieursdirections :

– Inclusion de la balance d’énergie dans le modèle original avec inertie de billes. En effet, pourl’instant, nous avons considéré seulement la balance du moment linéaire en supposant que lesolvant est isotherme. L’inclusion d’énergie permettra deconsidérer rigoureusement les situa-tion où la température (et donc l’intensité des impacts browniens) peut varier en espace et entemps. Le passage à la limitem → 0 nous obligera alors à considérer des termes d’ordreε2,mais la technique des développements asymptotiques sera toujours applicable.

– Expériences numériques plus poussés pour étudier les prévisions rhéologiques des modèlesavec inertie et avec la force réfléchie.

– Applications possibles de ces deux modèles pur raffiner la modélisation de globules rouges pardes dumbbells en écoulements sanguins, comme proposé dans [58]

69

Troisième partie

Autres travaux

71

11 Méthodes numériques pour lesécoulements biphasiques

11.1 Sur la discrétisation en temps en simulation desécoulements avec des particules


Une de difficultés liées aux simulations des écoulements desfluides avec des particules rigidesconsiste à prendre en compte les configurations où les deux particules (ou une particule et la paroi)sont presque en contact. En effet, en raison de forces de lubrification, une particule lisse enserrée dansun fluide visqueux ne peut pas toucher le mur en un temps fini. Unrésultat similaire est valable aussidans le cas de particules multiples. Une attention particulière doit être donc portée à la discrétisationpour bien prendre en compte les forces de lubrification d’uneamplitude énorme et empêcher lecontact artificiel entre les particules. En effet, en présence de force de lubrification, les équationsrégissant le mouvement d’une particule nécessitent des simulations avec un pas de temps minusculeafin d’obtenir une solution acceptable physiquement. Pour contourner ce problème, on a présentédans [27] une idée qui consiste à interdire à la particule de s’approcher trop près de la paroi lors d’unesimulation. On choisit donc un seuilqs pour la distanceq entre la particule et la paroi et on remplacela vraie trajectoire de la particule, dès qu’elle touche le seuil, par une approximation dans laquelle ladistanceq est maintenue égale àqs jusqu’à un possible rebond de la particule sur la paroi. Le momentdu rebond est prédit en utilisant une quantité auxiliaire (une approximation grossière de la vitesse)qui est calculée tout au long de l’intervalle de temps où la particule est bloquée à la distanceqs de laparoi. Cette approche rappelle le modèle de “gluey particles” de [54, 52] oùqs est effectivement misà zéro et on considère la limite de viscosité évanescente. Cependant, les motivations dans [27] sonttout à fait différente de celles derrière le modèle de “glueyparticles”. Ce dernier est conçu commeune alternative simple aux équations standards de type Navier-Stokes, éventuellement corrigées pourtenir compte de la rugosité de surface de la particule. Par contre, notre approche est de prendre leséquations classiques telles quelles et de fournir un outil pour leur discrétisation en temps robuste.

73

74 CHAPITRE 11. MÉTHODES NUMÉRIQUES POUR LES ÉCOULEMENTS BIPHASIQUES

Cela signifie, en particulier, que l’on construit une méthode assez précise et valable pour n’importequelle valeur de la viscosité, pas nécessairement petite. Anoter également, que le seuilqs dépendgénéralement du pas de discrétisation en temps. C’est donc un dispositif purement numérique qui n’aaucun sens physique.

11.2 Une méthode du type domaines fictifs pour desécoulements avec des bulles de gaz


Ce travail concerne la modélisation et la simulation numérique du mouvement des bulles de gazdans un fluide. On représente les bulles par des corps déformables de masse négligeable dont laposition et la forme sont décrites par un petit nombre de paramètres qui évoluent au cours du temps.Le mouvement de ces corps est régi par les forces hydrodynamiques provenant du fluide ambiantet des forces extérieures telles que la gravitation. Pour chaque bulle, les équations d’équilibre desforces sont obtenues en exigeant que le travail des forces appliquées sur la surface des bulles soit nulpour tous les déplacements virtuels, c’est-à-dire pour ceux qui sont compatibles avec les contraintesimposées par la forme des bulles. Pour tenir compte numériquement de la présence d’une quantitéimportante de bulles, on utilise la technique des domaines fictifs [46], c’est-à-dire qu’on remplitle domaine occupé par les bulles par le fluide (fictif) identique au fluide ambiant et on résout leséquations de Navier-Stokes pour une densité et une viscosité constantes dans le domaine tout entier.On impose l’égalité des vitesses du fluide et d’un corps déformable sur sa frontière par l’introductionde multiplicateurs de Lagrange. La méthode de jauge [43] estutilisée pour résoudre les équationsde Navier-Stokes, ce qui permet de ne résoudre que des équations du type de Laplace à chaque pasde temps dans un schéma évolutif. Les calculs sont effectuéssur une grille cartésienne, de sorte queles systèmes linéaires peuvent être résolus par FFT. On inclut la possibilité d’une coalescence desbulles en considérant celle-ci comme un processus se déroulant pendant un petit intervalle de temps.Pour le traitement numérique, on suppose que la coalescencecommence lorsque deux bulles sontsuffisamment proches et que la bulle qui est plus petite est “avalée” par la plus grande au cours dequelques pas de temps. On a considéré trois exemples de familles de forme de bulles, soit en 2D -des ellipses rigides (l’équilibre des forces se réduit dansce cas aux équations d’Euler pour le corpsrigide), des ellipses déformables dont l’aire est constante, ce qui permet d’avoir une contribution nontriviale des forces de la tension superficielle dans les équations des forces, et une famille de formesdéterminés par des polynômes d’ordre 4. Le logiciel développé permet de simuler des centaines desbulles dans ces deux situations. La méthode a été aussi appliquée à des ellipsoïdes rigides en 3D.

12 Méthodes numériques pour desproblèmes fortement anisotropes

12.1 Un schéma du type "Asymptotic-Preserving" pour uneéquation de diffusion anisotrope


Des problèmes fortement anisotropes appariassent dans plusieurs applications physiques et indus-trielles. C’est le cas, par exemple, dans la physique des plasmas lorsqu’on doit traiter de fortes valeursdu champ magnétique. Dans le contexte d’un plasma faiblement ionisé quasi-neutre, l’équation decontinuité, permettant de calculer le potentiel, contientune matrice de conductivité qui possède unetrès forte anisotropie dans la direction du champ magnétique. Noter que la direction peut varier enespace et en temps. Elle peut être donnée aussi par une résolution numérique d’autres équation dumodèle. D’où la nécessité de développer des méthodes robustes qui fonctionnent bien pour n’importequelle grandeur d’anisotropie et qui n’ont pas besoin d’adapter le maillage à la direction d’anisotro-pie.

On propose dans [28] une méthode pour une équation de diffusion anisotrope qui s’inscrit dansle cadre des schémas “Asymptotic Preserving” (AP). Cette technique est conçue pour des problèmesavec un petit paramètreε, ce qui est notre cas, où la diffusion dans la direction d’anisotropie estd’ordre1/ε et elle est d’ordre 1 dans les directions orthogonales. La méthode AP donne une solutionprécise pour toutes les valeurs deε sur un maillage fixé. La dérivation d’une telle méthode exiged’identifier le modèle limite quandε → 0. On reformule alors le problème original de telle sorte quele système d’équations obtenu contient, à la fois, le modèlelimite et le problème d’origine avec unetransition continue par rapport àε. Dans le cas de l’équation de diffusion anisotrope, la solution duproblème limite est constante le long des lignes de champ (direction d’anisotropie). L’idée est doncde décomposer la solutionφ pour ε > 0 en deux parties (φ = p + q) : une partie moyennep quiest constante le long des lignes de champ et une partie de fluctuationq qui donne une correctionde la moyennep. On utilise des maillages qui ne sont pas nécessairement adaptés à la direction

75

76 CHAPITRE 12. PROBLÈMES FORTEMENT ANISOTROPES

d’anisotropie. Cela est possible grâce à l’introduction demultiplicateurs de Lagrange assurant que lapartie moyennep est bien constante le long des lignes de champ.

Des cas plus complexes, quand l’anisotropie n’intervient que dans une partie du domaine ducalcul, sont maintenant à l’étude.

12.2 Méthodes adaptatives sur des maillages anisotropes pourles problèmes non stationnaires


L’article [23] présente des nouveaux estimateurs d’erreura posteriori pour l’équation de la cha-leur ∂u/∂t − ∆u = f discrétisée en temps par le schéma de Crank-Nicolson et en espace par deséléments finis continues linéaires sur des maillages hautement anisotropes. L’indicateur d’erreur dueà la discrétisation en espace est obtenu à l’aide d’estimations d’interpolation anisotrope, tandis quel’indicateur d’erreur due à la discrétisation en temps est obtenu à l’aide d’une reconstruction quadra-tique par morceaux de la solution discrète, à partir d’une approximation par différences finies pour∂2u/∂t2. L’utilisation de cette reconstruction dans l’estimationd’erreur est originale, même dans lecadre des éléments finis isotropes. Elle s’avère plus efficace qu’une autre reconstruction proposée an-térieurement dans la littérature [34] qui était basée sur une approximation par différences finies pour∂f/∂t + ∆(∂u/∂t). Des expériences numériques confirment que l’indicateur d’erreur et la méthodeadaptative proposés dans [23] sont d’ordre optimal par rapport aux pas de maillage en espace et entemps.

13 Méthodes numériques nonprobabilistes pour le modèle deschaînes

De nombreux phénomènes physiques, chimiques, biologiqueset autres sont décrits mathémati-quement par des EDP de très haute dimension. On peut penser, par exemples, à des réactions chi-miques impliquant des dizaines des composants, à des fluctuations des marchés financiers etc. Unexemple typique est le modèle bille-ressort dans la théoriecinétiques des polymères (voir Fig. 8, àgauche). Une molécule de polymère linéaire y est représentée par une chaîne constituée de(d + 1)billes rejointes successivement pard ressorts sans masse. Notons que le modèle des dumbmells (Fig.8, à droite) n’est qu’un cas particulier de celui-ci avecd = 1. Les chaînes de bille-ressort sont re-présentées pard vecteurs aléatoiresqj . Leur dynamique dans un écoulement homogène est donnéesoit par l’EDS du type (9.1) soit par l’équation de Fokker-Planck pour la fonction de densité deprobabilitéψ(t, q1, . . . , qd) :

∂ψ

∂t= Ld

FP (ψ) =d∑

k=1

Lkψ − 1

4λ

d∑

k=2

(Nk−1x Mk

x +Nk−1y Mk

y )ψ − 1

4λ

d−1∑

k=1

(Nk+1x Mk

x +Nk+1y Mk

y )ψ.

(13.1)où

Lk = ∇k · (−∇v · qk +1

2λF (qk) +

1

2λ∇k), M

kx =

∂

∂qk,x, Mk

y =∂

∂qk,y,

Nkx = Fx(qk) +

∂

∂qk,xetNk

y = Fy(qk) +∂

∂qk,y.

avec

qk =

(qk,x

qk,y

)et ∇k =

∂∂qk,x

∂∂qk,y

.

Les méthodes numériques traditionnelles sont pratiquement inapplicables dans ces cas (pourd > 3) car leur coût croît exponentiellement avec la dimension. Cependant, depuis quelques an-nées, plusieurs méthodes ont été proposées pour de tels problèmes. De manière très globale, on peut

77

78CHAPITRE 13. MÉTHODES NUMÉRIQUES NON PROBABILISTES POUR LEMODÈLE

DES CHAÎNES

les résumer en disant qu’on cherche une approximation

ψ ≈m∑

i=1

α(i)1 (q1) · · ·α(i)

d (qd) (13.2)

avec certaines fonctions uni-directionnellesα(1)i (q) ∈ VN et un nombre de termesm pas trop élevé.

VN désigne ici un espace de dimension finie utilisé pour l’approximation des fonctions d’une variableq. Dans la première approche, dite “Sparse Grids” [40] ou “Sparse Tensor Products”, on choisitapriori les espaces pour des combinaisons linéaires dans (13.2). L’idée de cette approche consiste àéliminer de l’espace de l’approximation les fonctions qui oscillent beaucoup dans toutes les variablesq1, . . . , qd). On a développé dans [17] (un des article annexés à la fin) une variante de cette approcheadaptée à l’équation de Fokker-Planck (13.1).

Dans la deuxième approche, dite “Proper Genralized Decomposition”, les fonctionsα(1)i (q) ∈ VN

sont choisisa posteriori de sorte que la somme ci-dessus donne une approximation optimale ouquasi-optimale deψ entre toutes les sommes dem termes. Noter que, contrairement à l’approche desSparse Grids, on n’impose pas des restrictionsa priori sur le choix de fonctionsα(i)

1 (q1) · · ·α(i)d (qd).

L’ensemble des sommes àm termes dans (13.2) ne forme pas un espace linéaire, de sorte que pourtrouver la meilleure approximation, on doit résoudre un problème d’optimisation non linéaire. Untel procédé peut être lié aux algorithmes gloutons [64] de l’approximation dans un espace de Hilbertd’un élément donné par des sommes des éléments d’un dictionnaire prédéfini [51]. Quelques résultatsnumériques utilisant cette approche ont été inclus dans [1].

14 Travaux pendant la thèse et avant

L’objectif de mon travail de thèse était de contribuer à la construction de méthodes numériquespour la simulation des écoulements des fluides viscoélastiques en utilisant les modèles issus de lathéorie cinétique des polymères. Les contributions principales étaient les suivantes :

1. Dans le cadre du modèle classique Oldroyd B, exprimé normalement par une équation dif-férentielle constitutive pour le tenseur des contraintes ?mais qui peut être interprété commeune suspension de dumbbells de Hooke dans un solvant Newtonien (cf. le début de Chapitre 8et Fig. 8, à droite), on a proposé des nouvelles méthodes numériques qui respectent certainespropriétés importantes des équations différentielles concernées. Dans [8], on a développé unschéma numérique basé directement sur la description stochastique du modèle ? qui respectela propriété de la définition positive du tenseur de conformation ? contrairement aux discrétisa-tions directes de l’équation constitutive Oldroyd-B. Ces méthodes ont été implémentées à l’aidede discrétisations spectrales pour l’écoulement bidimensionnel autour d’un cylindre placé dansun canal. Des expériences numériques démontrent que nos méthodes sont plus stables quecelles utilisant l’équation constitutive dans une plage denombres de Deborah modérés. Dans[11], on a continué à explorer cette voie en démontrant une estimation a priori de l’énergie eten proposant un schéma numérique pour lequel l’estimation est satisfaite automatiquement.

2. Une alternative courante aux dumbbells de Hooke consisteà considérer les dumbbells FENE(Finitely Extensible Non-linear Elastic) où la loi de forcede ressort est choisie de telle façonque la longueur du ressort ne puisse pas excéder une certainelimite conformément à la phy-sique (voir remarque 9.1). Ce modèle ne possède plus d’équations constitutives et peut être écritde deux façons formellement équivalentes : des équations différentielles stochastiques pour levecteurQ(t,x) ou une EDP de Fokker-Planck pour la densité de probabilitéψ(t,x,Q). Lapremière option peut être utilisée pour construire des méthodes numériques stochastiques, quisont devenues très populaires dans la communauté de rhéologie computationnelle. La secondeoption est relativement peu exploitée. Dans ma thèse, j’ai démontré que les méthodes numé-riques basées sur l’équation de Fokker-Planck peuvent êtreune alternative intéressante auxsimulations stochastiques. On a implémenté une méthode spectrale efficace pour l’équation de

79

80 CHAPITRE 14. TRAVAUX PENDANT LA THÈSE ET AVANT

Fokker-Planck correspondant au modèle des dumbbells FENE [9, 12]. Des expériences nu-mériques confirment que cette méthode est jusqu’à 50 fois plus rapide que des simulationsstochastiques au cas de l’écoulement autour d’un cylindre placé dans un canal. Elle est aussiplus précise en raison de l’absence du bruit stochastique. La même approche a été appliquéeaux écoulement non homogènes dans [14], ainsi qu’à un modèlede “reptation” pour les solu-tions concentrées de polymère [10].

Pendant les trois années passées au Centre de Calcul de l’Académie des Sciences de Russie(1997-2000), j’ai participé aux deux projets suivants :

1. Développement des nouvelles méthodes numériques pour unproblème du type Stokes. L’idéede ces méthodes, proposées par Prof. B.V. Pal’tsev, consiste à utiliser l’équation de Poissonpour la pression avec des conditions de bord qui proviennentd’un processus itératif. J’ai effec-tué une étude théorique de ces méthodes discrétisées par deséléments finis dans le cas d’unebande avec conditions périodiques. J’ai proposé égalementdes modifications des processus ité-ratifs ayant un taux de convergence plus élevé [5]. Les méthodes ont été testés numériquementdans le cas des anneaux non concentriques [6].

2. Dans le même temps, j’ai travaillé avec Prof. A.A. Mayer sur une méthode d’éléments finispour des EDP non linéaires qui décrivent la propagation de lalumière dans des fibres optiquescouplées par l’effet tunnel. Des phénomènes physiques importants, comme l’auto-transition etl’amplification des signaux quasi-soliton, ont été étudiésdans [2, 3, 4].

Bibliographie

[1] A. L OZINSKI, R. OWENS, AND T. PHILLIPS, The Langevin and Fokker-Planck Equations inPolymer Rheology, Special Volume ofHandbook of Numerical Analysis, Numerical Methodsfor Non-Newtonian Fluids, Vol. XVI (edited by R. Glowinski and J. Xu), 2010.

• Publication dans les ouvrages avec un comité des lecteurs

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[3] , Self-switching and amplification of unidirectional distributed-coupling solitons of ortho-gonal polarizations, Physics - Doklady, 43 (1998), pp. 74–79.

[4] , Self-switching of solitons in quadratically nonlinear, tunneling-coupled optical wave-guides, Physics - Doklady, 43 (1998), pp. 333–339.

[5] A. S. LOZINSKII, On the acceleration of finite-element realizations of iterative processes withsplitting of boundary conditions for a Stokes-type system, Zh. Vychisl. Mat. Mat. Fiz., 40(2000), pp. 1339–1363.

[6] , Finite-element realization of iterative processes with splitting of boundary conditions fora Stokes-type system in nonconcentric annuli, Zh. Vychisl. Mat. Mat. Fiz., 41 (2001), pp. 1203–1216.

[7] A. L OZINSKI, R. OWENS, AND A. QUARTERONI, On the simulation of unsteady flow of anOldroyd-B fluid by spectral methods, J. Sci. Comput., 17 (2002), pp. 375–383.

[8] C. CHAUVIÈRE AND A. LOZINSKI, An efficient technique for simulations of viscoelastic flows,derived from the Brownian configuration field method, SIAM J. Sci. Comput., 24 (2003),pp. 1823–1837.

[9] A. L OZINSKI AND C. CHAUVIÈRE, A fast solver for Fokker-Planck equation applied to vis-coelastic flows calculations : 2D FENE model, J. Comput. Phys., 189 (2003), pp. 607–625.

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[10] A. LOZINSKI, C. CHAUVIÈRE, J. FANG, AND R. G. OWENS, Fokker-planck simulations offast flows of melts and concentrated polymer solutions in complex geometries, Journal of Rheo-logy, 47 (2003), pp. 535–561.

[11] A. LOZINSKI AND R. G. OWENS, An energy estimate for the Oldroyd B model : Theory andapplications, J. Non-Newtonian Fluid Mech., 112 (2003), pp. 161–176.

[12] C. CHAUVIÈRE AND A. LOZINSKI, Simulation of complex viscoelastic flows using the Fokker-Planck equation : 3D FENE model, J. Non-Newtonian Fluid Mech., 122 (2004), pp. 201–214.

[13] , Simulation of dilute polymer solutions using a Fokker-Planck equation, Comput. Fluids,33 (2004), pp. 687–696.

[14] A. LOZINSKI, R. G. OWENS, AND J. FANG, A Fokker-Planck-based numerical method for mo-delling non-homogeneous flows of dilute polymeric solutions, J. Non-Newtonian Fluid Mech.,122 (2004), pp. 273–286.

[15] J. FANG, A. LOZINSKI, AND R. G. OWENS, Towards more realistic kinetic models for concen-trated solutions and melts, J. Non-Newtonian Fluid Mech., 122 (2004), pp. 79–90.

[16] R. GLOWINSKI , J. HE, A. LOZINSKI, J. RAPPAZ, AND J. WAGNER, Finite element approxi-mation of multi-scale elliptic problems using patches of elements, Numer. Math., 101 (2005),pp. 663–687.

[17] P. DELAUNAY, A. LOZINSKI, AND R. G. OWENS, Sparse tensor-product Fokker-Planck-based methods for nonlinear bead-spring chain models of dilute polymer solutions, in High-dimensional partial differential equations in science andengineering, vol. 41 of CRM Proc.Lecture Notes, Amer. Math. Soc., Providence, RI, 2007, pp. 73–89.

[18] J. HE, A. LOZINSKI, AND J. RAPPAZ, Accelerating the method of finite element patches usingapproximately harmonic functions, C. R. Math. Acad. Sci. Paris, 345 (2007), pp. 107–112.

[19] A. LOZINSKI, J. RAPPAZ, AND J. WAGNER, Finite element method with patches for Poissonproblems in polygonal domains, ESAIM Proceedings, 21 (2007), pp. 45–64.

[20] A. LOZINSKI AND M. V. ROMERIO, Motion of gas bubbles, considered as massless bodies,affording deformations within a prescribed family of shapes, in an incompressible fluid underthe action of gravitation and surface tension, Math. Models Methods Appl. Sci., 17 (2007),pp. 1445–1478.

[21] V. REZZONICO, A. LOZINSKI, M. PICASSO, J. RAPPAZ, AND J. WAGNER, Multiscale algo-rithm with patches of finite elements, Math. Comput. Simulation, 76 (2007), pp. 181–187.

[22] F. HECHT, A. LOZINSKI, A. PERRONNET, AND O. PIRONNEAU, Numerical zoom for multis-cale problems with an application to flows through porous media, Discrete Contin. Dyn. Syst.,23 (2009), pp. 265–280.

[23] A. LOZINSKI, M. PICASSO, AND V. PRACHITTHAM , An anisotropic error estimator for theCrank-Nicolson method : application to a parabolic problem, SIAM J. Sci. Comput., 31 (2009),pp. 2757–2783.

[24] A. BONITO, A. LOZINSKI, AND T. MOUNTFORD, Modeling viscoelastic flows using reflectedstochastic differential equations, Comm. Math. Sci., 8 (2010), pp. 655–670.

BIBLIOGRAPHIE 83

[25] P. DEGOND, A. LOZINSKI, AND R. G. OWENS, Kinetic models for dilute solutions of dumb-bells in non-homogeneous flows revisited, J. Non-Newtonian Fluid Mech., 165 (2010), pp. 509–518.

[26] A. LOZINSKI AND O. PIRONNEAU, Numerical zoom for localized multiscales, Numerical Me-thods for Partial Differential Equations, sous presse (2010) .

[27] M. H ILLAIRET , A. LOZINSKI, AND M. SZOPOS, On discretization in time in simulations ofparticulate flows, Discrete Contin. Dyn. Syst. B, sous presse (2010).

• Articles soumis à publication

[28] P. DEGOND, F. DELUZET, A. LOZINSKI, J. NARSKI, AND C. NEGULESCU, Duality-basedasymptotic-preserving method for highly anisotropic diffusion equations. Soumis à Comm.Math. Sci. Un preprint sur http ://arxiv.org/abs/1008.3405, 2010.

• Proceedings

[29] C. CHAUVIÈRE, J. FANG, A. LOZINSKI, AND R. G. OWENS, On the numerical simulation offlows of polymer solutions using high-order methods based onthe Fokker-Planck equation, Int.J. Mod. Phys. B, 17 (2003), pp. 9–14.

[30] M. ROMERIO, A. LOZINSKI, AND J. RAPPAZ, A new modelling for simulating bubble motionsin a smelter, in Light Metals 2005 as held at the 134 th TMS Annual Meeting ;San Francisco,CA, 2005, pp. 547–552.

[31] R. GLOWINSKI , J. HE, A. LOZINSKI, M. PICASSO, J. RAPPAZ, V. REZZONICO, AND

J. WAGNER, Finite element methods with patches and applications, in Domain decompositionmethods in science and engineering XVI, vol. 55 of Lect. Notes Comput. Sci. Eng., Springer,Berlin, 2007, pp. 77–89.

[32] F. HECHT, A. LOZINSKI, AND O. PIRONNEAU, Numerical zoom and the Schwarz algorithm,in Domain decomposition methods in science and engineeringXVIII, vol. 70 of Lect. NotesComput. Sci. Eng., Springer, Berlin, 2009, pp. 63–74.

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Quatrième partie

Articles annexés

87

NUMERICAL ZOOM FOR ADVECTION DIFFUSION PROBLEMS WITH LOCALIZED MULTISCALES

par

ALEXEI LOZINSKI AND OLIVIER PIRONNEAU

paru dans

Numerical Methods for Partial Differential Equations

Numerical Zoom for Advection Diffusion Problemswith Localized MultiscalesAlexei Lozinski,1 Olivier Pironneau2

1Institut de Mathématiques de Toulouse, Université Paul Sabatier, France2Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, France

Received 21 July 2010; accepted 26 July 2010Published online 1 October 2010 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/num.20642

We investigate the Schwarz’ domain decomposition algorithm for numerical zooms. Error estimates aregiven for nonmatching meshes for elliptic and parabolic partial differential equations. The method is appliedto the security assessment of the burial of nuclear waste which is a typical multiscale problem for porousmedia flow. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 197–207, 2011

Keywords: domain decomposition; finite elements; multiscale; numerical zoom; porous media flow

I. INTRODUCTION

Security assessment of nuclear waste repository sites requires the numerical simulations of ellip-tic and parabolic partial differential equations, namely Darcy’s law for the hydrostatic potentialand the convection diffusion equation of the radio-nucleides. These problems have multiscalesbecause the simulations are done over half a million years while the canister leaks over less than5,000 years; also the repository sites could pollute as much as 10 sq km of area while the lengthscales of a canister is of the order of the meter.

Strictly speaking the computations should be done with billions of mesh points on supercom-puters; however often enough engineers prototype their applications with a coarse calculation andthen a finer one on a subset (zoom) D of the whole domain .

We wish here to justify this approach, i.e., to study convergence and errors when the strategy isimplemented in the time loop of the convection diffusion equations and when the meshes used forthe zooms are not divided submeshes of the coarse mesh. One may argue that it is unnecessarilycomplex to do so; however many engineers use the shelves solvers which do not have a numericalzoom capacity built in and so it is quite complex—if even possible—to use them with a zoommesh made of divided triangles of the coarse mesh.

For Darcy’s law the situation is as follows: a coarse calculation is done in , then another onein a zoom D ⊂ ; a correction to the coarse problem must be found which takes into account the

Correspondence to: Olivier Pironneau, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, France (e-mail:[email protected])

© 2010 Wiley Periodicals, Inc.

Articles annexés 91

198 LOZINSKI AND PIRONNEAU

calculation in the zoom. In [1] and [2] two algorithms have been shown to converge: a subspacecorrection method (or Hilbert space Decomposition, equivalent to Schwarz domain decompositionon the continuous level) and the Harmonic Patch Iterator.

It is very natural to work with composite meshes, a coarse one for most of the domain and afine one in the region where the data are irregular; engineers do it spontaneously without evenbothering about convergence issues; this idea seems to be as old as the finite element method itselfand bears many names such as Chimera [3] in fluid mechanics, Global-local [4], composite grids[5], Arlequin [6] in structure mechanics, domain decomposition [7], Hilbert Space Decomposition[8], Mortars [9], etc. Therefore it is the analysis, more than the method, which is possibly new inthis article.

For nonmatching meshes the error estimations started with [8] and [1] for Hilbert space Decom-position but stumbled on the problem of quadrature errors for integrals involving products offunctions on coarse and fine meshes [10]. The problem was solved in [11] by working with thesup-norm. Here we present an extension of the same argument for parabolic problems. For theheat equation and the convection diffusion problems we will analyze the error for Schwarz’ whenonly one iteration is done at each time step.

II. CONVERGENCE OF SCHWARZ’ ALGORITHM FOR DARCY’S LAW ONARBITRARY NONMATCHING MESHES

In simple fully saturated cases the hydrostatic pressure H of the flow of water underground isgiven by

−∇ · (K∇H) = f in , H or (K∇H) · n given on (1)

where K is the permeability tensor, n the normal to := ∂. Assume that K has multiple scalesor is irregular in D ⊂ and consider the following Schwarz algorithm to zoom in D numerically.

A. The Schwarz-Zoom Method

Let H be a triangulated subdomain of such that ∂H = H ∪ SH with the property that SH isstrictly inside D and non empty. Let h be a triangulated approximation of D and call Sh = ∂h

(see Fig. 1). The triangumation of H (resp h) will be denoted TH (resp Th).Let

VH = v ∈ C0(H) : v|K ∈ P 1, ∀K ∈ TH

, V0H = v ∈ VH : v|∂H

= 0,and similarly with h. Denote by γH (resp γh) the interpolation operator on VH (resp Vh).

To simplify the notation, we choose the variant of (1) with the boundary condition H | = g

and set K to the identity matrix so that (1) becomes the Dirichlet problem for the Poisson equation.The solution of this problem will be denoted by u from now on. Let gH = γHg (to avoid furthertechnical difficulties we suppose here that all the boundary nodes of TH lying on H also belong to so that γHg is well defined). Starting from any u0

h ∈ Vh, the Schwarz-Zoom algorithm proceedsas follows: find um

H ∈ VH , umh ∈ Vh, m = 1, 2, . . ., such that ∀wH ∈ V0H , ∀wh ∈ V0h

aH

(um

H , wH

) = (f , wH), umH |SH

= γHum−1h , um

H |H= gH ,

ah

(um

h , wh

) = (f , wh), umh |Sh

= γhumH . (2)

Numerical Methods for Partial Differential Equations DOI 10.1002/num

92 Articles annexés

NUMERICAL ZOOM FOR ADVECTION DIFFUSION 199

FIG. 1. The drawing shows the fine mesh of the zoom region h overlapping the coarse mesh in ofH and its hole of boundary SH . [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

Hypothesis H1. Assume that all the off-diagonal elements in the matrices obtained on the lefthand side of the two problems in (2) are negative or zero so that the maximum principle holds forthese problems. Moreover, assume that νH ∈ VH solution of

aH (νH , wH) = 0, ∀wH ∈ V0H , νH |SH= 1, νH |H

= 0 (3)

satisfies λ := |νH |∞,Sh< 1.

Notice that the maximum principle is known to be true when all the angles of the triangulationare acute [12]. Error estimates in maximum norm of order h2 log 1

hwith respect to the mesh edge

size h for linear elements have been obtained in [13].

Theorem 1. Under Hypothesis H1 the Schwarz-Zoom algorithm converges towards u∗H , u∗

h,which satisfy

aH

(u∗

H , wH

) = (f , wH), ∀wH ∈ V0H , u∗H |SH

= γHu∗h, u∗

H |H= gH

ah

(u∗

h, wh

) = (f , wh), ∀wh ∈ V0h, u∗h|Sh

= γhu∗H . (4)

Provided the exact solution u of the Dirichlet problem for the Poisson equation is sufficientlysmooth and the distance between and H is of order H 2, one has

‖u − uH‖∞,H+ ‖u − uh‖∞,h

≤ C

(H 2 log

1

H‖u‖2,∞,H

+ h2 log1

h‖u‖2,∞,h

)(5)

The proof can be found in [11]. It shows also that the Schwarz algorithm converges linearly atspeed proportional to λ.

Remark 1. This results gives a rule to choose h/H because it is logical to have both pieces in(5) of the same size, giving

h2 log 1h

H 2 log 1H

∼ ‖u‖2,∞,H

‖u‖2,∞,h

(6)

and since D has been chosen so that ‖u‖2,∞,his large compared with ‖u‖2,∞,H

, it gives an h

smaller than H .




III. CONVERGENCE OF THE SCHWARZ-ZOOM ALGORITHM FORCONVECTION-DIFFUSION ON NON-MATCHING MESHES

To solve

∂u

∂t+ b · ∇u − u = f in with u = g on = ∂,

u|t=0 = u0,(7)

we proceed to analyze first the case b = 0.

A. Schwarz Method for the Heat Equation (b = 0)

To avoid further technical difficulties, we suppose that is a polygonal domain. As before wechoose two subdomains of , H , and h, and two triangulations TH of H , Th of h, such that

H ∪ h = , ∂h ⊂ H , ∂H\ ⊂ h,

and work with the P 1 finite element spaces VH and Vh on TH and Th. As in Fig. 1 we denote bySh the boundary of h and by SH the part of the boundary of H different from .

Euler-Schwarz algorithm starts from u0H = γHu0, u0

h = γhu0 and finds unH ∈ VH with

unH | = gn

H and unh ∈ Vh, n = 1, . . . , N such that ∀wH ∈ V0H , ∀wh ∈ V0h:

(un

H − un−1H

t, wH

)H

+ aH

(un

H , wH

) = (f n, wH

), un

H |SH= γHun−1

h ,

(un

h − un−1h

t, wh

)h

+ ah

(un

h, wh

) = (f n, wh

), un

h|Sh= γhu

n−1H , (8)

where f n = f (tn, ·), gnH = gH (tn, ·), gH = γHg, aH ,h(u, v) = ∫

H ,h∇u · ∇v and γH (resp γh) is

the interpolation operator on VH (resp Vh). The parentheses (·, ·) stand for the L2 inner productand (·, ·)H (resp (·, ·)h) denotes mass lumping on TH (resp Th).

Theorem 2. Under Hypothesis H1 and provided that u, solution to (7) with b = 0, is sufficientlysmooth, (un

h, unH ) in (8) solves approximately (7) with the following L∞ error estimate

max1≤n≤N

(∥∥unH − u(tn, ·)∥∥∞,H

,∥∥un

h − u(tn, ·)∥∥∞,h

)

≤ CT

((t + H 2 log

1

H

)‖u‖C2([0,T ],C(H ))∩C1([0,T ],H2,∞(H ))

+(

t + h2 log1

h

)‖u‖C2([0,T ],C(h))∩C1([0,T ],H2,∞(h))

). (9)

with T = Nt and a constant C depending only on the domains H and h.

Proof. Let u∗H = u∗

H (t , ·) ∈ VH and u∗h = u∗

h(t , ·) ∈ Vh for any t ∈ [0, T ] be the solutions of(4) with u = u(t , ·) and gH = gH (t , ·). By Theorem 1, the solution of (4) exists for all t and thefollowing estimate holds




∥∥u∗H − u

∥∥∞,H ×[0,T ] + ∥∥u∗

h − u∥∥

∞,h×[0,T ]

≤ C

(H 2 log

1

H‖u‖C([0,T ],H2,∞(H )) + h2 log

1

h‖u‖C([0,T ],H2,∞(h))

), (10)

and similarly for ‖u∗h − u‖∞,h×[0,T ]. Moreover, we can differentiate (4) with respect to time to

obtain the estimates for the derivatives ∂u∗H ,h/∂t :∥∥∥∥∂u∗

H

∂t− ∂u

∂t

∥∥∥∥∞,H ×[0,T ]

+∥∥∥∥∂u∗

h

∂t− ∂u

∂t

∥∥∥∥∞,h×[0,T ]

≤ C

(H 2 log

1

H‖u‖C1([0,T ],H2,∞(H )) + h2 log

1

h‖u‖C1([0,T ],H2,∞(h))

). (11)

It is now easy to obtain an a priori estimates for a fully implicit version of the Euler-Schwarzalgorithm (8): find un

H ∈ VH with unH | = gn

H and unh ∈ Vh such that ∀wH ∈ V0H , ∀wh ∈ V0h:(

unH − un−1

H

t, wH

)H

+ aH (unH , wH) = (f n, wH), un

H |SH= γH un

h,

(un

h − un−1h

t, wh

)h

+ ah(unh, wh) = (f n, wh), un

h|Sh= γhu

nH . (12)

Introduce the errors εnH ∈ VH , εn

h ∈ Vh as

εnH = un

H − u∗nH , εn

h = unh − u∗n

h ,

where u∗nH = u∗

H (tn, ·), u∗nh = u∗

h(tn, ·). After some simplification, the finite element problem for

εnH ,h reads: ∀wH ∈ V0H , ∀wh ∈ V0h(

εnH − εn−1

H

t, wH

)H

+ aH

(εn

H , wH

) = (rnH , wH

), εn

H |SH= γHεn

h, εnH |H

= 0,

(εn

h − εn−1h

t, wh

)h

+ ah

(εn

h, wh

) = (rnh , wh

), εn

h|Sh= γhε

nH , (13)

with

rnH = ∂u

∂t(tn, ·) − u∗n

H − u∗(n−1)

H

t, rn

h = ∂u

∂t(tn, ·) − u∗n

h − u∗(n−1)

h

t.

Estimating rnH with the help of the tringle inequality and (11) gives

∥∥rnH

∥∥∞,H

≤∥∥∥∥∂u

∂t(tn, ·) − u(tn, ·) − u(tn−1, ·)

t

∥∥∥∥∞,H

+ 1

t

∥∥∥∥∥∫ tn

tn−1

∂

∂t(u(s, ·) − u∗(s, ·))ds

∥∥∥∥∥∞,H

≤ Ct

∥∥∥∥∂2u

∂t2

∥∥∥∥∞,H

+∥∥∥∥ ∂

∂t(u − u∗)

∥∥∥∥∞,H

≤ C

(t‖u‖C2([0,T ],C(H )) + H 2 log

1

H‖u‖C1([0,T ],H2,∞(H ))

+ h2 log1

h‖u‖C1([0,T ],H2,∞(h))

). (14)




A similar bound holds for rnh . We observe also that the joint maximum of ‖εn

H‖∞,Hand ‖εn

h‖∞,h

is always attained in an iternal node of TH or Th. Indeed, the property εnh|Sh

= γhεnH entails

‖εnh‖∞,Sh

≤ ‖εnH‖∞,Sh

. Hence there is always an internal node of TH such that the absolute valueof un

H at this node is greater than ‖εnh‖Sh

, provided H is sufficiently small in comparison to the dis-tance between SH and Sh. Without loss of generality, we can now assume that the joint maximumof |εn

H | and |εnh| is attained at an internal node xn of Th and εn

h(xn) ≥ 0. Taking the hat function φn

associated to the node xn in the standard basis of Vh as the test function wh in (13) and observingthat ah(u

nn, φn) ≥ 0 by hypothesis H1, we see that

unh(xn) − un−1

h (xn)

t(1, φn) ≤ (rn

h , φn)

so that

max(∥∥εn

H

∥∥∞,H

,∥∥εn

h

∥∥∞,h

) = unh(xn) ≤ un−1

h (xn) + t∥∥rn

H

∥∥∞,H

≤ max(∥∥εn−1

H

∥∥∞,H

,∥∥εn−1

h

∥∥∞,h

) + t max(∥∥rn

H

∥∥∞,H

,∥∥rn

h

∥∥∞,h

). (15)

Summing up these inequalities on n = 1, . . . , N and combining with (10) and (14), we obtain theerror estimates for the implicit Schwarz algorithm

max1≤n≤N

(∥∥unH − u(tn, ·)∥∥∞,H

,∥∥un

h − u(tn, ·)∥∥∞,h

)

≤ CT

((t + H 2 log

1

H

)‖u‖C2([0,T ],C(H ))∩C1([0,T ],H2,∞(H ))

+(

t + h2 log1

h

)‖u‖C2([0,T ],C(h))∩C1([0,T ],H2,∞(h))

). (16)

Let us now return to our original Schwarz algorithm (8). Introduce the finite differencesdn

H = (unH −un−1

H )/t , dnh = (un

h −un−1h )/t . The equations for them are obtained by subtracting

(8) at time tn−1 from the same equation at time tn: dnH |H

= gnH − gn−1

H , ∀wH ∈ V0H , ∀wh ∈ V0h

(dn

H − dn−1H

t, wH

)H

+ aH

(dn

H , wH

) =(

f n − f n−1

t, wH

), dn

H |SH= γHdn−1

h ,

(dn

h − dn−1h

t, wh

)h

+ ah

(dn

h , wh

) =(

f n − f n−1

t, wh

), dn

h |Sh= γhd

n−1H , (17)

for n ≥ 2. By an application of the maximum principle, similar as above, we observe that

max(∥∥dn

H

∥∥∞,H

,∥∥dn

h

∥∥∞,h

) ≤ max

(∥∥gnH − gn−1

H

∥∥∞,

,

max(∥∥dn−1

H

∥∥∞,H

,∥∥dn−1

h

∥∥∞,h

) + t

∥∥∥∥f n − f n−1

t

∥∥∥∥∞,

). (18)

The norm of the finite difference in time for f in the last line can be bounded by the maximum of∂f

∂twhich, in turn, is bounded by ‖u‖C2([0,T ],C())∩C1([0,T ],H2,∞()) since ∂f

∂t= ∂2u

∂t2− ∂u

∂t. It is also




easy to obtain the appropriate estimates for ‖gnH − gn−1

H ‖∞, , ‖d1H‖∞,H

and ‖d1h‖∞,h

, so that wehave finally

max1≤n≤N

(∥∥dnH

∥∥∞,H

,∥∥dn

h

∥∥∞,h

) ≤ Cd := CT ‖u‖C2([0,T ],C())∩C1([0,T ],H2,∞()). (19)

The last step in the proof is to estimate the difference between the original Schwarz algorithm(8) and the implicit one (12). We thus introduce en

H = unH − un

H , enh = un

h − unh and observe that

they are solutions to the finite element problems: ∀wH ∈ V0H , ∀wh ∈ V0h(en

H − en−1H

t, wH

)H

+ aH

(en

H , wH

) = 0, enH |SH

= γHenh + tγHdn

h , enH |H

= 0,

(en

h − en−1h

t, wh

)h

+ ah

(en

h, wh

) = 0, enh|Sh

= γhenH + tγhd

nH , (20)

with initial conditions u0H = 0, u0

h = 0. We remind that all dnH and dn

h are bounded in the maximumnorm by Cd , take any number A > Cdt and consider sn

H = enH − AαH , sn

h = enh − Aαh where

the auxiliary functions αH ∈ VH , αh ∈ Vh satisfy

aH (αH , wH) = 0 ∀wH ∈ VH , αH |SH= γHαh + 1, αH | = 0,

ah(αh, wh) = 0 ∀wh ∈ Vh, αh|Sh= γhαH + 1. (21)

We now want to prove that snH and sn

h are nonpositive everywhere and for all n. If this is not thecase, we can suppose without loss of generality that the global maximum of sn

H , snh is attained

at the time step number k ≥ 1 on skh(xk) where xk is a node of Th. We have

(skh − sk−1

h

t, wh

)h

+ ah

(skh , wh

) = 0 ∀wh ∈ Vh,

skh|Sh

= γhskH + tγhd

kH − A. (22)

The first line in (22) tells us that xk cannot be an internal node of Th, while the second line impliesthat it cannot be a boundary node either since tdk

H − A < 0 on Sh. We obtain thus ∀n ≥ 1

supx∈

max(en

H (x), enh(x)

) ≤ ACα := A max(‖αH‖∞,H, ‖αh‖∞,h

).

Note that the constant Cα can be bounded independently of H and h. Repeating the same derivationwith en

H , enh replaced by −en

H , −enh and tending A to Cdt from above, we see that

max(∥∥en

H

∥∥∞,H

,∥∥en

h

∥∥∞,h

) ≤ CαCdt , ∀n ≥ 1. (23)

The desired result (9) is obtained from (16) and (23) by the triangle inequality.

Remark 2. The fully implicit Schwarz algorithm (12) can provide an interesting alternative toour basic algorithm (8) if several iterations of a standard Schwarz method like (2) are used onevery time step to find an approximation for the couple (un

H , unh). The convergence proof above

can be easily adapted for such an algorithm. It can be preferable to (8) in terms of the cost toaccuracy ratio.




Remark 3. One can associate two different time steps for the two subdomains, say t for theproblems with un

H and δt for the problems with unh. A variant of the Schwarz algorithm can be

then constructed provided t = mt with an integer m. In the zoom the problem is advanced mtimes and then the coarse problem is advanced once; the convergence proof is extensible to thiscase without difficulties.

Remark 4. Extension of our results to the operator ∇ · (K∇) in place of is straightforward.

B. Extension to Convection Diffusion (b = 0)

The convection term in (7) can be discretized by the characteristic-Galerkin scheme. Combina-tion with the Schwarz method gives the following algorithm: find un

H ∈ VH , unh ∈ Vh, such that

∀wH ∈ V0H , ∀wh ∈ V0h

1

t

(un

H − un−1H oXn−1, wH

)H

+ aH

(un

H , wH

) = (f , wH), unH |SH

= γHun−1h , un

H | = gnH ,

1

t

(un

h − un−1h oXn−1, wh

)h+ ah

(un

h, wh

) = (f , wh), unh|Sh

= γhun−1H , (24)

where Xn−1(x) = X(tn−1) and X is the solution of X = b(X(t), t), X(tn) = x. We consider thecase where Xn−1 can be computed exactly (b piecewise constant for instance).

As shown in [14] and [15], the accuracy of the characteristic-Galerkin method deteriorateswhen the time step t becomes too small before the mesh size. One cannot expect therefore toextend Theorem 2 to the general case b = 0. We have, however, a weaker result:

Theorem 3. Under Hypothesis H1 and provided that u, solution to (7), is sufficiently smooth,(un

h, unH ) in (24) solves approximately (7) with the following L∞ error estimate

max1≤n≤N

(∥∥unH − u(tn)

∥∥∞,H

,∥∥un

h − u(tn)∥∥

∞,h

) ≤ CT

(t + H 2 log 1

H

t

)‖u‖H2,∞(H ×[tn−1,tn])

+(

t + h2 log 1h

t

)‖u‖H2,∞(h×[tn−1,tn]),

(25)

with T = Nt and a constant C depending only on the domains H and h.

Proof. Denote un = u(tn, ·) and consider unH ∈ VH such that ∀wH ∈ V0H

aH

(un

H , wH

) = aH (un, wH), unH |SH

= γHu(tn), unH | = gH (tn, ·).

By the results in [13], we have

∥∥unH − un

∥∥∞,H

≤ CH 2 log1

H‖un‖2,∞,H

. (26)

We also introduce unh ∈ Vh in a similar way and observe that the same estimate as (26) holds for

unh replacing H by h.




Denote

εnH = un

H − unH , εn

h = unh − un

h.

Then εnH ∈ VH solves ∀wH ∈ V0H(

εnH − εn−1

H oXn−1

t, wH

)H

+ aH

(εn

H , wH

) = (rnH , wH

),

εnH |SH

= γHεn−1h , un

H |H= 0 (27)

with

rnH = Du

Dt(tn, ·) − un

H − un−1H oXn−1

t,

where Du

Dt= ∂u

∂t+ b∇u is the material derivative. By the triangle inequality and (26) we have

∥∥rnH

∥∥∞,H

≤∥∥∥∥Du

Dt(tn, ·) − un − un−1oXn−1

t

∥∥∥∥∞,H

+∥∥∥∥un− un−1oXn−1

t− un

H − un−1H oXn−1

t

∥∥∥∥∞,H

≤ Ct‖u‖H2,∞(H ×[tn−1,tn]) + 1

t

∥∥un − unH

∥∥∞,H

+ 1

t

∥∥(un−1 − un−1

H

)oXn−1

∥∥∞,H

≤ C

[(t + H 2 log 1

H

t

)‖u‖H2,∞(H ×[0,T ]) + h2 log 1

h

t‖u‖H2,∞(h×[0,T ])

].

Note here the use of the inequality∥∥(un−1 − un−1

H

)oXn−1

∥∥∞,H

≤ ∥∥un−1 − un−1H

∥∥∞,H

+ ∥∥un−1 − un−1h

∥∥∞,h

.

One cannot leave only the norm in H in the right hand side of this inequality since a characteristicstarting in H at time tn can go back to h at time tn−1.

Writing down an estimate for rnh similar to what is done above for rn

H and noting that

max(∥∥εn

H

∥∥∞,H

,∥∥εn

h

∥∥∞,h

) ≤ max(∥∥εn−1

H

∥∥∞,H

,∥∥εn−1

h

∥∥∞,h

) + t max(∥∥rn

H

∥∥∞,H

,∥∥rn

h

∥∥∞,h

),

we conclude the proof in the same way as that of the estimate (16) for the implicit Schwarz methodin the proof of Theorem 2.

IV. NUMERICAL TESTS

We shall apply the method to solve one of Andra’s COUPLEX test cases (see [16]). but beforethat let us verify the error estimate on a simpler example. A closed form solution u = tr−2,r2 = 0.01 + x2 + y2 is computed in a square (−1, 1)2 by solving the heat equation with an appro-priate source term f = (r−2 + 4tr−4 − 8t(x2 + y2)r−6. The subdomains are the square minus thedisk of radius 0.25 and a disk of radius 0.3. The left (resp right) part of Table I displays the L∞

errors on the coarse (resp fine) subdomain for several values of H and t when h = H/8. The




TABLE I. The left (resp right) part displays the L∞ errors on the coarse (resp fine) subdomain for severalvalues of H and t when h = H/8.

t\H 0.2 0.1 0.05 0.2 0.1 0.05

0.25 0.124378 0.124378 0.124378 9.31509 9.70629 9.88340.125 0.0621891 0.0621891 0.0621891 7.26138 8.28583 8.755520.0625 0.0310945 0.0310945 0.0310945 3.83163 5.8366 6.723920.03125 0.0155473 0.0155473 0.0155473 0.355462 2.9373 4.326950.015625 3.02283 0.00777363 0.00777363 4.10578 0.442487 2.29146If t is too small the quadrature errors become too large; this is why the errors for H = 0.25 and t = 0.015625 arelarge (the same happens to the problem without domain decomposition).

COUPLEX test case is made first of an elliptic problem ∇ · (K∇H) = 0 to obtain the hydrostaticpressure H from the porosity K and convection velocity as −K∇H and then of a convectiondiffusion equation for the concentration c(x, y, t) of the radionucleide in the underground. Thedifficulties come from the large differences of K in each geological layer and from the differentlength scales in the problem. The details of the calculation are not very important but Fig. 2 showsthat the Schwarz-zoom algorithm works for this problem.

FIG. 2. The top pictures shows detailed level lines of the concentration of radionucleide nearby the canisters(the seven circles) after 600 time steps. A concentration is given at initial time in the canister which are allsupposed to leak due to rust and spread their content by convection and diffusion in the clay. The calculationis obtained from a Schwarz-zoom refinement. The bottom picture shows the full domain, its decompositioninto two subdomains and a superposition of both fields of level lines of cH and ch; both fields match fairlywell. The domain is physically a rectangle (0,10 km) × (0,500 m) but it is rescaled for the simulation; thelong horizontal lines are the separations between (from top to bottom), marl, limestone, clay, and doggerlayers. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]




V. CONCLUSION

The article has explained why the standard Schwarz algorithm can be used for a numerical zoomprocedure, with standard interpolations at the boundaries. The method works for time dependentproblems as well and it is not necessary to iterate the Schwarz algorithm because the time steppingprocedure has the same effect. A compatibility condition between the time step and the mesh sizemay be necessary in the case of convection diffusion problems. However, the same feature ispresent at the standard finite element characteristic-Galerkin simulations on a single mesh so thatit cannot be attributed to the Schwarz method.

Application to flows through porous media is covered by the two theorems of this article; itwould be nice to extend the results to other fluid flows such as Navier-Stokes flow but the proofshere rely heavily on the maximum principle which is not valid for the generalized Stokes operator.

References

1. R. Glowinski, J. He, J. Rappaz, and J. Wagner, Approximation of multi-scale elliptic problems usingpatches of finite elements, C R Math Acad Sci Paris 337 (2003), 679–684.

2. J. He, A. Lozinski, and J. Rappaz, Accelerating the method of finite element patches using approximatelyharmonic functions, C R Math Acad Sci Paris 345 (2007), 107–112.

3. J. L. Steger and J. A. Benek, On the use of composite grid schemes in computational aerodynamics,Comput Methods Appl Mech Eng 64 (1987), 301–320.

4. C. D. Mote, Global-local finite element method, Int J Num Methods Eng 3 (1971), 565–574.

5. J. Fish and T. Belytschko, Elements with embedded localization zones for large deformation problems,Comput Struct 30 (1988), 247–256.

6. H. B. Dhia and G. Rateau, Mathematical analysis of the mixed Arlequin method, C R Math Acad SciParis 332 (2001), 649–654.

7. P. L. Lions, On the Schwarz alternating method, I,II,III. International Symposium on DomainDecomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, 89–90.

8. J.-L. Lions and O. Pironneau, Domain decomposition methods for CAD, C R Acad Sci Paris Sér I Math328 (1999), 73–80.

9. Y. Achdou and Y. Maday, The mortar element method with overlapping subdomains, SIAM J NumerAnal 40 (2002), 601–628.

10. F. Brezzi, J. L. Lions, and O. Pironneau, Analysis of a chimera method, C R Math Acad Sci Paris 332(2001), 655–660.

11. F. Hecht, A. Lozinski, and O. Pironneau, Numerical zoom and the Schwarz algorithm, M. Bercovier,M. Ganders, R. Kornhuber, and O. Widlund, editors, In Proceeding Domain Decomposition Conferencedd18, Jerusalem 2008, Lecture Notes in Computational Science and Engineering, Vol. 70, SpringerVerlag, Berlin, 2009, pp. 63–73.

12. P. G. Ciarlet and P.-A. Raviart, Maximum principle and uniform convergence for the finite elementmethod, Comput Methods Appl Mech Engrg 2 (1973), 17–31.

13. A. H. Schatz and L. B. Wahlbin, On the quasi-optimality in L∞ of the H1-projection into finite elementspaces, Math Comp 38 (1982), 1–22.

14. O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations,Numerische Mathematik 38 (1982), 309–312.

15. O. Pironneau and M. Tabata, Galerkin-characteristics finite element method, Int J Numer MethodsFluids, to appear.

16. ANDRA, Couplex test cases (2001). Available at: http://www.andra.fr/couplex.



ON DISCRETIZATION IN TIME IN SIMULATIONS OF PARTICULATE

par

MATTHIEU HILLAIRET, ALEXEI LOZINSKI AND MARCELA SZOPOS

à paraître dans

Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XX

ON DISCRETIZATION IN TIME IN SIMULATIONS OF

PARTICULATE FLOWS

Matthieu Hillairet

Ceremade, UMR CNRS 7534, Universite DauphinePlace du Marechal De Lattre De Tassigny

F-75016 Paris, France

Alexei Lozinski

Institut de Mathematiques de ToulouseUMR 5219, Universite de Toulouse et CNRS

118 route de NarbonneF-31062 Toulouse Cedex 9, France

Marcela Szopos

Institut de Recherche Mathematique AvanceeUMR 7501, Universite de Strasbourg et CNRS

7 rue Rene DescartesF-67084 Strasbourg Cedex, France

(Communicated by the associate editor name)

Abstract. We propose a time discretization scheme for a class of ordinary dif-ferential equations arising in simulations of fluid/particle flows. The scheme isintended to work robustly in the lubrication regime when the distance betweentwo particles immersed in the fluid or between a particle and the wall tends tozero. The idea consists in introducing a small threshold for the particle-walldistance below which the real trajectory of the particle is replaced by an ap-proximated one where the distance is kept equal to the threshold value. Theerror of this approximation is estimated both theoretically and by numerical

experiments. Our time marching scheme can be easily incorporated into a fullsimulation method where the velocity of the fluid is obtained by a numericalsolution to Stokes or Navier-Stokes equations. We also provide a derivationof the asymptotic expansion for the lubrication force (used in our numericalexperiments) acting on a disk immersed in a Newtonian fluid and approachingthe wall. The method of this derivation is new and can be easily adapted toother cases.

1. Introduction. One of the challenges for fluid/particle flow simulations is toprovide an accurate resolution of the lubrication regime when the distance betweentwo particles immersed in the fluid or between a particle and the wall becomes verysmall. The presence of extremely high gradients of the velocity in the narrow gapbetween the particle and the wall (or between the two particles) results in verystrong drag forces (which can be referred to as the lubrication forces in this case)

2000 Mathematics Subject Classification. Primary: 65Z05, 76T99; Secondary: 65L04.Key words and phrases. Particulate flow, near contact, lubrication force.The work of the first author was supported by ANR project ANR-08-JCJC-0104-01. The work

of the last two authors was partially supported by the project “BQR AO1” of Universite PaulSabatier.

1


2 ALEXEI LOZINSKI, MATTHIEU HILLAIRET AND MARCELA SZOPOS

and thus requires special attention when one attempts to construct an efficient andaccurate discretization in space. The corrections specific to the lubrication forceswere proposed in the context of the boundary integral methods [22, 12, 17], in thatof the Stokesian Dynamics [1, 23], and in that of the force-coupling method [4].Short-range repulsion forces mimicking the lubrication ones were also used in [8] inthe context of fictitious domain finite element methods. Taking aside the formidableproblems related to the discretization in space, we focus our attention in this articleon the discretization in time. Indeed, the presence of lubrication force of very highmagnitude in the equations governing the motion of a particle necessitates the useof very small time steps in the lubrication regime in order to obtain a physicallyacceptable solution. Our idea is to prohibit the particle from approaching too closelythe wall during a simulation. We shall thus choose a threshold qs for the distanceq between the particle and the wall and replace the true trajectory of the particleby an approximated one, in which the distance q is kept equal to qs until a possiblerebound of the particle from the wall. The moment of rebound is predicted usingan auxiliary quantity (a crude approximation of the velocity) that is computed allalong the period of time when the particle is stuck at the distance qs. This approachreminds the gluey particle model of [18, 15] where qs is set to zero and the limit ofvanishing viscosity is considered. However, our motivations are quite different fromthat behind the gluey particle model. This model is intended as a simple alternativeto the standard governing equations of Navier-Stokes type, possibly corrected bytaking into account the roughness of the particle surface. On the other hand, ourapproach is to take the standard fluid equations for granted (assuming the particlesurface to be smooth) and to provide a tool for a robust time discretization of them.It means, in particular, that we would need an accurate enough method valid forany given value of the viscosity, not necessarily small. Note also, that our thresholdqs will typically depend on the time step size, so that it is indeed a numerical deviceand it has no physical meaning.

The plan of the article is as follows: we start by reminding the governing equa-tions in a general setting and by explaining in more detail the difficulties related tothe simulations in the lubrication regime in the next section. Section 3 is the coreof the paper. The idea of the threshold is rigorously introduced and studied therefor the model ordinary differential equation representing the essence of the generalsetting in the simplest case of a circular (or spherical) particle approaching thewall. The discussion is held on the continuous level in Section 3. The discretizationin time is introduced in Section 4 where several implementations of our idea areproposed on the discrete level followed by numerical experiments. We also includean appendix detailing a derivation of the asymptotic expansion for the lubricationforce acting on a disk approaching the wall. The method of this derivation is newand can be easily adapted to other cases.

2. Motivations: governing equations for the fluid/particle flows and somedifficulties arising in their simulations. The general setting of this work is astudy of the motion of a rigid particle immersed in a viscous, incompressible fluidwith a particular emphasis on situations when the particle approaches the plane.To set the notations, we assume in general that the fluid (with the particle inside)fills a fixed domain Ω⊂Rd with d = 2 or 3, while the region occupied by the particleBt ⊂ Ω varies with time t. We denote the time-dependent fluid domain Ft so thatΩ = Bt ∪ Ft at any time t. Supposing that the inertial effects are negligible in the


TIME DISCRETIZATION FOR PARTICULATE FLOWS 3

fluid and the no-slip conditions are valid on the boundaries of Ω and Bt, the fluidmotion is governed by the Stokes equations

R

y

x

Bt

q

V

Ft = Ω \ Bt

Figure 1. Notations

−ν∆u + ∇p = ρfg, in Ft

∇ · u = 0, in Ft

u = 0, on ∂Ω

u = V + ω × r on ∂Bt

(1)

where u and p are the velocity and the pressure in the fluid, ν and ρf are the viscosityand the density of the fluid, g is the external force, V = V(t) and ω = ω(t) are thetranslational and angular velocities of the rigid body Bt, r = x − G is the vectorpointing from the center of mass of the particle G to a point x on its boundary. Themore realistic Navier-Stokes equations may be accommodated into the framework(1) by including the convective term u · ∇u into g.

The fluid exerts a net force F and a torque T on the particle given by

F = F(Bt,V, ω) =

∫

∂Bt

(2νD(u) − pI)ndσ, (2)

T = T(Bt,V, ω) =

∫

∂Bt

r × (2νD(u) − pI)ndσ,

where D(u) stands for the symmetric gradient of u and n is the unit normal vectoron ∂Bt directed towards the fluid domain. Note that F and T are indeed functionsof only the placement of the particle Bt and its translational and angular velocitiessince the velocity u and the pressure p are uniquely determined in the fluid by theparameters Bt, V and ω as the solution to the Stokes equations (1). Moreover, thedependence of F and T on V and ω is linear. Using these notations we write outthe equations of motion of the particle as follows

mdVdt = F(Bt,V, ω) +mg,

Itdωdt + ω × Itω = T(Bt,V, ω),

(3)

where m is the mass of the particle and It is its inertia tensor, expressed in the fixedCartesian frame and thus dependent on time. Equations (3) are coupled with the



equations describing the propagation of the particle, i.e. G = V for the center ofmass and ri = ω× ri, i = 1, . . . , d, for the vectors ri fixed in the particle. The forceF is a sum of the Archimedes force due to gravity and of the drag force which ispurely hydrodynamic, i.e. obtained from (1) by setting g = 0. The particularityof the drag is that it tends very rapidly to ∞ when the particle approaches thewall, thus preventing collisions between them. Indeed, it has been proved in [10](2D case), [11] (3D case), that a smooth rigid body embedded in a viscous fluidcannot touch the wall in finite time. In the regime of very small distance betweenthe particle and the wall, the drag force is also known as the lubrication force andit is notoriously difficult to take into account in a numerical simulation.

It is noteworthy that these considerations are valid only for smooth surfaces;however, modelling surface roughness of the wall or particle is a much more delicateissue. Several experimental ([24], [14], [13], [25]) and theoretical ([13], [21]) workspropose in this case to modify the expression for the lubrication drag force byintroducing a shift to the distance to the wall, of magnitude strictly lower than theroughness size. This physically means that roughness decreases the dissipation inthe system, and that the interaction is similar to that between equivalent smoothsurfaces, located at some intermediate position, between the peaks and the valleysof asperities.

To simulate the motion of the particle governed by equation described above,one should be able to solve numerically the Stokes system (1) for any given particleposition Bt and for any given velocities V and ω. This is a formidable task in itselfespecially because the position of the particle is not known a priori but changes withtime. We do not make precise the choice of the numerical method for the Stokessystem. We just assume for the moment that the force F and the torque T canbe computed for any Bt, V, ω but this computation is in general very expensive.Our primary goal in this article is to devise an efficient discretization in time of (3)using as few as possible solutions of the Stokes system. The simplest idea is to usethe following scheme

mVk − Vk−1

∆t= F(Btk−1

,Vk, ωk) +mg(tk), (4)

Itk−1

ωk − ωk−1

∆t+ ωk−1 × Itk−1

ωk = T(Btk−1,Vk, ωk), (5)

Gk − Gk−1

∆t= Vk, (6)

rik − ri

k−1

∆t= ri

k × ωk, i = 1, . . . , d. (7)

We have introduced here the uniform grid in time tk = k∆t and have denoted thequantities computed at the time step tk by the superscript k. The idea behind thescheme (4)–(7) is to compute first the velocities by (4)–(5) and then to propagatethe particle using the last available values of the velocities. The equations (4)–(5) are thus coupled together and also coupled with the solution of the Stokessystem on a fixed geometry given by position of the particle Btk−1

on the previoustime step tk−1. The cost of such a computation is normally essentially the sameas that of the Stokes system (1) with prescribed V and ω. We need thus onesolution of the Stokes system per time step. This approach was successfully used,for example, in [16] in conjunction with a fictitious domain discretization of theStokes system as in [8]. However, independently from the discretization in space,



one would encounter problems when the particle approaches the wall. Indeed, inthis lubrication regime the force F explodes and thus the scheme (4)–(5) is nolonger valid unless an extremely low value for the time step is used that makes thesimulation prohibitively expensive. A commonly used cure for this problem is tointroduce short-range repulsion forces between the particle and the wall, as in [8],for example. However, the influence of these (not necessarily realistic) forces onthe accuracy of a simulation is not well understood. Another simple idea is just tostop the particle when it tries to penetrate the wall during a numerical simulation.However, it is then not necessarily clear what criterion should be chosen to decideif the particle should eventually bounce off the wall and when should it happen.These questions have a partial answer in the articles [18, 15] on the gluey particlemodel. It is shown there that the particle trajectory satisfies an integro-differentialequation in the limit of vanishing viscosity, which is easy to discretize in time usingmoderate time steps and which predicts the moment of an eventual rebound fromthe wall. We pursue a similar idea in this article but our aim is to construct anapproximated trajectory of the particle in the lubrication regime that would beaccurate enough for any given value of the viscosity, not necessarily small.

3. A model ordinary differential equation with lubrication forces.

3.1. The model. Let us consider the simplest setting of the problem describedin the previous section: assume Ω ⊂ R2 is the half-plane (x, y), y > 0 and theparticle is a disk of radius R. Let moreover g(t) = g(t)e2 and assume the particleis at rest at the initial time. The x-component of the particle velocity and itsangular velocity will then vanish at all time. The position of the particle is fullydetermined by its distance q from the bottom, as in Figure 1. The net force F is thesum of the drag, which is a function of q and V, linear in V, and of the Archimedesforce:

F = −n(q)V +mag(t)e2, with ma = m− ρf |Bt|,where n(q) is the drag coefficient computed by the Stokes equations (1). Denotingthe y-component of the velocity by v, we are thus led to the following differentialequations

mv = −n(q)v +mag,

q = v.(8)

We are especially interested in the lubrication regime of small q. The asymptotic

of n(q) when q → 0 is given by 3√

2πν(

Rq

) 32

(see Appendix A) in the 2D case.

After eliminating v from the system (8) and going to non-dimensional variables(see Appendix B for the details) we obtain the following equation for q(t)

q = −ε qq

32

+ g, (9)

with ε = 3ν

ρsR32

√2ρs

gchar(ρs−ρf )where ρs is the density of the solid disk and gchar is

the characteristic value of g.In the same way, we can consider the analogous three dimensional problem

setting Ω to be the half-space (x, y, z), z > 0 and the particle to be a ball ofradius R. The asymptotic expression is well known in this case (cf. Remark 5) and



is given by 6πV ν R2

q . Performing the same non-dimensionalizations as in the 2D

case, we arrive at the equation for q(t) (the distance from the particle to the wall):

q = −ε qq

+ g, (10)

with ε = 9ν

2ρsR32

√ρs

gchar(ρs−ρf ).

liquid ρs/ρf R (mm) ε in (9) ε in (10)

water 1.1 1 0.13 0.141.5 0.1 1.55 1.64

glycerin 1.1 1 152 1611.5 0.1 1843 1954

Table 1. Typical values of the parameter ε in (9) or in (10) takingthe viscosity and density of either water or glycerin and differentvalue of the particle radius R and density ρs.

We remind that equations of the type (9)–(10) are at the basis of the glueyparticle model of [18, 15]. The model consists in fact in considering the limit ε → 0,which is physically the limit of vanishing viscosity. We see, however, that ε is notnecessarily small.

3.2. Estimates and an approximated solution for the model ODE. Con-sider the ordinary differential equation

q = −n(q)q + g (11)

or, equivalently, the system of first-order equationsv = −n(q)v + g

q = v(12)

where n(q) is a given differentiable decreasing positive function on (0,∞) with

n(q) = N ′(q) such thatN(q) → −∞ as q → 0. Note that this is valid for n(q) = ε/q32

or n(q) = ε/q, which give the asymptotic of the lubrication force in 2D and 3Drespectively. We assume g ∈ L1

loc(R+) from now on. Under this hypothesis, Problem(11), completed with appropriate initial conditions, is well-posed, as proved forinstance in [18, Prop. 1.1] for the case n(q) = ε/q. The proof is generalized withoutproblem to any n(q) satisfying the above hypotheses, as mentioned in [18, Section5.1].

Proposition 1. Given (q0, v0) ∈ R2 with q0 > 0 there exists a unique positive global

solution q ∈ W 1,∞loc (R+) to (11) with initial conditions:

q(0) = q0 q(0) = v0. (13)

In the following, we are especially interested in the lubrication regime of smallq → 0, when the drag coefficient n(q) tends to infinity, which can create difficultieswhen computing numerical solutions of (11). In order to overcome this problem, oneshould either use a very small time step when discretizing the equation, but this canbecome computationally heavy, or propose a strategy to prevent q from becomingtoo small. The approach we propose here, inspired by the latter idea, consists



in introducing a threshold value qs for the distance q, below which, instead ofsolving (11), we introduce an alternative model and construct a sufficiently accurate“approximate” solution q instead of the “true” solution q. We will first pursue thisstrategy at the continuous level and then implement it at the discrete level in thenext section.

Our construction is the following: we first choose some threshold value qs andnext suppose that q goes below this value after some time t1 when q(t) hits thethreshold qs for the first time. Our aim is to find a suitable approximation of qafter t1 which enables to predict the time t2 when q goes above the threshold qswithout solving (11). To this end, we introduce

v(t) = q(t1) +

∫ t

t1

g(s)ds (14)

for t ≥ t1 and note that v(t2) = q(t2). Indeed, integrating (11) from t1 to t > t1shows that

q(t) = q(t1) +N(qs) −N(q(t)) +

∫ t

t1

g(s)ds = N(qs) −N(q(t)) + v(t). (15)

Setting here t = t2 and noting that q(t2) = qs gives the desired result. Our approx-imation of the trajectory q(t), denoted by q(t), stems from the assumption (verifiedafterwards in Proposition 3) that the velocity q(t2) at the return point t2 is smallprovided the threshold qs is small. Consequently, a good approximation for the timet2 should be provided by the time t2 defined as the first time larger than t1 whenv(t) = 0. The construction of the approximated trajectory is hence the following:we first assume that q(t) is the same as q(t) until the latter hits qs for the first timeat t = t1. Next, the trajectory q(t) is frozen until the time t = t2 and resumes thenagain as a solution to (11) starting from qs with zero velocity:

q(t) =

q(t), for 0 < t < t1qs, for t1 ≤ t < t2solution to (11) with q(t2) = qs, q(t2) = 0, for t ≥ t2

(16)

(see Figure 2).In order to study the error committed by introducing the approximated trajectory

q(t), we start by infering the following estimate:

Proposition 2. The time t2 provides a lower bound for t2, i.e. t2 ≤ t2.

Proof. There are three cases. If t2 does not exist, there is nothing to prove. Ift2 = t1 this means t1 is a local minimum for q so that q(t1) = 0 and t1 = t2 = t2.Finally, if t2 exists and satisfies t1 < t2, since q(t1) = q(t2) = qs the function q(t)has a local minimum tmin ∈ (t1, t2) where q(tmin) = 0. Evaluating (15) at t = tmin

shows then v(tmin) = N(q(tmin))−N(qs) ≤ 0. On the other hand, v(t2) = q(t2) ≥ 0since the function q(t) is increasing at t2 by the definition of t2. The intermediatevalue theorem tells now that there exists t ∈ [tmin, t2] such that v(t) = 0. Thus thefirst time t = t2 when v(t) = 0 is certainly in (t1, t] ⊂ (t1, t2].

We have proved, in particular, that if the true trajectory q(t) ever returns tothe values above the threshold qs, i.e. t2 < ∞, the indicator t2 will show it, i.e.t2 < ∞. In Proposition 5, we compute an upper bound t2 for t2. In particular,under suitable assumptions on g the time t2 exists.



q q

qs

q

t1 t2t2 t2

Figure 2. Construction of the approximated trajectories q(t) and q(t).

To see moreover that t2 may be a good approximation to t2 we need, as mentionedbefore, to control q(t2). The following proposition proves that q(t2) is not too largewith respect to qs.

Proposition 3. Let q+ (resp. g+) be the positive part of q (resp. g): q+ =max(q, 0) (resp. g+ = max(g, 0)). Then

‖q+‖ ≤ ‖g+‖n(qs)

, (17)

where ‖ · ‖ denotes the supremum on [t1, t2]. In particular, q(t2) ≥ 0, so that

q(t2) ≤ ‖g+‖n(qs)

.

Proof. Let q be non-negative on an interval [tmin, tmax] ⊂ [t1, t2]. Without loss ofgenerality, we can thus assume that tmin is a local minimum of q(t) and tmax iseither a local maximum or tmax = t2. Then, we introduce

f(t) := q(t) − ‖g+‖n(q(t))

, ∀ t ∈ [t1, t2].

If f(t0) > 0 at some t0 ∈ (tmin, tmax) then f(t) remains positive on a small intervalaround t0 so that q < 0 by invoking (11) and thus q(t) is decreasing on this interval.Since q(t) is increasing on [tmin, tmax] and n(q) is decreasing, we obtain that f(t) isdecreasing locally around t0 so that f(t) ≥ f(t0) > 0 for all t ∈ [t0−δ, t0] with someδ > 0. Repeating this argument for times down to tmin we see that f(t) should bepositive on [tmin, t0] but this is impossible since q(tmin) = 0 and f(tmin) < 0. Thisproves that f(t) ≤ 0 on [tmin, tmax].

The results above allow us to fully characterize the error of approximating q(t)by q(t) up to time t2 and to bound it by certain quantities depending on q alonefor time after t2, at least in the special case when g(t) is given by

g(t) =

g−(t) < 0, for t ≤ t0,

g+ > 0, for t > t0,(18)



with some positive constants t0, g+ and a negative function g−(t). Indeed, we provein the following proposition that the error τ = t2 − t2 is bounded by 1/n(qs) andwe remind that n(qs) → ∞ as qs → 0.

Proposition 4. If g(t) is given by (18), then

0 ≤ t2 − t2 ≤ 1

n(qs),

supt∈[0,t2]

|q(t) − q(t)| ≤ qs, (19)

|q(t2) − ˙q(t2)| = q(t2) ≤ g+n(qs)

.

Moreover, both q(t) and q(t) are non-decreasing for t ≥ t2 and

|q(t) − q(t)| ≤q(t), if t2 ≤ t < t2,

q(t) − q(t− τ), if t ≥ t2,(20)

with τ = t2 − t2.

Proof. We first note that t2 ≥ t2 ≥ t0. The first inequality here is already proved inProposition 2. The second inequality follows from v(t2) = 0, which can be rewrittenas

−q(t1) =

∫ t2

t1

g(s)ds.

Since q(t1) ≤ 0, the last equality cannot hold if g is negative everywhere on [t1, t2].Since t2 ≥ t0, we have g(t) = g+ > 0 for t ≥ t2. Evaluating (15) at t = t2 gives

q(t2) =

∫ t2

t2

g(s)ds = g+(t2 − t2). (21)

Recalling Proposition 3, we see that

t2 − t2 =q(t2)

g+≤ 1

n(qs),

which is the first inequality in (19). The second inequality in (19) is obvious since0 < q(t2) ≤ qs and q(t2) = qs. The third inequality follows from Proposition 3 sinceq(t2) ≥ 0. Indeed, (15) evaluated at t = t2 gives q(t2) = N(qs) −N(q(t2)).

We now turn to the study of q(t) and q(t) for time t ≥ t2, in order to prove thatq(t) and q(t) are non-decreasing on this interval. We first observe that q(t) cannothave local maximums at t > t0. Indeed, at any local extremum te > t0, equation(11) would imply q(te) = g+ > 0. Since q(t) is non-decreasing at t2 > t0, it shouldbe non-decreasing also everywhere on [t2,∞). The same reasoning applies to q(t).The estimate (20) is now evident in the case t2 ≤ t ≤ t2 since q(t) ≥ qs and q(t) ≤ qsfor such t.

In order to prove (20) in the case t ≥ t2 we will show that q(t) is squeezed betweenqτ (t) and q(t) for t ≥ t2 where qτ (t) = q(t−τ). Indeed both qτ (t) and q(t) satisfy thesame equation (11) with the same initial condition at t = t2 (q(t2) = qτ (t2) = qs)but possibly different initial velocities q(t2) ≥ 0, ˙qτ (t2) = 0. Writing (11) for qτ (t)and q(t), taking the difference thereof and integrating from t2 to t yields

q(t) − ˙qτ (t) = q(t2) −N(q(t)) +N(qτ (t)).



This shows that if the trajectories of q(t) and qτ (t) intersect at some time t′ thenq(t′) ≥ ˙qτ (t′) so that q(t) ≥ qτ (t) at least for some time after t′. Since q(t) ≥ qτ (t)holds also for some time after t2 it should hold for all t ∈ [t2,∞).

It remains to prove that q(t) ≤ q(t) for all t ∈ [t2,∞). To this end, we integrate(11) for q(t) from t2 to t and that for q(t) from t2 to t. This yields with the aid of(21)

q(t) = q(t2) +N(qs) −N(q(t)) +

∫ t

t2

g(s)ds = N(qs) −N(q(t)) +

∫ t

t2

g(s)ds, (22)

˙q(t) = N(qs) −N(q(t)) +

∫ t

t2

g(s)ds. (23)

We see now that if q(t′) = q(t′) at some time t′ ≥ t2 then also q(t′) = ˙q(t′). It meansby the uniqueness of solutions to (11) that the trajectories of q(t) and q(t) eithercoincide or do not intersect on [t2,∞). Since q(t2) ≤ q(t2) this implies q(t) ≤ q(t)on [t2,∞).

It is possible to relax the hypotheses of the last Proposition on the particularform of the function g in several ways, if we still remain in the case when q goesbelow the threshold value qs a finite number of times (in particular, g should beallowed to change sign only a finite number of times). Although we do not havean analogue of these results for a general g(t), we can always provide an easilycomputable upper bound for t2 alongside the lower bound t2.

Proposition 5. Let t2 be the first time after t2 such that∫ t2

t2

(t2 − s)g(s)ds = qs. (24)

Then t2 ≥ t2.

Proof. Consider q(t) defined for t ≥ t2 as the solution to

˙q(t) =

∫ t

t2

g(s)ds, q(t2) = 0

We recall now the property (22) of the original solution and note N(q(t)) < N(qs)for t ∈ (t2, t2) so that

q(t) ≥∫ t

t2

g(s)ds ≥ ˙q(t).

Since q(t2) > q(t2), the trajectory of q(t) lies above that of q(t) for t ∈ (t2, t2) sothat q(t) hits the threshold qs before q(t). In other words, t2 < t2, where t2 is thefirst time after t2 such that q(t) = qs. This definition of t2 is equivalent to that in(24).

4. Discretization in time of the model ODE. We are now going to give adiscretized in time version of some of the ideas elaborated for the model ODE (11)in the previous section. We shall start by a straightforward Euler discretization intime of (11) rewritten as the system (12) to remind that this scheme (referred to asAlgorithm 1) does not always work in the near-contact regime. We then turn to adiscrete version of the method constructing the approximate trajectory q(t) definedin (16) and suitable in the near-contact case. This is done in Algorithm 2, where an



a priori threshold value qs > 0 is used to compute an approximated solution to theproblem. We pursue by discussing the strategy to choose an appropriate value for qswhich will typically depend on the time step ∆t. Finally, we combine an adaptativestrategy for the choice of the time step with the threshold approximation given byAlgorithm 2. This gives Algorithm 3.

Although all these developments are done in this article only for the model equa-tion (11), we have in mind a possible incorporation of our time marching schemesinto a full simulation of flow with solid particles. The coefficient n(q) will be thengiven by a solution to the Stokes system at each time step where q represents thesurface-wall distance in the case of a particle approaching to the wall or the inter-particle distance in the case of two approaching particles. The “external” forceg(t) in (11) will then include the components of the physical external forces, likegravitation, in the direction connecting the closest points on the surfaces of nearlytouching particles, plus other corrections. Most importantly, these include the partof hydrodynamic drag due to the convective terms in Navier-Stokes, if the con-vective terms in the fluid motion equation are not neglected. The time marchingscheme should thus work robustly for any right hand side g(t), in particular thatchanging the sign. Indeed, a possible scenario even in the simplest test case of aparticle falling under its weight on a horizontal wall (so that g(t) is negative in thebeginning) could be that the convective terms in Navier-Stokes equations becomeso important at some point in time that they prevail over the gravitation force andg(t) becomes positive, which may lead to the rebound of the particle.

4.1. Three schemes for ODE (11). Introducing the time step ∆t and the dis-crete times tk = k∆t, k = 1, 2, . . . we first describe in Algorithm 1 a straightforwardEuler discretization in time of (11) rewritten as the system (12).

Algorithm 1 (straightforward Euler)

Step 0. Initialize (v0, q0).Step 1. For k = 1, 2, . . . update (vk, qk) as follows

vk−vk−1

∆t = −n(qk−1)vk + g(tk)qk−qk−1

∆t = vk.(25)

Algorithm 1 is inspired by the real fluid-particle simulations in which one firstfinds the new velocity at each time step and then moves the particle with thisvelocity. An immediately evident drawback of the scheme in Algorithm 1 is that itdoes not necessarily provide a positive approximation qk. We remind that negativevalues of q are unphysical. Moreover, even if approximations qk remain positive butbecome tiny at some time steps, one would require very small stepsize to obtain anaccurate solution.

Remark 1. One may argue that this can be cured by resorting to a fully implicitscheme that couples the evaluation of the velocity with the displacement of theparticle:

vk−vk−1

∆t = −n(qk)vk + g(tk)qk−qk−1

∆t = vk.(26)

Indeed, this scheme provides a positive solution for q, for example, in the importantcase n(q) = ε/q. To see this we eliminate vk from (26), which gives an equation for



qk only

qk

∆t− εqk−1

qk=qk−1

∆t− ε+ vk−1 + ∆tg(tk)

The function of qk in the left-hand side is increasing form −∞ to ∞ when qk goesfrom 0 to ∞. This means that there is a unique positive solution qk for any givenvk−1 and qk−1. However, scheme (26) would be too expensive in a real fluid/particlesimulation where n(qk) should be recomputed after any change in qk through anumerical approximation of the Stokes equations.

Remark 2. Difficulties related to the discretization of these differential equationshave already been pointed out in [18] (see Remark 3.1): it is shown that verysmall values of q can be reached (below the smallest real number which can bestored by standard numerical softwares and also below intermolecular distances).Different approaches have been proposed in previous related works to deal withsimilar problems: in [18, Section 5.2], the author suggests the introduction of acut-off function for the microscopic distance γ = ε ln(q), defined in terms of theroughness of the surface (see also [15, Section 2.5]); in [15, Section 2.4.3], for thecase of fluid/particle simulation, a constraint for the distance q is defined in termsof the mesh size of the fluid domain.

In order to overcome these difficulties, in this work we propose to introduce asmall threshold value qs > 0, to replace the exact solution q(t) by the approximatedone q(t), as summarized in (16) on the continuous level. Discretization in time ofthis idea introducing also an approximation of v(t) defined in (14) is detailed inAlgorithm 2. Note that the exact threshold is replaced in Step 2 of this algorithmby the last available value of qk before passing below qs, i.e. qk1−1, so that thepassage time t1 is legitimately approximated by tk1−1.

Algorithm 2 (with an a priori fixed threshold)

Step 0. Initialize (v0, q0) and choose qs > 0.Step 1. For k = 1, 2, . . . update (vk, qk) by (25) until qk ≤ qs at some k = k1. We

then redefine vk1 = 0, qk1 = qk1−1, initialize v by vk1 = vk1−1 + g(tk1)∆tand switch to Step 2.

Step 2. For k = k1 + 1, k1 + 2, . . . update vk as follows

vk = vk−1 + g(tk)∆t (27)

keeping qk = qk1−1, vk = 0 until vk ≥ 0 at some k = k2. We then abandonthe calculation of v and switch to Step 3.

Step 3. For k = k2 + 1, k2 + 2, . . . update (vk, qk) as in Step 1. If qk goes againunder qs at some time step, then reintroduce v and keep switching betweenSteps 1 and 2 back and forth.

As suggested by Proposition 4 a good choice for qs would be such that 1/n(qs) =C∆t with some constant C so that the criterion for switching from Step 1 to Step2 can be rewritten as n(qk) ≥ 1/(C∆t). Indeed, extrapolating the result of Propo-sition 4 to a general g(t), we see then that the error of approximation of the exactreturn time t2 by t2 ≈ k2∆t is of order ∆t. Moreover, the error |q(t) − q(t)| isdominated by |q(t) − q(t − τ)| with τ of order ∆t. Thus the error caused by theintroduction of the threshold should be of of the same order as that of Euler scheme



itself. However, an optimal choice of the constant C above would require some aposteriori error indicators to control the error of the Euler discretization.

Algorithm 3 (with adapted variable stepsize and automatically chosen threshold)

Step 0. Initialize (v0, q0) for t0 = 0, choose the first time step ∆t1 > 0, the minimaltolerated time step ∆tmin > 0 and the error tolerance per time step tol > 0.

Step 1. For k = 1, 2, . . . set tk = tk−1 + ∆tk, update (vk, qk) by

vk−vk−1

∆tk= −n(qk−1)vk + g(tk)

qk−qk−1

∆tk= vk.

(28)

and calculate the approximation for the error committed on this time stepby ek = max(|q(tk)|, |v(tk)|)∆t2k/2 where q(tk), v(tk) are computed using(29)–(30). If ek ≤ tol and qk > 0 then we accept the just computedvalues (vk, qk) and proceed to the next time step increasing the time step

to ∆tk+1 =√

tolek

∆tk. Otherwise, if ek > tol or qk ≤ 0, we reject the

approximation (vk, qk) and try to recompute it by (28) with a smaller

time step ∆tk := min(√

tolek

∆tk,∆tk/2). If the new approximation (vk,

qk) is still not sufficiently accurate, we try to diminish the time step againand again until an acceptable approximation (vk, qk) is computed andproceed only then to the next time step taking ∆tk+1 equal to the smallestvalue of ∆tk used on step k. However, if in the process of reducing ∆tkwe come to a time step smaller than ∆tmin at some k = k1, we abandonthe calculation of (vk1 , qk1) and switch to Step 2 taking the initial valuesvk1−1 = vk1−1, qs = qk1−1 and setting ∆t to the current value of ∆tk.

Step 2. For k = k1, k1 + 1, . . . update vk by (27) keeping qk = qs, vk = 0 until

vk ≥ 0 at some k = k2. We then abandon the calculation of v and switchto Step 3.

Step 3. For k = k2 + 1, k2 + 2, . . . update (vk, qk) as in Step 1. If qk goes againunder qs at some time step, then reintroduce v and keep switching betweenSteps 1 and 2 back and forth.

Generally speaking, a discretization of an ODE using the constant time step isoften not optimal concerning the CPU time needed to achieve a desired accuracy.This is especially true for our ODE (11) since the coefficient n(q(t)) can changeenormously during a simulation. One could try therefore to modify our algorithmsby introducing a non constant stepsize chosen at each step using an error indicator.A general recipe for an optimal adaptation of the stepsize, according to [2] is tochoose the stepsizes so that the discretization error per time step remains approxi-mately constant during the whole simulation. As is well known, the error per timestep in a Euler scheme like that of Algorithm 1, is given to the leading order bymax(|q(tk)|, |v(tk)|)∆t2k/2 where ∆tk = tk − tk−1 is the stepsize on step k which canbe varying. Assuming that (qk−1, vk−1) and (qk, vk) are Euler approximations attimes tk−1 and tk respectively, the second derivatives of q and v can be computedapproximately by deriving the equations in (12) with respect to time and replacing



the missing derivatives by finite differences. This gives

q(tk) ≈ −n(qk)vk + g(tk) (29)

v(tk) ≈ n2(qk)vk − n(qk)g(tk) − n(qk) − n(qk−1)

∆tkvk +

g(tk) − g(tk−1)

∆tk(30)

Thus, denoting Ek = max(|q(tk)|, |v(tk)|)/2 where the second derivatives are re-placed by the formulas above, the strategy to choose the time step would be torequire Ek∆t2k = tol = const where tol is prescribed tolerance. As this recipe canlead to extremely small ∆tk we combine it with the threshold approximation byswitching to q(t) only when ∆tk becomes smaller than some minimal admissibletime step size. This idea is implemented in Algorithm 3.

4.2. Some numerical tests. As an illustration, we consider the ODE (11) withn(q) = ε/q3/2 corresponding to the lubrication force in 2D. We report several nu-merical results setting

g(t) =

−2, for t ≤ 2,

2, for t > 2,(31)

and the initial conditions q(0) = 1, q(0) = 0. In all the cases, we have at ourdisposal a very accurate reference solution, which we call “exact” in what follows.

Remark 3. The source term g(t) represents an important and relevant test casein view of the possible incorporation of our time marching schemes into the fullfluid/particle simulations, as explained in the beginning of Section 4. Note that(31) was also used in [15] to test the gluey particle model.

We first compare Algorithm 1 (without threshold) vs. Algorithm 2 with thethreshold such that n(qs) = 1/(20∆t). The choice of the coefficient 20 in the lastformula is somewhat arbitrary, but it works fine in our test cases. The first seriesof numerical experiments is performed taking ε = 0.1. The results are reported inFigure 3. Note that, although the solution obtained with Algorithm 1 is positive at∆t = 0.01 and smaller, the introduction of the threshold in Algorithm 2 seems toenhance the quality of the solution at large time.

Let us now turn to another series of numerical experiments taking ε = 10−3.The results are reported in Figure 4. Algorithm 1 is now unable to produce aphysically acceptable solution even with very small ∆t = 0.0001. Algorithm 2, onthe other hand, works fine. In order to observe better the quasi-contact region andthe effect of introducing the threshold, we zoom on small distances by passing toa log-scale in Figure 5 (left). We report also some numerical experiments aimingat the determination of an optimal value of the threshold qs. As said already, ourstrategy is to choose it so that n(qs) = 1/(C∆t) taking C = 20 in all the precedingexperiments. Figure 5 (left) illustrates the results obtained fixing ∆t = 0.001 andtaking three different values for qs by varying C. These results confirm two generalobservations: taking qs too small may deteriorate the accuracy of the solution(indeed, the original goal of the introduction of qs was to avoid the extremely smallvalues of q). On the other hand, taking qs too large may unnecessarily perturbthe solution in the regions where a straight discretization could give more preciseresults. Thus, there should be an optimal choice of qs which seems to be not toofar from the formula mentioned above with C = 20.



t

q

0 2 4

0

0.2

0.4

0.6

0.8

1

1.2

exact0.10.010.0010.0001

t

q

0 2 4

0

0.2

0.4

0.6

0.8

1

1.2

exact0.10.010.0010.0001

Figure 3. Solution to the ODE (11) with n(q) = ε/q3/2, ε = 0.1.Left: the exact solution and solutions obtained by Algorithm 1with ∆t ranging from 0.1 to 0.0001. Right: solutions obtained byAlgorithm 2 with ∆t in the same range. The results obtained with∆t = 0.0001 are visually indistinguishable from the exact solution.

t

q

0 2 4

-6

-5

-4

-3

-2

-1

0

1

exact0.10.010.0010.0001

t

q

0 2 4

0

0.2

0.4

0.6

0.8

1

1.2

exact0.10.010.0010.0001

Figure 4. Solution to the ODE (11) with n(q) = ε/q3/2, ε = 10−3.Left: the exact solution and solutions obtained by Algorithm 1with ∆t ranging from 0.1 to 0.0001. Right: solutions obtained byAlgorithm 2 with ∆t in the same range. The results obtained with∆t ≤ 0.01 on the right are visually indistinguishable from the exactsolution.

We conclude by a series of experiments using Algorithm 3 with adapted stepsize.We take again the same governing equation with ε = 10−3 and run Algorithm 3setting the tolerance to tol = 10−5 and taking a number of values for ∆tmin. Theresults are reported in Figure 6 with the evolution of q(tk) on the left and thatof ∆tk on the right. We give in particular the results with ∆tmin = 0 so thatAlgorithm 3 is reduced to a standard time marching scheme with automaticallyadapted step sizes. It is interesting to note that step sizes ∆t of the order 10−4 areactually sufficient to satisfy our error tolerance criterion almost everywhere apart



t

q

0 2 4

10-6

10-5

10-4

10-3

10-2

10-1

100

exact0.10.010.0010.0001

t

q

0 2 4

10-6

10-5

10-4

10-3

10-2

10-1

100

exactC = 1C = 20C = 200C = 2000

Figure 5. Left: same results as on Figure 4, i.e. varying ∆t andchoosing qs so that n(qs) = 1/(20∆t). Right: same test with ∆t =0.001 fixed and different values of qs calculated so that n(qs) =1/(C∆t) and C varying in the range [1, 2000].

t

q

0 2 4

10-6

10-5

10-4

10-3

10-2

10-1

100

exact00.0010.0001

t

step

size

0 1 2 3 4 510-8

10-7

10-6

10-5

10-4

10-3

00.0010.0001

Figure 6. Results obtained by Algorithm 3 with tol = 10−5 anddifferent values of ∆tmin specified in the legend. Left: evolution ofq(tk) as compared to the exact solution. Right: evolution of ∆tk.

from a very small range of time around 1 where the Algorithm chooses ∆t of theorder 10−8. Setting ∆tmin to 10−3 or 10−4 avoid such small time steps and givesfairly good results.

Acknowledgements. We would like to thank the organizers of the conferenceECM’09 (International Conference of Microfluidics and Complex flow) Lassaad ElAsmi and Mourad Ismail for giving us the opportunity to present this work. Wehad a number of useful discussions during this conference, especially with AlineLefebvre-Lepot and Francois Feuillebois, to whom we are very grateful.

Appendix A. An estimate for the drag (lubrication) force. This part isdevoted to computing the first order expansion of the viscous drag exerted by afluid on a disk approaching a plane wall. Let Ω ⊂ R2 be the half plane Ω =



(x, y) with y > 0, the boundary of Ω is ∂Ω = (x, 0), x ∈ R (see Figure 1). Weassume that the fluid fills the domain Fq := Ω \ Bq where Bq := B((0, R+ q), R) isthe domain occupied by the solid disk of radius R > 0 and placed at the distanceq > 0 from the boundary ∂Ω. Our notations are lightly different here from that ofSection 2. Indeed, domains Fq and Bq are now indexed by q rather than t since theeventual dependence of q on time does not interest us in this Appendix. Assumemoreover that the disk has a prescribed velocity V = V e2 and zero angular velocity.The motion of the fluid is governed by Stokes equations (1) with g = 0 and zeroboundary conditions at the infinity for u. It is classical that (1) is well-posed in asuitable framework and has a unique solution (u, p) (the pressure p is defined upto a constant). In particular, the viscous drag force F exerted by a fluid flow (u, p)on Bq and computed by (2) is well-defined as a function of V and q. Our aim is toprove the following result:

Theorem A.1. The viscous drag force satisfies F = −n(q)V where

n(q) = 3√

2πν

(R

q

) 32

[1 + ε(q)] ,

with ε(q) → 0 when q → 0.

Remark 4. The equivalent result in the three-dimensional setting is well-knownbut it is new in two dimensions. Whatever the novelty of this result, the maincontribution in this section is that we provide a new way for computing the firstorder expansion of the viscous drag. This new method is more robust than theknown computations [3, 5, 19, 20]. In particular, we claim it extends to the three-dimensional setting with small changes and can be adapted to other boundaryconditions. As an illustration, our method is a well-designed tool for estimating thedistance between solid bodies in solutions to the fluid-structure interaction systemwith the full newtonian Navier Stokes equation on the fluid domain [7, 10, 11].

The proof of Theorem A.1 is divided into three steps. First, we recall the varia-tional formulation for the solution (u, p) to (1) and apply it to compute the associ-ated drag coefficient n(q) in the expression for viscous drag F = −n(q)V. Secondly,we deduce from the variational formulation the lower and upper bounds for n(q).We conclude the proof by computing an asymptotic expansion of the bounds of thisrange.1. The variational formulation for the Stokes system (1). As (u, p) in (1) and theformula (2) for the drag depend linearly on V, we set V = e2 in what follows. It isclassical to extend the velocity u to the whole domain Ω by setting it to e2 insideBq and to look for u in the following space

D1,20 (Ω) :=

w ∈ L1

loc(Ω) with ∇w ∈ L2(Ω), div w = 0 in Ω and w = 0 on ∂Ω.

The solution of (1) is associated with the minimization of Dirichlet integral D(w) =∫Ω |∇(w)(x, y)|2dxdy over the subset Yq of D1,2

0 (Ω):

Yq =w ∈ D1,2

0 (Ω) such that w|Bq= e2

.

Let us denote this minimum by ED(q), i.e.

ED(q) := min D(w) ; w ∈ Yq , (32)



As the Dirichlet integral D is a strictly convex functional on Yq it has a uniqueminimizer uq for any q > 0. It is easy to see that uq gives the solution of (1).Indeed, we have for any w ∈ C∞

c (Fq) such that div w = 0,∫

Ω

∇u(x, y) : ∇w(x, y)dxdy = 0,

which is the variational formulation of (1). The pressure pq ∈ L2loc(Ω) can be then

recovered as the Lagrange multiplier corresponding to the constraint div u = 0. Wenotice that pq is unique up to a constant and the ellipticity of the Stokes problem

implies that (uq, pq) is smooth on Fq so that it furnishes a classical solution to (1).We refer to [6] for more details and also for a proof of the converse implication.

Given w ∈ C∞c (Ω) such that div w = 0 and w = e2 on Bq, a straightforward

integration by parts yields (recall that n is the unit normal looking into Fq):

ν

∫

Ω

∇uq(x, y) : ∇w(x, y)dxdy = 2ν

∫

Fq

D(uq)(x, y) : ∇w(x, y)dxdy

= −∫

∂Bq

[2νD(uq) − pqI2]ndσ · e2.

Taking a suitable family of approximation of uq (see [6] for details) we obtain inthe limit:

νED(q) = −∫

∂Bq

[2νD(uq) − pqI2]ndσ · e2.

The drag F is parallel to the velocity V for symmetry reasons, i.e. F = (F · e2)e2.We have thus the following result.

Lemma A.2. For any given q > 0 and V parallel to e2, the drag force is given byF = −n(q)V with n(q) = νED(q).

2. Upper and lower bounds for ED(q). We apply the above variational formulationto bound ED(q). To this end, given q > 0, we first prove that the Dirichlet integralof any u ∈ Yq is greater than some mD(q) depending only on q. This gives us a lowerbound for ED(q). Then, we construct a suitable uq ∈ Yq and compute its Dirichletintegral MD(q). This gives an upper bound for ED(q). We finally compare mD andMD for small values of q to prove Theorem A.1.

As the scale invariance of our problem implies that n(·) is actually a function ofq/R, it is sufficient to consider the case R = 1, which we admit in what follows. For

any x ∈ (−1, 1), y < 1+ q such that (x, y) ∈ ∂Bq there holds y = 1+ q−√

1 − x2 :=δq(x) and we denote

Ωq := (x, y) ∈ R2 such that x ∈ (−1/2, 1/2) and y ∈ (0, δq(x)).Given a smooth u ∈ Yq, we introduce

ψ(x, y) = −∫ y

0

u1(x, z)dz.

Then, straightforward computations imply that ψ ∈ C∞(F q) and satisfy u =∇⊥ψ = (−∂yψ, ∂xψ), so that we have the boundary conditions:

ψ(x, y) = 0, ∂yψ(x, y) = 0, on ∂Ω,

∂xψ(x, y) = 1, ∂yψ(x, y) = 0, on ∂Bq.(33)



On the other hand, we compute the Dirichlet integral D(u) with respect to ψ. Thisyields:

D(u) ≥∫

Ωq

|∂yyψ(x, y)|2dxdy.

We denote I(ψ) the integral on the right-hand side of the above inequality. Thus,D(u) is greater than the minimum of I(ψ) over smooth ψ satisfying the boundaryconditions (33). This minimum is computed in the following lemma:

Lemma A.3. Given q > 0 and ψ ∈ C∞(Ωq), satisfying boundary conditions (33)there holds I(ψ) ≥ I(ψq) where :

ψq(x, y) = x

[y

δq(x)

]2 (3 − 2

[y

δq(x)

]), ∀ (x, y) ∈ Ωq.

Proof. Given any ψ(x, y) ∈ C∞(Ωq), satisfying (33), boundary conditions on ∂Bq

imply ψ(x, δq(x)) = C + x, for all x ∈ (−1/2, 1/2), with some C ∈ R. Thus ψsatisfies for all x ∈ (−1/2, 1/2)

ψ(x, 0) = 0, ψ(x, δq(x)) = C + x,∂ψ

∂y(x, 0) = 0,

∂ψ

∂y(x, δq(x)) = 0.

Then, for arbitrary x ∈ (−1/2, 1/2), with the possible exception of x = −C, weintroduce χx(t) = ψ(x, tδq(x))/(x + C), t ∈ (0, 1), which satisfies

χx(0) = 0, χx(1) = 1, χ′x(0) = 0, χ′

x(1) = 0. (34)

By a standard optimization argument we see that the minimum of∫ 1

0|χ′′

x(t)|2dt over

all smooth χx(t) satisfying (34) is attained on the function η(t) such that η(4)(t) = 0on (0, 1), which is given by η(t) = t2(3 − 2t). This yields

∫

Ωq

|∂yyψ(x, y)|2dxdy =

∫ 1/2

−1/2

∫ 1

0

(x+ C)2

|δq(x)|3|χ′′

x(t)|2dtdx

≥∫ 1/2

−1/2

∫ 1

0

(x+ C)2

|δq(x)|3|η′′(t)|2dtdx. (35)

Minimizing the last integral with respect to C, we obtain C = 0. The last inequalitybecomes equality if we take ψ(x, y) = ψq(x, y) = xη(y/δq(x)).

Finally, a lower bound is given by

mD(q) =

∫

Ωq

|∂yyψq(x, y)|2dxdy.

On the other hand, it is always possible to extend ψq on the whole Ωq to some

ψq ∈ C∞(Fq) ∩ C∞(Bq) ∩ C(Ω) and such that uq := ∇ψq ∈ Yq. For instance onemight interpolate (in the x-variable) ψq with a suitable truncation of ψ0(x, y) = x.As, outside Ωq, the solid disk remains at a positive distance of ∂Ω, the truncationand interpolation function can be made independent of q. In particular there holds:

∫

Ω

|∇uq(x, y)|2dxdy

= mD(q) + r(q) +

∫

Ωq

[|∂xxψq(x, y)|2 + |∂xyψq(x, y)|2

]dxdy := MD(q),

with r a bounded function of q.



3. Asymptotic expansion of mD(q) and MD(q).

Lemma A.4. There holds, for small values of q :

mD(q) = 3√

2π1 + ε(q)

q3/2, (36)


Proof. As already seen in the proof of the preceding Lemma, mD(q) is given by theintegral (35) with C = 0 and η(t) = t2(3 − 2t). Integrating with respect to t andperforming a change of variables x → √

qx yields

mD(q) = 12

∫ 1/2

−1/2

x2

|δq(x)|3dx =

12

q3/2

∫ 1/(2√

q)

−1/(2√

q)

x2dx

(δq(√qx)/q)3

Expanding to the first order, we have:

limq→0

x2

(δq(√qx)/q)3

= limq→0

x2

[(1 + q −√

1 − qx2)/q]3=

x2

(1 + x2/2)3.

We remark moreover that standard geometric arguments imply that δq(x) ≥ q+x2/2or δq(

√qx)/q ≥ 1 + x2/2 so that we can apply the Lebesgue theorem to obtain:

limq→0

∫ 1/(2√

q)

−1/(2√

q)

x2dx

(δq(√qx)/q)3

=

∫ ∞

−∞

x2dx

(1 + x2/2)3=

π

2√

2.

One can also check that ∂yyψq is the second derivative of ψq which diverges thefastest when q goes to 0. Thus, MD(q) −mD(q) = o(mD(q)) as q → 0 so that (36)holds also for MD(q), which proves Theorem A.1.

Remark 5. The same method can be easily adapted to the three dimensionalcase. More precisely, the result of Lemma A.2 is still valid : the viscous dragexerted by a fluid on a sphere can be computed as the result of a minimizationproblem of the same type as the one given by equation (32). Next, since theproblem is axisymmetric, one can use cylindrical coordinates and perform the sameasymptotical analysis as in the 2D case, in order to obtain that

n(q) = 6πV νR2

q[1 + ε(q)] ,


Numerical evaluation of the drag force. In order to compare the theoreticalestimate given by Proposition A.1 to a numerical evaluation of the force F, wecarry out the following simulations: we begin with solving the Stokes problem in afluid with viscosity ν = 1 and then compute the corresponding force exerted by thefluid on a particle of radius R = 0.1 situated above a plane at a distance q. Thecomputations are performed by using the Finite-Element solver FreeFem++ [9] andthe results are plotted in Figure 7 (dashed line with squares). They agree with the

asymptotic expansion F ∼ 3√

2πV ν(

Rq

) 32

(solid line).



10

100

1000

10000

100000

1e+06

0.001 0.01 0.1

log(

F)

log(q)

Lubrication force - 2D

Lubrication force : asymptotic expression Lubrication force : numerical result

Figure 7. Approximation of the lubrication force

Appendix B. Non-dimensional form of the governing equations (8) in-volving the lubrication force. Let us first consider the 2D case. Eliminatingthe velocity v from (8) and specifying m = ρsπR

2 with ρs the density of the solid,we arrive at

ρsπR2q = −3

√2πν

(R

q

) 32

q + (ρs − ρf )πR2g.

In order to understand the typical values of parameters in this ODE, we shouldpass to non-dimensional variables, which can be introduced as follows:

q = Rq′, t = T t′, g = gcharg′

where the radius R is used as the length scale, the time scale is denoted by T andgchar is a typical value of the external force density, so that gchar ∼ 10m/s2. Wechoose then the time scale T so that the non-dimensional external force becomesof order 1, i.e. T 2

Rρs−ρf

ρsgchar = 1 and obtain the non-dimensional ODE (dropping

the primes) (9).

The asymptotic in the 3D case is is given by n(q) = 6πν R2

q (cf. Remark 5) so

that starting from the dimensional ODE

ρs4

3πR3q = −6πνR2 q

q+ (ρs − ρf )

4

3πR3g (3D).

and using the similar non-dimensionalizations as before we obtain (10).

REFERENCES

[1] J. F. Brady and G. Bossis, Stokesian dynamics, Annual Review of Fluid Mechanics, 20 (1988),111–157.

[2] J. Butcher, “Numerical methods for ordinary differential equations,” John Wiley & Sons Ltd.,Chichester, 2003.

[3] M. Cooley and M. O’Neill, On the slow motion generated in a viscous fluid by the approachof a sphere to a plane wall or stationary sphere., Mathematika, 16 (1969), 37–49.

[4] S. L. Dance and M. R. Maxey, Incorporation of lubrication effects into the force-couplingmethod for particulate two-phase flow, Journal of Computational Physics, 189 (2003), 212–238.



[5] W. R. Dean and M. E. O’Neill, A slow motion of viscous liquid caused by the rotation of asolid sphere, Mathematika, 10 (1963), 13–24.

[6] G. P. Galdi, “An introduction to the mathematical theory of the Navier-Stokes equations. Vol.I, Linearized steady problems,” vol. 38 of Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1994.

[7] D. Gerard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interac-tion, Arch. Ration. Mech. Anal., 195 (2010), 375–407.

[8] R. Glowinski, T.-W. Pan, T. I. Hesla, and D. D. Joseph, A distributed lagrange multi-plier/fictitious domain method for particulate flows, Int. J. Multiphase Flow, 24 (1999),755–794.

[9] F. Hecht, O. Pironneau, A. L. Hyaric, and K. Ohtsuka, Freefem++, ver. 3.7,http://www.freefem.org/ff++, (2009).

[10] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm.Partial Differential Equations, 32 (2007), 1345–1371.

[11] M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interactionproblems, SIAM J. Math. Anal., 40 (2009), 2451–2477.

[12] M. S. Ingber, A. Mammoli, P. Vorobieff, T. McCollam, and A. Graham, Experimental andnumerical analysis of irreversibilities among particles suspended in a couette device, Journalof Rheology, 50 (2006), 99–114.

[13] N. Lecocq, R. Anthore, B. Cichocki, P. Szymczak, and F. Feuillebois, Drag force on a spheremoving towards a corrugated wall, J. Fluid Mech., 513 (2004), 247–264.

[14] N. Lecocq, F. Feuillebois, N. Anthore, R. Anthore, F. Bostel, and C. Petipas, Precise mea-surement of particle-wall hydrodynamic interactions at low reynolds number using laser in-terferometry, Phys. Fluids A, 5 (1993), 3–12.

[15] A. Lefebvre, Numerical simulation of gluey particles, M2AN Math. Model. Numer. Anal., 43(2009), 53–80.

[16] A. Lozinski and M. Romerio, Motion of gas bubbles, considered as massless bodies, affordingdeformations within a prescribed family of shapes, in an incompressible fluid under the actionof gravitation and surface tension, M3AS Math. Mod. Meth. Appl. Sci., 17 (2007), 1445–1478.

[17] A. A. Mammoli, The treatment of lubrication forces in boundary integral equations, RoyalSociety of London Proceedings Series A, 462 (2006), 855–881.

[18] B. Maury, A gluey particle model, ESAIM: Proceedings, 18 (2007), 133–142.[19] M. O’Neill, A slow motion of viscous liquid caused by a slowly moving solid sphere, Mathe-

matika, 11 (1964), 67–74.[20] M. O’Neill and K. Stewartson, On the slow motion of a sphere parallel to a nearby plane

wall, J. Fluid Mech., 27 (1967), 705–724.[21] L. Pasol, M. Chaoui, S. Yahiaoui, and F. Feuillebois, Analytical solutions for a spherical

particle near a wall in axisymmetrical polynomial creeping flows, Phys. Fluids, 17 (2005),1–13.

[22] F. Qi, N. Phan-Tien, and X. J. Fan, Effective moduli of particulate solids: The completed dou-ble layer boundary element method with lubrication approximation, Zeitschrift AngewandteMathematik und Physik, 51 (2000), 92–113.

[23] A. Sierou and J. F. Brady, Accelerated Stokesian Dynamics simulations, Journal of FluidMechanics, 448 (2001), 115–146.

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Received xxxx 20xx; revised xxxx 20xx.

E-mail address: [email protected]




MOTION OF GAS BUBBLES, CONSIDERED AS MASSLESS BODIES, AFFORDING DEFORMATIONS

WITHIN A PRESCRIBED FAMILY OF SHAPES, IN AN INCOMPRESSIBLE FLUID UNDER THE

ACTION OF GRAVITATION AND SURFACE TENSION

par

ALEXEI LOZINSKI AND MICHEL ROMERIO

paru dans

Mathematical Models and Methods in Applied Sciences

August 14, 2007 15:33 WSPC/103-M3AS 00234

Mathematical Models and Methods in Applied SciencesVol. 17, No. 9 (2007) 1445–1478c© World Scientific Publishing Company

MOTION OF GAS BUBBLES, CONSIDERED AS MASSLESS

BODIES, AFFORDING DEFORMATIONS WITHIN A

PRESCRIBED FAMILY OF SHAPES, IN AN INCOMPRESSIBLE

FLUID UNDER THE ACTION OF GRAVITATION AND

SURFACE TENSION

ALEXEI LOZINSKI

Department of Mathematics, University of Houston,4800 Calhoun Rd., Houston TX 77204-3476, USA

MICHEL V. ROMERIO

Institut d’Analyse et de Calcul Scientifique,Ecole Polytechnique Federale de Lausanne,

1015 Lausanne, Switzerland

Received 26 May 2006Revised 19 April 2007

Communicated by N. Bellomo

A model allowing to describe motion and coalescence of gas bubbles in a liquid underthe action of gravitation and surface tension is proposed. The shape of the bubblesis described by a pre-defined family of mappings, for example ellipsoids with a fixedvolume and the effects of the gas motions inside the bubbles are neglected. The motionof a bubble is obtained in a Lagrangian form using the D’Alembert principle of virtualworks. The set of equations is numerically solved with the help of the fictitious domaintechnique in which the Navier–Stokes equations in the domain formed by both fluidand gas are considered. The equations governing the bubbles motion are imposed byintroducing Lagrange multipliers on the bubbles boundaries. Numerical results in 2Dand 3D are presented.

Keywords: Gas bubbles; Lagrangian description; fictitious domain.

1. Introduction

The problem we are dealing with in this paper originates from the production of

aluminum in a reduction cell. This reduction process gives birth to a large amount

of carbon oxide which is released in the so-called channels between the anodes

(Fig. 1). These anodes, which are formed of a porous agglomerate mainly consti-

tuted in carbon, are burnt during the process; the gas produced is stocked into

the alveolate structure of this material. The bubbles are generated in small cavities

located at the bottom of the anodes. Once they reach some critical size they can

1445


August 14, 2007 15:33 WSPC/103-M3AS 00234

1446 A. Lozinski & M. V. Romerio

cathode

anodes

bath

aluminum

anodes

cathodic bar

interface

Fig. 1. The aluminum reduction cell.

detach from the surface of the anode and then move freely on a small layer of fluid.

This peculiar behavior of bubbles called detachment phenomena has been studied

both experimentally and theoretically, see for example Refs. 10 and 12.

As already pointed out in Ref. 6, when, in such a situation, a large set of

bubbles, initially uniformly distributed is growing through coalescence only the

uniform character is rapidly lost in the course of time. This implies that, in this

case, the use of a homogenization method for the description of the gas flux is

certainly inappropriate.

It is the purpose of this work to describe the motion of such a gas bubble flux

in an incompressible fluid under the action of both gravitation and surface tension.

Here the action of the Lorentz forces are disregarded for their effects are small with

respect to those resulting from the other forces.

It should be noted that in order to get a description of this nonhomogeneous

gas flux an accurate description of each bubble and of its generation is certainly not

necessary. Having this in mind we make the assumption that at some initial time all

the bubbles which constitute the gas flux, have the same size and are located under

a portion of a plane, representing the bottom of the anode. We suppose that this

size corresponds to the critical one given by the detachment phenomena, so that

the bubbles can move freely on a thin layer of fluid (around 100 microns thick).

It is also assumed that they are no longer growing during their motions but by a

coalescence procedure.

As already pointed out since our goal is to describe a non-homogeneous gas flux

constituted of a large number of bubbles, we make use of a modeling in which the

bubble shapes are described by a few parameters only. This means that we renounce

an exact description of the bubble shapes but keep what we consider as the main

features of the gas flux corresponding to the situation under consideration. This

implies that we also disregard methods like level set (see Refs. 11 and 14) or the

volume of fluid (see Refs. 3 and 9) which could hardly deal with the large number

of bubbles which are present in the problem mentioned above.


August 14, 2007 15:33 WSPC/103-M3AS 00234

Motion of Gas Bubbles 1447

Taking these remarks into account we derive our modeling under the following

assumptions.

• In the absence of coalescence the gas amount contained in a bubble at some initial

time is preserved at later time.

• Bubbles are massless deformable bodies the shapes of which belong to some

predefined family of surfaces. In particular, during the deformation processes, all

the successive shapes are elements of this family.

It will be shown below that under these assumptions the motion of a bubble is

described by a restricted number of equations, namely

• barycenter equations,

• rotation equations (rotation around the barycenter),

• equations describing the body deformations, compatible with the constraints

imposed by the belonging to the predefined family of surfaces.

These equations are naturally expressed in terms of the barycenter coordinates,

the rotation coordinates, and the parameters describing the predefined family of

surfaces subject to the volume conservation constraint. We stick to the incompress-

ibility assumption on the gas bubbles which is justified by the weak variation of the

hydraulic pressure in our situation.10 Another possible assumption would be to con-

sider that pressure P and volume V satisfy the following relation for an adiabatic

transformation, PV γ = const, where γ is the adiabatic constant for ideal gas.

The fluid, in which the bubbles are moving, is assumed to be governed by

Navier–Stokes equations. Its motion results from the bubbles displacements, which

are produced by the Archimedes drag, from the deformations resulting from the

surface tension and from some velocity conditions on the boundary of the domain

containing the fluid and the bubbles. The mathematical well-possedness of a similar

problem is well studied in the case of rigid bodies moving in a fluid with two

space variables. Existance of gobal weak solutions is obtained in Ref. 13, however

the solution is non-unique after the collision between the bodies.15 To the best

of our knowledge, no similar study exists in the case of gas bubbles immersed in

a fluid.

Since the bubble shapes in our modelling belong to a predefined family of sur-

faces the coalescence phenomena, which takes place when two bubbles are getting

into contact, requires a particular treatment. In order to preserve the constraint on

the surfaces we assume that the coalescence process starts as soon as two bubbles

are sufficiently close. The process then takes places along a few time steps dur-

ing which the smallest bubble empty itself into the largest one. The intermediate

shapes of the two bubbles remain evidently elements of the predefined family of

surfaces. Our simple model of coalescence does not take into account the fact that

bubbles may stay in contact without immediately coalescing. From the observation

performed on industrial cells one knows that large bubbles are generated under the

anodes before the release of the gas in the channels. Since we are mainly interested


August 14, 2007 15:33 WSPC/103-M3AS 00234


in the effect of this gas motion on the fluid velocity we assume that this motion

is not strongly affected by the fact that some of the bubbles did not coalesce but

stayed in contact with a large bubble. In this last situation the volume of gas which

is released remains the same whereas the shape of the domain occupied by the two

bubbles in contact is only slightly modified.

In order to avoid numerical problems related to the time dependence of the

fluid domain we make use of a fictitious domain technique already introduced in

Ref. 7 in a similar context. The fluid is thus considered to occupy the fixed domain

Ω, namely that of both fluid and gas. A Lagrange multiplier is moreover intro-

duced to take care of the velocity requirement at the boundaries between fluid

and gas.

This paper is organized as follows: original governing equations are presented

in Sec. 2. Section 3 is devoted to a description of the kinematics and dynamics of

a deformable body in the frame of analytical mechanics. The general motion equa-

tions corresponding to such a situation are obtained with the help of D’Alembert’s

principle. Our numerical approach (based on the method of the fictitious domain) is

considered in Sec. 4 and our implementation of the coalescence process is explained

in Sec. 5. Finally, numerical results in two- and three-dimensional situations are

reported in Sec. 6.

2. Original Governing Equations

We consider a system consisting of fluid and gas bubbles. We denote by Ω the

domain (not changing in time) occupied by the whole system (fluid and gas) and

by P (t) the time-dependent sub-domain of Ω occupied by the gas bubbles. We

assume that the fluid is Newtonian with density ρ and viscosity η. It is described

by Navier–Stokes equations in the domain Ωl(t) = Ω \ P (t). Together with the

conditions on the boundary Γ = ∂Ω they read:

ρDu

Dt− η∆u + ∇p = ρg, (2.1)

∇ · u = 0, (2.2)

u|Γ = 0, (2.3)

where u and p are respectively the velocity and the pressure in the fluid and g is

some prescribed body force, such as gravity. D/Dt stands for the material deriva-

tive, i.e. D/Dt = ∂/∂t + u · ∇.

The gas in the bubbles is supposed to be inviscid, isothermal and incompressible

with density ρg. Its dynamics is thus governed by the Euler equations

ρgDv

Dt+ ∇P = ρgg, (2.4)

∇ · v = 0, (2.5)

where v and P are respectively the velocity and the pressure in the gas.


August 14, 2007 15:33 WSPC/103-M3AS 00234


In order to close the system we should add some relations on the liquid–gas

interface ∂P (t). The choice of these relations depends on the chemical properties

of both the gas and the liquid. One can consider two extreme cases:

• “Clean” bubbles. We impose the immiscibility

u · n|∂P (t) = v · n|∂P (t) (2.6)

and the force balance

σfn|∂P (t) = (−Pn + αHn)|∂P (t), (2.7)

where n is the unit normal to ∂P (t) pointing towards the fluid domain, α is the

surface tension coefficient, H is the mean curvature of ∂P (t) and σf denotes the

Cauchy stress tensor in the fluid

σf = −pI + η(∇u + ∇uT ). (2.8)

Equation (2.7) thus expresses the equilibrium between the hydrodynamic force in

the liquid, the force produced by the pressure inside the bubble and the surface

tension.

• “Dirty” bubbles, i.e. those containing a surface active film which makes them

behave very much like a rigid body. We impose therefore the no-slip condition in

this case:

u|∂P (t) = v|∂P (t). (2.9)

Requiring in addition the force balance Eq. (2.7) would lead to an over-

determined system. We should choose instead some “relaxed” interface condi-

tions. To this end, we revisit the derivation of the motion equation of an ideal

gas immersed into a fluid.

Take any volume ω inside P (t) and write the momentum equation for the gas

contained in it, namely,

d

dt

∫

ω

ρgvdx =

∫

ω

ρggdx −∫

∂ω∩Int(P (t))

Pnds +

∫

∂ω∩∂P (t)

(σfn − αHn)ds

or∫

ω

(ρg

Dv

Dt+ ∇P

)· wdx =

∫

ω

ρgg · wdx+

∫

∂ω∩∂P (t)

(Pn + σfn − αHn) · wds

for any vector w.

Let ωi, i = 1, . . . , N be a partition of P (t). Taking, in each ωi, a vector wi

in the above formula, we sum up the contributions of all the subdomains. At the

limit where the number of subdomains tends to infinity and where their respective

diameters tend to zero, we get∫

P (t)

(ρg

Dv

Dt+ ∇P

)· wdx =

∫

P (t)

ρgg · wdx+

∫

∂P (t)

(Pn + σfn − αHn) · wds,


August 14, 2007 15:33 WSPC/103-M3AS 00234


which holds for any vector function w in H1(P (t)). If we require furthermore

that ∇ · w vanishes in P (t) and integrate by parts, we obtain

∫

P (t)

ρgDv

Dt· wdx =

∫

P (t)

ρgg · wdx+

∫

∂P (t)

(σfn − αHn) · wds, (2.10)

which is the weak form of the Euler equations with the boundary conditions

suited for the dirty bubbles.

In what follows we will deal only with the second option (2.9)–(2.10).

In view of constructing an approximation for the bubble dynamics, we adopt

from now on a Lagrangian viewpoint on the equations of gas motion (2.10) and

(2.5). We restrict ourselves to the case of a single bubble, the generalization to

the case of multiple bubbles being straightforward if one disregards the coalescence

which will be considered later. Let us index the gas particles by the parameter χ

running over a fixed connected domain X ⊂ R3, i.e. consider the set of mappings

M = x : X → P | P is a connected domain in R3,

x is one-to-one and onto as well as volume preserving, (2.11)

and let x(·, t) : X → P (t) be the mappings such that:

x(χ; 0) = x0(χ),

∂tx(χ, t) = v(x(χ, t), t),(2.12)

where x0 ∈ M : X → P (0) is some arrangement of the initial configuration of gas

particles. We see by (2.5) that x(·, t) ∈ M, the latter set can therefore be called

the configurational space of the gas bubble.

Note that Dv/Dt = x where the dots stand for the derivation in time, hence

(2.10) can be rewritten as

∫

P (t)

ρgx · wdx =

∫

P (t)

ρgg · wdx +

∫

∂P (t)

(σfn − αHn) · wds. (2.13)

Recall that (2.13) holds for any w ∈ H1(P (t)) such that ∇ · w = 0 in P (t).

3. Approximate Representation of Bubble Dynamics by

Deformable Bodies

In this section our purpose is to derive a set of equations which describes the motion

of a bubble approximated by a deformable body whose possible deformations are

described by a finite number of parameters. In order to motivate such an approxi-

mation we start from a rather non-rigorous remark. Following the spirit of Ref. 1,

the configurational space M can be formally regarded as an “infinite dimensional

manifold” with the tangent space TMx at any x ∈ M : X → P consisting of the


August 14, 2007 15:33 WSPC/103-M3AS 00234


divergence free vector fields over P . Hence, Eq. (2.13) can be formally rewritten as

the relation∫

P (t)

ρgx · δxdx =

∫

P (t)

ρgg · δxdx +

∫

∂P (t)

(σfn − αHn) · δxds, (3.1)

which should hold for any element δx of the tangent space TMx(t). Our idea is then

to choose a subset Mn of M, which can be considered as a (finite-dimensional)

submanifold of M, and look for the approximate solution x(·, t) ∈ Mn such that

(3.1) holds for any δx ∈ TMnx(t). The latter manifold can be thought of as a

configurational manifold of a deformable body.

The first step in this direction consists of getting an expression for the position

vector of an arbitrary point of the deformable body. In order to get such an expresion

we are inspired by the classical description of a rigid body in which a frame of

reference is bound to the body. The positions of the different points are then fixed,

with respect to this moving frame, and the position of this last one is described,

from a fixed frame of reference, by a translation and a rotation around the origin,

usually the barycenter, of the moving frame.

In the general case the deformation is described in a frame of reference the origin

of which is the barycenter of the deformable body. A general displacement is then

obtained by a translation and a rotation as in the case of the solid body.

Let us introduce some notations.

3.1. Notations

• G: barycenter of the deformable body; G is independent of the mass density

which is supposed to be homogeneous.

• O, ei, i = 1, 2, 3, an orthonormal frame of reference, with origin O, of the

three-dimensional Euclidean space.

• G, εj , j = 1, 2, 3, a moving orthonormal frame of reference with origin G.

• f(χ; q1, . . . , qn), where χ runs over the domain X of R3 : the family of map-

pings, depending on the parameters q1, . . . , qn. We suppose that the mappings

are volume preserving, i.e.

det

∣∣∣∣Df(χ; q1, . . . , qn)

Dχ

∣∣∣∣ = 1.

Moreover∫

X

f(χ; q1, . . . , qn)dχ = 0. (3.2)

• P (t): domain occupied by the deformable body. P (t) is supposed to belong to

the family f taken in the moving frame G, εj , j = 1, 2, 3 at any time t.

• Ωl(t) = Ω \ P (t): domain occupied by the fluid.


August 14, 2007 15:33 WSPC/103-M3AS 00234


3.2. The configuration manifold

Let f j(χ; q1, . . . , qn) be the components of f(χ; q1, . . . , qn). With the above nota-

tions, any point x ∈ P (t) can be expressed in the form:

x = G +3∑

j=1

f j(χ; q1, . . . , qn)εj (3.3)

for some values of χ ∈ X and q1, . . . , qn.

We introduce some notations.

• Gi = qn+i for i = 1, 2, 3,

• α = qn+4, β = qn+5, γ = qn+6, being the Euler’s angles describing the rotation

from the fixed frame ei, i = 1, 2, 3 to the moving one εi, i = 1, 2, 3,

• q = q1, . . . , qn+6,

• q = q1, . . . , qn.

We can now consider the manifold Mn of dimension n+6 consisting of mappings

χ → x of the form (3.3) and depending on n + 6 parameters q. By construction

Mn is suited for the description of a body which can afford some prescribed types

of deformations, defined by f(·;q), and which is submitted to some external forces.

The last manifold will be called the configuration manifold of the deformable body.

The motion of this body is fully determined by a knowledge of the values the

parameters q take in the course of time.

The elements of the tangent space TMnq at the point q are usually called virtual

displacements. With (3.3) the elements of TMq can be written in the form

n+6∑

k=1

∂x(χ; q)

∂qkδqk, (3.4)

where the δqk, for k = 1, . . . , n + 6, have arbitrary real values. Since the bubbles

are assumed to be incompressible the divergence of (3.4) is vanishing

∇ · ∂x(χ; q)

∂qk= 0, for k = 1, . . . , n + 6. (3.5)

3.3. The velocity field

We now allow x ∈ Mn to evolve in time, i.e. q1, . . . , qn+6 are now regarded as

functions of t. In order to perform the time derivation of (3.3) we introduce the

angular velocity ω(t) corresponding to the rotation of the frame G, εj, j = 1, 2, 3.

In order to do this, we note that

d

dt(εi · εj) = εi · εj + εi · εj = 0, i, j = 1, 2, 3,

hence the matrix Ω : Ωi,j = εi · εj , i, j = 1, 2, 3 is antisymmetric and one can

consider the action of the corresponding operator as given by the vector product


August 14, 2007 15:33 WSPC/103-M3AS 00234


with an axial vector ω(t). In particular,

εi(t) = Ωεi(t) = ω(t) ∧ εi(t). (3.6)

We are now ready to derive the expression for the velocity field. Differentiating

(3.3) with respect to the time variable and taking (3.6) into account we get

x = V + ω ∧3∑

j=1

f j(χ;q(t))εj +

3∑

j=1

n∑

k=1

∂f j(χ;q(t))

∂qkqkεj , (3.7)

where V = G.

3.4. Lagrange equations

Our next task is thus to derive the equations satisfied by the qk, k = 1, . . . , n + 6,

considered as functions of t. We recall that the dynamics of our deformable body

is governed by the variational principle (3.1) restricted to Mn. More precisely we

require that ∫

P (t)

ρgx · δxdx =

∫

P (t)

ρgg · δxdx +

∫

∂P (t)

F · δxds, (3.8)

holds for any δx ∈ Mnx(t). In (3.8)

F = σfn − αHn (3.9)

denotes the resultant of the hydrodynamic forces and of the surface tension.

Remark 3.1. The relation (3.8) can be identified with d’Alembert’s principle

in classical mechanics.8 We consider here F as the exterior force applied to the

deformable body P (t) and postulate the existence of some interior constraint forces

such that their work on virtual displacements is zero.

Let us now recast (3.8) in the more convenient form of the Lagrange equations.

As a first step in this direction we compute the expression of the kinetic energy.

3.4.1. The kinetic energy

By definition the kinetic energy is given by the expression

Ekin =

∫

P (t)

ρgx2dx =

∫

X

ρgx2dχ.

In order to express Ekin in a more usual way we need the following results.

Lemma 3.1. For any V, ω, qk and qk, k = 1, . . . , n we have∫

X

V · ω ∧ f(χ;q)dχ = 0,

∫

X

V ·n∑

k=1

∂f(χ;q)

∂qkqkdx = 0.


August 14, 2007 15:33 WSPC/103-M3AS 00234


Proof. Making use of (3.2) the first relation is trivial for ω and V are both

independent of χ. We write the second one in the formn∑

k=1

qkV · ∂

∂qk

∫

X

f(χ;q)dχ,

which is also vanishing by (3.2).

Let

L =

3∑

i,j=1

∫

X

ρgfi(χ;q(t))εi ∧ f j(χ, t)εjdχ,

where

f j(χ, t) =n∑

k=1

∂f j(χ;q)

∂qkqk.

L is formally the angular momentum of the deformable body one would get when

the basis vectors εj , j = 1, 2, 3, are time independent. Let

Edef =1

2

∫

X

ρg|f (χ)|2dχ.

Edef be the kinetic energy due to the deformation inside the moving frame.

With these definitions we can state the following result.

Proposition 3.1.

Ekin(t) =1

2MV(t)2 +

1

2(ω, Iω) + ω · L + Edef , (3.10)

where M is the mass of the deformable body and I is the tensor of inertia the

coefficients of which are given by

Ii,j =

∫

X

ρg(f2(χ;q)δi,j − f i(χ;q)f j(χ;q))dx.

Proof. From Lemma 3.1 when the expression of x, given by (3.7), is introduced

into Ekin, four terms only are remaining, namely:

(1) 12

∫X ρgV

2(t)dχ = 12V

2(t)∫

X ρgdχ = 12MV2(t).

(2) 12

∫X ρg

(ω(t) ∧∑3

i,j=1 f i(χ;q)εi . ω(t) ∧∑3i,j=1 f i(χ;q)εi

)dχ

= 12

∫X ρg(ω(t) ∧ f(χ;q) . ω(t) ∧ f(χ;q))dχ

= 12ωi(t)ωj(t)

∫X ρg(f

2(χ;q)δi,j − f i(χ;q)f j(χ;q))dχ = 12ω(t) · Iω(t).

(3) 12

∫X

ρg

(ω(t) ∧∑3

i,j=1 f i(χ;q)εi

)· f j(χ;q)εjdχ = ω(t) · L.

(4) 12

∫X

ρg

∑3i=1 f i(χ;q)f i(χ;q)dχ.

We therefore obtain expression (3.10).


August 14, 2007 15:33 WSPC/103-M3AS 00234


3.4.2. Generalized forces

In order to bring the derivation of Lagrange equations to an end we substitute (3.4)

into (3.8). This yields

n+6∑

1

(Pj − Qj)δqj = 0, (3.11)

where, for j = 1, . . . , n + 6,

Qj =

∫

P (t)

ρgg · ∂x(χ; q)

∂qjdx +

∫

∂P (t)

F · ∂x(χ; q)

∂qjds,

are the generalized forces and

Pj =

∫

P (t)

ρx · ∂x(χ; q)

∂qjdx =

∫

X

ρx(χ; q) · ∂x(χ; q)

∂qjdχ

are the generalized inertial forces.

Since the δqi, i = 1, . . . , n + 6, are independent (3.11) leads to

(Pj − Qj) = 0, j = 1, . . . , n + 6. (3.12)

The usual form of Lagrange equations then results from the following lemma.

Lemma 3.2. Considering the kinetic energy Ekin as a function of qj and qj ,

j = 1, . . . , n + 6, we have

Pj =d

dt

∂Ekin

∂qj− ∂Ekin

∂qj, j = 1, . . . , n + 6.

Proof. ∫

X

ρx(χ; q) · ∂x(χ; q)

∂qjdχ

=

∫

X

ρd

dt

(x(χ; q) · ∂x(χ; q)

∂qj

)−∫

X

ρX(χ; q) · d

dt

∂x(χ; q)

∂qjdχ.

It is an easy matter to see that

∂x(χ; q)

∂qα=

∂x(χ; q)

∂qα

and that

d

dt

∂x(χ; q)

∂qα=

∂x(χ; q)

∂qα.

The proof then follows by noticing that the integration domain X is time

independent.

From this lemma and from (3.11) the Lagrange equations read

d

dt

∂Ekin

∂qi− ∂Ekin

∂qi= Qi, for i = 1, 2, . . . , n + 6. (3.13)


August 14, 2007 15:33 WSPC/103-M3AS 00234


3.5. The case of a massless deformable body in a viscous and

incompressible fluid

From now on we make the assumption that the deformable bodies which represent

the bubbles have negligible masses. With this assumption the kinetic energy is

vanishing so that, from (3.13), the motion equations reduce to

Qj =

∫

∂P (t)

F · ∂x(χ; q)

∂qjds = 0, for j = 1, . . . , n + 6.

Taking (3.3) and (3.9) into account, these equations can be rewritten as∫

∂P (t)

(σfn − αHn)ds = 0, (3.14)

∫

∂P (t)

r ∧ (σfn − αHn)ds = 0, (3.15)

∫

∂P (t)

fk · (σfn − αHn)ds = 0, k = 1, . . . , n, (3.16)

where we make use of the notations

r = x − G =

3∑

j=1

f j(χ;q(t))εj (3.17)

and

fk =∂x(χ; q(t))

∂qk=

3∑

j=1

∂f j(χ;q(t))

∂qkεj . (3.18)

4. The Numerical Approach

Before presenting the numerical scheme let us recapitulate the governing equations

and rewrite them in a weak form. We recall that we are looking for the parameters

defining the bubble propagation, i.e. V(t), ω(t) and qk(t), k = 1, . . . , n, as well as

fluid velocity u and pressure p in Ω \ P (t).

The governing equations include the Navier–Stokes equations

ρ

∫

Ω\P (t)

Du

Dt· vdx −

∫

Ω\P (t)

p∇ · vdx + η

∫

Ω\P (t)

∇u : ∇vdx

=

∫

Ω\P (t)

g · vdx, ∀ v ∈ H10 (Ω \ P (t))

∫

Ω\P (t)

u · ∇q = 0, ∀ q ∈ L20(Ω \ P (t))

combined with the boundary conditions

u|Γ = 0, (4.1)

u|∂P (t) = V(t) + ω(t) ∧ r +

n∑

k=1

fk qk(t), (4.2)


August 14, 2007 15:33 WSPC/103-M3AS 00234


where notations (3.17) and (3.18) are used, and Eqs. (3.14)–(3.16) for the

deformable body.

The bubble domain P (t) is given by the parameters G(t), εj(t), j = 1, 2, 3 and

qk(t), k = 1, . . . , n which are to be found from the equations

dG

dt= V(t), (4.3)

dεj

dt= ω(t) ∧ εj , j = 1, 2, 3, (4.4)

dqk

dt= qk(t), k = 1, . . . , n. (4.5)

4.1. Fictitious domain technique and Lagrange multiplier

In this subsection we rewrite the Navier–Stokes equations with two technical

changes which will be useful for a numerical approach of the solutions. For the

numerical reasons mentioned above it is convenient to consider that the domain

P (t) is also filled with a fluid having the same density but another stress tensor

which is chosen in such a way that the conditions imposed on ∂P (t) are fitted. In

other words, the motion of the fluid in Ωl is not affected by the presence of the

other fluid in P (t). This approach is usually called fictitious domain technique (see

for example Ref. 7 for more details).

The second change consists in introducing a Lagrange multiplier, we name λ,

which takes care of relation (4.2) relative to the matching of the velocities on ∂P (t).

The term containing the Lagrange parameter has the following form:∫

∂P (t)

λ · vds.

λ can thus be viewed as a force field which allows to satisfy condition (4.2) on

∂P (t).

In view of the above changes the equations for u(·, t) ∈ H10 (Ω), p(·, t) ∈ L2

0(Ω)

and λ ∈ H− 12 (∂P (t)) take the following form:

ρ

∫

Ω

Du

Dt· vdx −

∫

Ω

p∇ · vdx + η

∫

Ω

∇u : ∇vdx

=

∫

Ω

g · vdx +

∫

∂P (t)

λ · vds, ∀ v ∈ H10 (Ω) (4.6)

∫

Ω

u · ∇q = 0, ∀ q ∈ L20(Ω) (4.7)

∫

∂P (t)

u · µds =

∫

∂P (t)

(V(t) + ω(t) ∧ r +

n∑

k=1

fk qk(t)

)· µds, ∀ µ ∈ H− 1

2 (∂P (t)).

(4.8)


August 14, 2007 15:33 WSPC/103-M3AS 00234


The system should be combined again with the force balance Eqs. (3.14)–(3.16)

and the propagation Eqs. (4.3)–(4.5).

We now have to specify under which condition the presence of the fluid in the

domain P (t) does not affect the motion in Ωl. In order to do that we split, in

Eq. (4.6), the domain of integration Ω into the two sub-domains Ωl and P (t). Then

performing standard integration by part we draw the following condition on the

interface ∂P (t)

λ =

(−pn + η

∂u

∂n

)∣∣∣∣gas

fluid

= (−pn+ η(∇u)n)|gasfluid . (4.9)

Let us take any point O on the interface and consider the Cartesian coordinates

(ξ1, ξ2, ξ3) attached to this point such that the axis ξ1 is in the direction of the

normal vector n and the axes ξ2 and ξ3 are tangential to ∂P (t). Let u1, u2, u3

denote for the moment the components of u with respect to this frame. We have

then (in the same frame):

(∇uT)n =

∂u1

∂ξ1

∂u1

∂ξ2

∂u1

∂ξ3

=

−∂u2

∂ξ2− ∂u3

∂ξ3

∂u1

∂ξ2

∂u1

∂ξ3

and we see that (∇uT)n is continuous for it contains only tangential derivatives

of the continuous field u. This remark allows us to rewrite the equation for λ as

(cf. (2.8))

λ = (σg − σf )n, (4.10)

where σg is the fictitious stress tensor in P (t).

4.2. The gauge reformulation

The numerical treatment of the Navier–Stokes equations is prone to some well-

known difficulties, most notably the need to satisfy the inf-sup condition or to use

some stabilization. One of the techniques to get around these difficulties and to

construct an efficient time marching scheme is the so-called Gauge method.5 This

approach consists in reformulating the Navier–Stokes equations by introducing the

new variables a and φ, which are related to the fluid velocity u through the relation

u = a + ∇φ (4.11)

in such a way that the pressure is eliminated and Laplace-like equations only have

to be solved for a and φ on each time step. In order to do this we substitute (4.11)

into (4.6) and split the terms there, into two groups, namely

ρ

∫

Ω

∂a

∂t· vdx + η

∫

Ω

∇a : ∇vdx =

∫

∂P (t)

λ · vds −∫

Ω

((u · ∇)u) · vdx (4.12)


August 14, 2007 15:33 WSPC/103-M3AS 00234


and

ρ

∫

Ω

∇∂φ

∂t· vdx −

∫

Ω

p∇ · vdx + η

∫

Ω

∇∇φ : ∇vdx = ρ

∫

Ω

g · vdx. (4.13)

We then require that both relations (4.12) and (4.13) be satisfied for all v ∈ H10 (Ω)

for some a(t, ·) ∈ H1(Ω) and φ(t, ·) ∈ H2(Ω). In order to see that (4.13) is solvable

for φ we perform an integration by parts in all the terms of (4.13). In particular,

∫

Ω

∇∇φ : ∇vdx =

3∑

i,j=1

∫

Ω

∂2φ

∂xi∂xj

∂vi

∂xjdx

=

3∑

i,j=1

∫

Γ

ni∂φ

∂xj

∂vi

∂xjds −

3∑

i,j=1

∫

Ω

∂φ

∂xj

∂2vi

∂xi∂xjdx.

If we require furthermore that

∂φ

∂n

∣∣∣∣Γ

= 0 (4.14)

then the boundary term∑3

i,j=1

∫Γ

ni∂φ∂xj

∂vi

∂xjds vanishes. Indeed, we have at any

point on Γ

3∑

i,j=1

ni∂φ

∂xj

∂vi

∂xj=

3∑

i=1

2∑

j=1

ni(τ j · ∇φ)(τ j · ∇vi) = 0,

where τ 1 and τ 2 are two orthonormal vectors tangential to Γ. We have thus∫

Ω

∇∇φ : ∇vdx = −∫

Ω

∇φ · ∇(∇ · v)dx,

so that, in the case where g has a constant value, Eq. (4.13) is equivalent to∫

Ω

p∇ · vdx = −ρ

∫

Ω

∂φ

∂t∇ · vdx − η

∫

Ω

∇φ · ∇(∇ · v)dx + ρ

∫

Ω

(g · x)(∇ · v)dx.

It is now clear that (4.13) is satisfied if we require that

p = −ρ∂φ

∂t+ η∆φ + ρg · x.

Finally, the equation for φ is obtained by substituting (4.11) into (4.7) and

taking (4.14) into account. This yields∫

Ω

∇φ · ∇q = −∫

Ω

a · ∇q. (4.15)

Hence, the Navier–Stokes equations (4.6)–(4.7) are replaced by the relations (4.12)

and (4.15) which should hold for any v ∈ H10 (Ω) and q ∈ H1(Ω).

The boundary conditions for a are obtained from (4.1) and (4.14) in the form

a · τ |Γ = − ∂φ

∂τ

∣∣∣∣Γ

, a · n|Γ = 0.


August 14, 2007 15:33 WSPC/103-M3AS 00234


Substituting (4.11) into (4.8) yields the interface conditions

∫

∂P (t)

a · µds =

∫

∂P (t)

−∇φ + V(t) + ω(t) ∧

3∑

j=1

f jεj

+3∑

j=1

n∑

k=1

∂f j

∂qkqk(t)εj

· µds, ∀ µ ∈ H− 1

2 (∂P (t)).

4.3. Force equations

A direct discretization of the force equations (3.14)–(3.16) would involve some dif-

ficulties for the Cauchy tensor contains the normal derivatives of the velocity on

∂P (t), i.e. precisely where these last ones are discontinuous and consequently not

straightforward to compute. For this reason we reformulate them in a way where

the computation of the Cauchy tensor on the bubbles boundaries can be avoided.

Lemma 4.1. If the relations (4.10) and (4.12) hold, then Eqs. 3.14–3.16 are equiv-

alent to

∫

∂P (t)

λds = ρ

(d

dt

∫

P (t)

adx +d

dt

∫

∂P (t)

φnds − g|P (t)|)

− α

∫

∂P (t)

Hnds,

(4.16)

∫

∂P (t)

r ∧ λds = ρ

(d

dt

∫

P (t)

r ∧ adx +d

dt

∫

∂P (t)

φr ∧ nds + V ∧∫

P (t)

udx

)

− α

∫

∂P (t)

Hr ∧ nds (4.17)

and

∫

∂P (t)

fk · λds = ρ

(d

dt

∫

P (t)

fk · adx +d

dt

∫

∂P (t)

φfk · nds

−∫

P (t)

∂fk∂t

· udx −∫

P (t)

u · ((u · ∇)fk)dx

)

+ η

(∫

∂P (t)

(n · ((u · ∇)fk) + u · ∂

∂nfk

)ds−

∫

P (t)

u · ∆fkdx

)

− α

∫

∂P (t)

Hfk · nds. (4.18)

The field fk is defined by (3.18) and is treated in (4.18) as a function of x and t.


August 14, 2007 15:33 WSPC/103-M3AS 00234


Proof. First, we rewrite the Eqs. (3.14)–(3.16) by using (4.10)∫

∂P (t)

σgnds =

∫

∂P (t)

(λ + αHn)ds, (4.19)

∫

∂P (t)

r ∧ σgnds =

∫

∂P (t)

r ∧ (λ + αHn)ds, (4.20)

∫

∂P (t)

fk · σgnds =

∫

∂P (t)

fk · (λ + αHn)ds. (4.21)

Then the terms containing σg in (4.19)–(4.21) can be rewritten in a form containing

the integrals of a, φ and u only, but not their derivatives. To this end, we use the

momentum equation

ρ

(Du

Dt− g

)= ∇ · σg (4.22)

in the fictitious domain P (t) and apply the divergence and transport theorems.

Thus, (4.19) yields with (4.11)∫

∂P (t)

σgnds =

∫

P (t)

∇ · σgdx

= ρ

∫

P (t)

(Du

Dt− g

)dx

= ρ

(d

dt

∫

P (t)

udx − g|P (t)|)

= ρ

(d

dt

∫

P (t)

adx +d

dt

∫

∂P (t)

φnds − g|P (t)|)

so that (4.19) can be replaced by (4.16).

Analogously for the angular momentum equation (4.20) we have∫

∂P (t)

r ∧ σgnds =

∫

P (t)

r ∧ (∇ · σg)dx = ρ

∫

P (t)

r ∧(

Du

Dt− g

)dx

= ρ

(d

dt

∫

P (t)

r ∧ udx −∫

P (t)

Dr

Dt∧ udx

)

since∫

P (t) rdx = 0. We now note that DrDt = D

Dt (x − G) = u − V and substitute

(4.11) for u to conclude∫

∂P (t)

r ∧ σgnds = ρ

(d

dt

∫

P (t)

r ∧ (a + ∇φ)dx +

∫

P (t)

V ∧ udx

)

= ρ

(d

dt

∫

P (t)

r ∧ adx +d

dt

∫

∂P (t)

φr ∧ nds + V ∧∫

P (t)

udx

)

so that (4.20) can be replaced by (4.17).


August 14, 2007 15:33 WSPC/103-M3AS 00234


Finally, the left-hand side in the remaining force Eq. (4.21) is rewritten as∫

∂P (t)

fk · σgnds =

∫

P (t)

fk · (∇ · σg)dx +

∫

P (t)

σg : ∇fkdx. (4.23)

We now treat the two terms in the last relation separately. Note first that

∇ · fk = 0 (4.24)

by comparing (3.5) and (3.18). For the first term in (4.23) we get now with (4.22),

(3.2) and (4.24)∫

P (t)

fk · (∇ · σg)dx = ρ

∫

P (t)

fk ·(

Du

Dt− g

)dx

= ρ

(d

dt

∫

P (t)

fk · udx −∫

P (t)

DfkDt

· udx

)

= ρ

(d

dt

∫

P (t)

fk · adx +d

dt

∫

∂P (t)

φfk · nds

−∫

P (t)

∂fk∂t

· udx −∫

P (t)

u · ((u · ∇)fk)dx

). (4.25)

For the second term in (4.23) we get

∫

P (t)

σg : ∇fkdx = η

3∑

j=1

∫

P (t)

(∇u + ∇uT )εj · ∇∂f j

∂qkdx (4.26)

since, with (4.24)

∫

P (t)

pI : ∇fkdx =

∫

P (t)

p

3∑

j=1

εj · ∇∂f j

∂qkdx =

∫

P (t)

p∇ · ∂x

∂qkdx = 0.

We now denote the Cartesian coordinates in the frame (G, εj) by ξj and the com-

ponents of the vectors u and n in the same frame by uj and nj . We then write the

right-hand side of (4.26) in an explicit form and integrate it by parts using (4.24)

for a simplification. This yields with (3.18)

3∑

j=1

∫

P (t)

(∇u + ∇uT )εj · ∇∂f j

∂qkdx

=

3∑

i,j=1

∫

P (t)

(∂ui

∂ξj+

∂uj

∂ξi

)∂

∂ξi

∂f j

∂qkdx

=3∑

i,j=1

∫

∂P (t)

(uinj + uj ni)∂

∂ξi

∂f j

∂qkds


August 14, 2007 15:33 WSPC/103-M3AS 00234


−3∑

i,j=1

∫

P (t)

uj∂2

∂ξ2i

∂f j

∂qkdx

=

∫

∂P (t)

(n · ((u · ∇)fk) + u · ∂

∂nfk

)ds

−3∑

j=1

∫

P (t)

u · ∆fkdx. (4.27)

Putting (4.23), (4.25) and (4.27) together we see that (4.21) can be replaced

by (4.18).

4.4. Time marching scheme

Let us now present our time marching scheme. Assuming that all the unknowns

an−1, φn−1, Gn−1, εn−1j , qn−1

k , Vn−1, ωn−1, qn−1k (j = 1, 2, 3 and k = 1, . . . , n) are

already computed at time tn−1 = (n − 1)∆t we proceed with their calculation at

time tn = n∆t through the four following stages:

(1) Propagation of bubbles

Gn = Gn−1 + Vn−1∆t

εnj = εn−1

j + ωn−1 ∧ εn−1j ∆t, j = 1, 2, 3

qnk = qn−1

k + qn−1k ∆t, k = 1, . . . , n.

The bubble domain Pn at time tn has now been determined.

(2) Updating the variable φ∫

Ω

∇φn · ∇q = −∫

Ω

an−1 · ∇q, ∀ q ∈ L20(Ω). (4.28)

(3) Updating the vector field an, the Lagrange multiplier λn and the velocity

parameters of the bubbles Vn, ωn and qnk :

ρ

∆t

∫

Ω

an · vdx + η

∫

Ω

∇an : ∇vdx −∫

∂Pn

λn · vds

=ρ

∆t

∫

Ω

an−1 · vdx − ρ

∫

Ω

(un−1 · ∇)un−1 · vdx, ∀ v ∈ H10 (Ω), (4.29)

an · τ |Γ = − ∂φn

∂τ

∣∣∣∣Γ

, an · n|Γ = 0,

∫

∂Pn

an·µds =

∫

∂Pn

(−∇φn + Vn + ωn ∧ r +

n∑

k=1

fk qnk

)·µds, ∀ µ ∈ H− 1

2 (∂Pn)

and force Eqs (4.16)–(4.18).


August 14, 2007 15:33 WSPC/103-M3AS 00234


(4) Calculating the velocity

un = an + ∇φn.

For the discretization in space, both an and φn are approximated by bilin-

ear finite elements on a Cartesian grid of size Nx × Ny on Ω. This allows us

to use the Fast Fourier Transform to solve the linear systems resulting from

Eq. (4.28) for φn and Eq. (4.29) for an (with a given λn). The boundary ∂Pn

is represented by Nλ segments and the Lagrange multiplier λn is approximated

by linear finite elements on these segments. The linear system for λn, Vn, ωn

and qnk resulting from the equations in the stage 3 of the above algorithm is

solved by the GMRES method. Note that the linear system (4.29) for an should

be solved on each iteration of GMRES, which can make the overall resolution

rather expensive. To rectify this problem we have used in situations with an

important number of bubbles a Schwartz-type domain decomposition method.

Namely, rather than solving the equations of the stage 3 directly, we can find

iteratively the approximations for the Lagrange multiplier λn and velocity param-

eters Vn, ωn and qnk separately on each bubble restricting the domain to a par-

allelipiped (or a rectangle in 2D) containing this bubble. All the Lagrange mul-

tipliers and velocity parameters are then matched in the iterative process, in

which the linear system (4.29) for an is solved on the whole domain Ω on each

iteration.

5. The Coalescence Process

Coalescence is an instantaneous transition in which two bubbles, or more, in contact

along a common boundary transform themselves into a single one. We choose to

model this transition in considering that it takes place during a finite but small

interval of time, which is suited for a numerical approach. More precisely we assume

that

• When two bubbles are “sufficiently close” the coalescence process is starting.

• In a short time (numerically a small number of time steps) the small bubble

empty itself into the large one.

Let V and respectively W be the volume of the large and respectively small bubble

before coalescence. When they have the same size, one of the bubble, arbitrarily

chosen, plays the role of the large one.

During the coalescence process their volumes are represented by

τV and µW,

where the parameters τ and µ are related by the following mass conservation

equation:

τV + µW = V + W.


August 14, 2007 15:33 WSPC/103-M3AS 00234


In this relation µ varies between 0 and 1 whereas τ varies between 1 and 1 + W/V .

In other words,

µ = 1 + (1 − τ)V/W

and, in a few time steps, the volume of the larger bubble passes from V to V + W

whereas the smallest one disappears.

In order to obtain a criteria telling us when two bubbles are considered as

sufficiently close, we derive a method allowing to compute the distance separating

them.

5.1. Distance between two bubbles

Let P1 and P2 be two domains representing the bubbles and d(P1, P2) be the

distance between them. Our algorithm is an iterative process. More precisely, we

will generate two sequences of points xk and yk on the boundaries of P1 and P2,

respectively, which become closer and closer to each other so that |xk − yk| is

expected to tend to d(P1, P2) as k → ∞.

Let us choose some points G1 ∈ Int(P1) and G2 ∈ Int(P2). We will assume that

for any given points z = G1 and y we are able to find the intersection of ∂P1 with

the line passing through G1 and z and to choose the point out of this intersection

that is the closest to y. We will denote the point described above by Pr∂P1(z|y).

A similar construction gives us Pr∂P2(z|y) for any z, y = G2.

We set x0 = Pr∂P1(G2,G2) and y0 = Pr∂P2(G1,G1) as the initial approxi-

mations. At the kth iteration, having two points xk ∈ ∂P1 and yk ∈ ∂P2, we try

to find a new point xk+1 ∈ ∂P1, which is closer to yk than xk. To this end, we

construct first the plane L passing through xk and tangential to ∂P1. We denote

then z(0)k the projection of yk on the plane L and set x

(0)k = Pr∂P1(z

(0)k ,yk). If the

distance |x(0)k − yk| is less than |xk − yk|, we accept x

(0)k as xk+1. Otherwise, we

start another inner loop

z(l)k =

xk + z(l−1)k

2, x

(l)k = Pr∂P1(z

(0)k ,yk), l = 1, 2, . . .

until |x(l)k − yk| is less than |xk − yk| and then accept x

(l)k as xk+1. To construct

yk+1 we proceed in the similar way, trying to find a point yk+1 ∈ ∂P2, which is

closer to xk than yk.

6. Numerical Results

6.1. The two-dimensional problem for rigid ellipses

Let us present some results in 2D. The modeling and the numerical approach remain

the same as in the 3D case. The only difference is that the barycenter is now given by

a 2-vector G = (G1, G2) and the rotation is described by a single scalar parameter,

which can be taken as the angle θ between the axes e1 and ε1. The angular velocity


August 14, 2007 15:33 WSPC/103-M3AS 00234


is reduced to the scalar ω(t) = θ(t) as well as the corresponding force Eq. (4.17).

We take the following physical parameters: the density of the fluid ρ = 1000 kg/m3,

the viscosity η = 0.01 Pa · s, the gravitation force g = −10ey(m/s2). The time step

will be fixed to ∆t = 10−3 s in all our simulations.

Our first example is the motion of a massless rigid ellipse in a fluid. Hence, the

system of force equations reduces to (4.16)–(4.17).

We begin with the results of simulations of a single rigid circular bubble of

radius R = 1mm rising in a 2 cm×2 cm container. We make use of a Cartesian grid

of N × N points to represent a and φ and a regular grid of Nλ points on ∂P (t) to

represent λ. Figure 2 shows the initial configuration of the bubble, its configuration

and the fluid velocity at t = 0.1 s and a zoom of the bubble and the fixed grid for

N = 128.

0 0.005 0.01 0.015 0.020

0.005

0.01

0.015

0.02

0 0.005 0.01 0.015 0.020

0.005

0.01

0.015

0.02

0.009 0.01 0.011

0.015

Fig. 2. One circular rigid body. Initial configuration, configuration and the fluid velocity att = 0.1s and a zoom with the grid.


August 14, 2007 15:33 WSPC/103-M3AS 00234


We now study the influence of the discretization parameters, i.e. N and Nλ. We

choose Nλ as a function of N for Nλ = [kλR/h] where kλ is some constant, h is the

grid step corresponding to N and the brackets denote the integer part. Figures 3

and 4 present the evolution of the y-coordinate of the bubble barycenter with time.

On Fig. 3, we fix N = 128 and vary kλ. The ratio of the size h of the fixed grid and

the size hλ of the moving grid is also indicated. We see that the convergence (in an

eyeball norm) is achieved for kλ = 4, i.e. when hλ is about 1.5 times greater than

h. This rule for choosing Nλ will be adopted in all our subsequent simulations. On

Fig. 4, we fix kλ = 4 and vary N . The convergence can also be observed when N

gets sufficiently high.

t (sec)

y(m

m)

0 0.025 0.05 0.075 0.10

5

10

k = 2k = 4k = 6k = 8

λ

λλ

λ

Fig. 3. One circular rigid body rising in a container; dependence on the size of the grid on ∂P (t):solid line – kλ = 2, hλ/h = 3.06; dashed line – kλ = 4, hλ/h = 1.54; dash-dotted line – kλ = 6,hλ/h = 1.02; dotted line – kλ = 8, hλ = 0.77/h. The last three lines are practically superimposed.

t (sec)

y(m

m)

0.025 0.05 0.075 0.10

5

10

N = 32N = 64N = 128N = 256

Fig. 4. One circular rigid body rising in a container; dependence on the size of the grid on Ω.


August 14, 2007 15:33 WSPC/103-M3AS 00234


To conclude this section, we present two more complicated simulations. The

first one deals with three circular bubbles of radii from 0.8 to 1.5mm evolving in a

1 cm × 3 cm container. The Cartesian grid of 64 × 200 points was used to represent

a and φ. The results are shown in Fig. 5 as snapshots of bubbles at some time

moments increasing from left to right and from top to bottom.

Our last example in this section involves a much greater amount of bubbles and

illustrates our treatment of the coalescence process. We start with a random config-

uration of 200 elliptic bubbles with sizes from 0.5 to 3 mm placed in a 5 cm× 10 cm

container. The results are shown in Fig. 6 as snapshots of bubbles at increasing

time moments. The bubbles rise so that the bigger ones swallow or sweap away

the smaller ones. The Cartesian Grid consisted of 256 × 512 points. The domain

decomposition technique described at the end of Sec. 4 was used in this case. The

simulation took 501 seconds of CPU time for 300 time steps on a Pentium M760

laptop computer.

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

0 0.010

0.01

0.02

0.03

Fig. 5. Three circular rigid bodies. Snapshots at times 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 · 10−2 s.


August 14, 2007 15:33 WSPC/103-M3AS 00234


0 0.050

0.05

0.1

0 0.050

0.05

0.1

0 0.050

0.05

0.1

0 0.050

0.05

0.1

0 0.050

0.05

0.1

0 0.050

0.05

0.1

Fig. 6. Initial configuration of 200 elliptic bubbles and their evolution. Snapshots at timemoments 0, 6, 12, 18, 24, 30 · 10−2 s.

6.2. The two-dimensional problem for a family of deformable

elliptic surfaces

Let us now allow the ellipses from the preceding section to change their shapes in

the course of time, but preserving their area. It means that we introduce a one

parameter family of area preserving mappings f(·, a) : χ ∈ X → x ∈ Pa, X being

a circle of area V centered at the origin and Pa an ellipse with axes a√

V/π and

1/a√

V/π:

x1 = aχ1,

x2 =1

aχ2.

Obviously, the force equations (4.16)–(4.17) remain the same as for the rigid ellipses.

We should however add one more equation corresponding to (4.18) for q1 = a. At


August 14, 2007 15:33 WSPC/103-M3AS 00234


the point f(χ, a) = x we have clearly

∂f

∂a=

χ1

− 1

a2χ2

=

1

a

(x1

−x2

).

Hence

fa =

2∑

j=1

∂f j

∂aεj =

1

ar∗

with

r∗ = (x − G, ε1)ε1 − (x − G, ε2)ε2

Equation (3.7) for the velocity of the deformable body can be now explicitly

rewritten as

x = V + ω ∧ r +a

ar∗ (6.1)

and the force Eq. (4.18) reduces to

∫

∂P (t)

r∗ · λds = ρ

(d

dt

∫

P (t)

r∗ · adx +d

dt

∫

∂P (t)

φr∗ · nds

+

∫

P (t)

(a

ar∗ + V∗ − 2ω ∧ r∗

)· udx

−∫

P (t)

((u · ε1)2 − (u · ε2)2

)dx

)

+ 4V ηa

a− α

∫

∂P (t)

Hr∗ · nds (6.2)

with

V∗ = (V, ε1)ε1 − (V, ε2)ε2.

We made use of the relations

∂r∗

∂t= −V∗ + 2ω ∧ r∗,

∫

P (t)

u · ((u · ∇)r∗)dx =

∫

P (t)

((u · ε1)

2 − (u · ε2)2)dx

and∫

∂P (t)

(n · ((u · ∇)r∗) + u · ∂

∂nr∗)

ds = 4Va

a.


August 14, 2007 15:33 WSPC/103-M3AS 00234


0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

0 0.005 0.010

0.01

0.02

0.03

Fig. 7. Three elliptical deformable bubbles. Snapshots at times 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 ·10−2 s.

The last computation can be easily checked by using the divergence theorem and

by extending u|∂P (t) into P (t) by u = x where x is given by (6.1).

We now present some results for this family of surfaces. We begin with a simu-

lation starting from the configuration of three circular bubbles used in Fig. 5. The

results in the case of deformable elliptic bubbles are shown in Fig. 7. The surface

tension coefficient was chosen as α = 0.05. Comparing Figs. 5 and 7, we observe very

similar bubble evolution, however a coalescence occurs in the latter case. A more

detailed representation of a coalescence of two bubbles is given in Fig. 8, where two

elliptic bubbles are initially chosen. The surface tension coefficient is again equal

to 0.05.

Figure 9 demonstrates the influence of the surface tension. We follow here

the evolution of one elliptic bubble under three different values of the surface

tension coefficient α. The less α is, the flatter is the final form of the rising

bubble.


August 14, 2007 15:33 WSPC/103-M3AS 00234


0 0.005 0.01 0.0150

0.005

0.01

0.015

0 0.005 0.01 0.0150

0.005

0.01

0.015

0 0.005 0.01 0.0150

0.005

0.01

0.015

0 0.005 0.01 0.0150

0.005

0.01

0.015

0 0.005 0.01 0.0150

0.005

0.01

0.015

0 0.005 0.01 0.0150

0.005

0.01

0.015

Fig. 8. Two elliptical deformable bubbles. Snapshots at times 0, 2, 3.8, 4, 4.2, 4.6 · 10−2 s.

6.3. The two-dimensional problem for a family of surfaces given

by polynomials of degree 4

Let us now allow for one more degree of freedom, which we add to the ellipses used

in the preceding section. Our family of mappings f now depends on two parameters

q1 = a and q2 = c, i.e. f(·, a, c) : χ ∈ X → x ∈ Pa,c, X being again a circle of


August 14, 2007 15:33 WSPC/103-M3AS 00234


area V centered at the origin:

x1 = aχ1,

x2 =1

aχ2 − cχ2

1 +cV

4π.

(6.3)

Obviously, the force equations (4.16)–(4.17) remain the same as for the rigid

ellipses. It is easy to check that the force Eq. (6.2) corresponding to q1 = a remains

also valid. We should add the force equation corresponding to q2 = c. To do this,

we compute

∂f

∂c=

0

−χ21 +

V

4π

=

0

−x21

a2+

V

4π

.

Hence

fc =2∑

j=1

∂f j

∂cεj =

(− 1

a2(r, ε1)

2 +V

4π

)ε2.

The equation for the velocity of the deformable body now reads

x = V + ω ∧ r +a

ar∗ + c

(− 1

a2(r · ε1)2 +

V

4π

)ε2 (6.4)

and the force Eq. (4.18) reduces to∫

∂P (t)

fc · λds = ρ

(d

dt

∫

P (t)

fc · adx +d

dt

∫

∂P (t)

φfc · nds −∫

P (t)

∂fc∂t

· udx

+2

a2

∫

P (t)

(u · ε1)(u · ε2)(r · ε1)dx

)

+2η

a2

∫

P (t)

u · ε2dx − 2V η

a2V · ε2 +

V 2η

πa3c − α

∫

∂P (t)

Hfc · nds,

0 0.005 0.010

0.01

0.02

0 0.005 0.010

0.01

0.02

0 0.005 0.010

0.01

0.02

0 0.005 0.010

0.01

0.02

Fig. 9. One elliptical deformable bubble: influence of the surface tension. Initial configurationand final configurations at t = 0.1 s and for α = 0.5, 0.05 and 0.0005.


August 14, 2007 15:33 WSPC/103-M3AS 00234


0 0.005 0.010

0.01

0.02

0 0.005 0.010

0.01

0.02 0.02

0 0.005 0.010

0.01

0 0.005 0.010

0.01

0.02

Fig. 10. A bubble described by mapping (6.3). Initial configuration and final configurations att = 0.10 s and for α = 0.5, 0.05 and 0.0005.

where we should substitute

∂fc∂t

= ω

(1

a2(r, ε1)

2 − V

4π

)ε1 +

2(r, ε1)

a2

(a

a(r, ε1) + (V, ε1) − ω(r, ε2)

)ε2.

We made use of the relations∫

P (t)

u · ((u · ∇)fc)dx = − 2

a2

∫

P (t)

(u · ε1)(u · ε2)(r · ε1)dx

and∫

∂P (t)

(n · ((u · ∇)fc) + u · ∂

∂nfc

)ds = −2V

a2V · ε2 +

V 2

πa3c.

Figure 10 shows some results obtained for the family of mappings described

above. We follow here the evolution of a bubble under three different values of the

surface tension coefficient α. Note that the elongated form of the bubble in the

experiment with the smallest value of the surface tension α = 5 · 10−4 allows it to

rise faster (cf. Fig. 9 as well).

6.4. The two-dimensional problem for rigid ellipsoidal surfaces

For the three-dimensional setting, we restrict ourselves in this paper to the gen-

eralization of the rigid body family considered in Sec. 6.1, i.e. we represent the

three-dimensional bubbles by rigid ellipsoids.

We begin with the results of the simulations of a single spherical bubble of radius

R = 1mm rising in a 1 cm×1 cm×1 cm container. In order to be able to compare our

results with Stokes law for the terminal settling velocity of a sphere in unbounded

viscous liquid Vterm = 2gρR2/9η we reduced the fluid velocity to ρ = 100 kg/m3

in

this simulation. We have then Vterm ≈ 22mm/s and the Reynolds number based

on the sphere radius is 0.22 so that Stokes law should be applicable. The results

are shown in Fig. 11. We present here two levels of discretization: N = 32 and


August 14, 2007 15:33 WSPC/103-M3AS 00234


t

z(m

m)

0.05 0.1 0.15 0.20

5

10

n = 32n = 64

t

v z(m

m/s

)

0.05 0.1 0.15 0.2

10

15

20

n = 32n = 64

Fig. 11. Rise of a spherical rigid bubble in 3D. Evolution in time of the z-coordinate of thebarycenter and z-component of its velocity.

N = 64, where N stands for the N × N × N Cartesian grid in Ω. The moving grid

on the sphere with 66 vertices was used in both cases. We observe that the settling

value of the rising velocity becomes closer to that predicted by Stokes law Vterm

as the mesh is refined. In both simulations, after reaching the settling velocity, the

bubble is impeded when it comes too close to the upper wall of the container. We

note as well the presence of high oscillations in time in the velocity. This is clearly a

numerical artifact which becomes less pronounced when the grids are refined. It can

be mainly attributed to the imperfections of the numerical interpolation between

the moving grid and the fixed one.

The snapshots of another simulation in 3D are shown in Fig. 12. We present here

two ellipsoidal bubbles with axes ranging from 1 to 2 mm rising in a 1 cm × 1 cm ×1.5cm container, the fluid density being reset to ρ = 1000 kg/m3. The Cartesian

grid of 64 × 64 × 64 points was used in this simulation and it took 420 seconds of

CPU time for 90 time steps.

7. Conclusion

The numerical results exhibited in the last section show that the model which has

been derived in this paper yields on the one hand a description of the motion of a

large number of bubbles in an incompressible fluid under the action of gravitation

and on the other hand the velocity field within the fluid. It moreover accounts for

possible coalescence between bubbles which are sufficiently close. The calculations

are relatively fast: in 2D setting one can simulate up to about 2000 bubbles on

a laptop computer taking about 20 seconds per time step. In 3D, we have exper-

imented with 12 bubbles which takes about 40 secs per times step. One way to

decrease the cost could be the use of domain decomposition techniques such as the

Fat Boundary Method Ref. 2 for the case of rigid bodies moving in a fluid. In this

method, a moving mesh is introduced not only on the bodies surfaces but also in


August 14, 2007 15:33 WSPC/103-M3AS 00234


0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

z

0

0.005

0.01

x

0

0.0025

0.005

0.0075

0.01

y

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

z

0

0.005

0.01

x

0

0.0025

0.005

0.0075

0.01

y

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

z

0

0.005

0.01

x

0

0.0025

0.005

0.0075

0.01

y

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

z

0

0.005

0.01

x

0

0.0025

0.005

0.0075

0.01

y

Fig. 12. Two rigid bubbles in 3D. Snapshots at times 0, 3, 6, 9 · 10−2 s.

some layers surrounding them. This permits to better represent the velocity around

the bodies and thus to coarsen the fixed mesh on Ω.

The use of a predefined family of surfaces yields bubble shapes which are regular

functions. It should be noted that with a suited family, i.e. with a sufficiently large

number of parameters, it is in principle possible to reach a given degree of adequacy

with respect to the exact shape.

Some difficulties still have to be overcome. The first one concerns the conditions

imposed on the matching of the velocity field on the boundary between the fluid


August 14, 2007 15:33 WSPC/103-M3AS 00234


and the bubble. This paper sticks to the dirty bubble case. For numerical reasons

the clean bubble case seems more difficult to be handled than this first one.

The second problem is related to the simulation of the fluid layer on which

the bubbles are freely moving after detachment. In our first attempt this physical

phenomena has been described by the introduction of a short range repulsive force

in the vicinity of the anode. This problem is also connected with the choice of the

family of predefined surfaces used. This choice must in fact keep enough freedom

to allow a displacement of the bubbles under the anode in which this last one

essentially sticks to the layer.

As mentioned in the Introduction, our interest in the bubble motions arises from

the fact that it enters the more general problem of the stability of the aluminum

electrolysis cell, i.e. the stability of the motion of the electrolytic bath and of the

aluminum already produced. The present work is thus intended to obtain a correct

calculation of the velocity field which enters the equations used to derive a stability

criterion, as in Ref. 4. Our approach consists in using 2D bubble simulations to

calculate the velocity induced by the bubbles on the bath in the directions tangential

to the anode inferior surface. This velocity is then imposed as boundary conditions

to calculate the fluid mothion in the bath and melted aluminum.

Acknowledgments

This research was performed in the framework of scientific collaboration between

the Chair of Analysis and Numerical Simulations, IACS-EPFL, and the company

ALCAN. The authors thank Jacques Antille, Michel Flueck, Olivier Martin, Marco

Picasso and Jacques Rappaz for numerous helpful discussions, insightful remarks

and interest in their work. The first author was supported by CTI Project 6437.1

IWS-IW. The second author would like to acknowledge very fruitful discussions on

coalescence and related problems with J. Hammerberg during his visit to the Los

Alamos National Laboratory.

References

1. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Math-ematical Sciences, Vol. 125 (Springer-Verlag, 1998).

2. S. Bertoluzza, M. Ismail and B. Maury, The fat boundary method: semi-discretescheme and some numerical experiments, in Domain Decomposition Methods in Sci-ence and Engineering, Lect. Notes Comput. Sci. Eng., Vol. 40 (Springer, 2005),pp. 513–520.

3. A. Caboussat, Numerical simulation of two-phase free surface flows, Arch. Comput.Meth. Eng. 12 (2005) 165–210.

4. J. Descloux, M. Flueck and M. V. Romerio, A modelling of the stability of aluminiumelectrolysis cells, in Nonlinear Partial Differential Equations and Their Applications.College de France Seminar, Vol. XIII (Paris, 1994/1996), Pitman Res. Notes Math.Ser., Vol. 391 (Longman, 1998), pp. 117–133.

5. E. Weinan and J.-G. Liu, Gauge method for viscous incompressible flows, Commun.Math. Sci. 1 (2003) 317–332.


August 14, 2007 15:33 WSPC/103-M3AS 00234


6. S. Poncsak et al., Mathematical modelling of the collective behaviour of gas bubblesunder the anode, CIM Light Metals (1999) 139–146.

7. R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph and J. Periaux, A fictitious domainapproach to the direct numerical simulation of incompressible viscous flow past movingrigid bodies: Application to particulate flow, J. Comput. Phys. 169 (2001) 363–426.

8. H. Goldstein, Classical Mechanics, 2nd edn. (Addison-Wesley, 1980).9. C. W. Hirt and B. D. Nichols, Volume of fluid (vof) method for the dynamics of free

boundaries, J. Comput. Phys. 39 (1981) 201–225.10. P. Kloucek and M. V. Romerio, The detachment of bubbles under a porous rigid

surface during aluminum electrolysis, Math. Mod. Meth. Appl. Sci. 12 (2002) 1617–1652.

11. S. Osher and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces(Springer-Verlag, 2003).

12. S. Poncsak, L. I. Kiss and R. T. Bui, Mathematical modelling of the growth of gasbubbles under the anode in the aluminum electrolysis cells, TMS Light Metals (1999)57–72.

13. J. A. San Martın, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch.Rational Mech. Anal. 161 (2002) 113–147.

14. S. A. Sethian, Level Set Methods, Evolving Interfaces in Geometry, Fluid Mechanics,Computer Vision, and Material Science (Cambridge Univ. Press, 1996).

15. V. N. Starovoıtov, On the nonuniqueness of the solution of the problem of the motionof a rigid body in a viscous incompressible fluid, Zap. Nauchn. Sem. S.-Peterburg.Otdel. Mat. Inst. Steklov. (POMI) 306 (2003) 199–209; 231–232.


DUALITY-BASED ASYMPTOTIC-PRESERVING METHOD FOR HIGHLY ANISOTROPIC DIFFUSION

EQUATIONS

par

PIERRE DEGOND, FABRICE DELUZET, ALEXEI LOZINSKI, JACEK NARSKI AND CLAUDIA

NEGULESCU

soumis à

Communications in Mathematical Sciences

arX

iv:1

008.

3405

v1 [

mat

h.N

A]

19

Aug

201

0

DUALITY-BASED ASYMPTOTIC-PRESERVING METHOD FOR HIGHLYANISOTROPIC DIFFUSION EQUATIONS

PIERRE DEGOND†‡ , FABRICE DELUZET†‡ , ALEXEI LOZINSKI† , JACEK NARSKI† , AND CLAUDIA

NEGULESCU§

Abstract. The present paper introduces an efficient and accurate numerical scheme for the solution of a highlyanisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based onan asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of theanisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedureis the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preservingmethod generalizes the method investigated in a previous paper [11] to the case of an arbitrary anisotropy direction field.

1. IntroductionAnisotropic problems are common in mathematical modeling of physical problems. They occur in

various fields of applications such as flows in porous media [3, 20], semiconductor modeling [29], quasi-neutral plasma simulations [10], image processing [37, 38], atmospheric or oceanic flows [36] and so on,the list being not exhaustive. The initial motivation for this work is closely related to magnetized plasmasimulations such as atmospheric plasma [24, 26], internal fusion plasma [4, 12] or plasma thrusters [1].In this context, the media is structured by the magnetic field, which may be strong in some regions andweak in others. Indeed, the gyration of the charged particles around magnetic field lines dominates themotion in the plane perpendicular to magnetic field. This explains the large number of collisions in theperpendicular plane while the motion along the field lines is rather undisturbed. As a consequence themobility of particles in different directions differs by many orders of magnitude. This ratio can be ashuge as 1010. On the other hand, when the magnetic field is weak the anisotropy is much smaller. Asthe regions with weak and strong magnetic field can coexist in the same computational domain, oneneeds a numerical scheme which gives accurate results for a large range of anisotropy strengths. Therelevant boundary conditions in many fields of application are periodic (for instance in simulations ofthe tokamak plasmas on a torus) or Neumann boundary conditions (atmospheric plasma for example[5]). For these reasons we propose a strongly anisotropic model problem for wich we wish to introducean efficient and accurate numerical scheme. This model problem reads

−∇·A∇φε = f in Ω,

n ·A∇φε =0 on ∂ΩN ,

φε =0 on ∂ΩD ,

(1.1)

where Ω⊂R2 or Ω⊂R3 is a bounded domain with boundary ∂Ω=∂ΩD ∪∂ΩN and outward normal n.The direction of the anisotropy is defined by a vector field B, where we suppose divB=0 and B 6=0.The direction of B is given by a vector field b=B/|B|. The anisotropy matrix is then defined as

A=1

εA‖b⊗b+(Id−b⊗b)A⊥(Id−b⊗b) (1.2)

and ∂ΩD = x∈∂Ω | b(x) ·n=0. The scalar field A‖> 0 and the symmetric positive definite matrixfield A⊥ are of order one while the parameter 0<ε< 1 can be very small, provoking thus the highanisotropy of the problem. This work extends the results of [11], where the special case of a vector fieldb, aligned with the z-axis, was studied. An extension of this approach is proposed in [6] to handle more

†Universite de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathematiques de Toulouse, F-31062 Toulouse, France‡CNRS, Institut de Mathematiques de Toulouse UMR 5219, F-31062 Toulouse, France§CMI/LATP, Universite de Provence, 39 rue Frederic Joliot-Curie 13453 Marseille cedex 13

1


realistic anisotropy topologies. It relies on the introduction of a curvilinear coordinate system with onecoordinate aligned with the anisotropy direction. Adapted coordinates are widely used in the frameworkof plasma simulation (see for instance [4, 14, 31]), coordinate systems being either developped to fitparticular magnetic field geometry or plasma equilibrium (Euler potentials [35], toroidal and poloidal[18, 23], quasiballooning [15], Hamada [19] and Boozer [7] coordinates). Note that the study of certainplasma regions in a tokamak have motivated the use of non-orthogonal coordinates systems [21]. Incontrast with all these methods, we propose here a numerical scheme that uses coordinates and meshesindependent of the anisotropy direction, like in [33]. This feature offers the capability to easily treat timeevolving anisotropy directions. This is very important in the context of tokamak plasma simulation,the anisotropy being driven by the magnetic field which is time dependent.

One of the difficulties associated with the numerical solution of problem (1.1) lies in the fact thatthis problem becomes very ill-conditioned for small 0<ε≪1. Indeed, replacing ε by zero yields anill-posed problem as it has an infinite number of solutions (any function constant along the b fieldsolves the problem with ε=0). In the discrete case the problem translates into a linear system whichis ill-conditioned, as it mixes the terms of different orders of magnitude for ε≪1. As a consequencethe numerical algorithm for solving this linear system gives unacceptable errors (in the case of directsolvers) or fails to converge in a reasonable time (in the case of iterative methods).

This difficulty arises when the boundary conditions supplied to the dominant O(1/ε) operatorlead to an ill-posed problem. This is the case for Neumann boundary conditions imposed on the partof the boundary with b ·n 6=0 as well as for periodic boundary conditions. If instead, the boundaryconditions are such that the dominant operator gives a well-posed problem, the numerical difficultyvanishes. One can resort to standard methods, as the dominant operator is sufficient to determine thelimit solution. This is the case for Dirichlet and Robin boundary conditions. The problem addressedin this paper arises therefore only with specific boundary conditions. It has however a considerableimpact in numerous physical problems concerning plasmas, geophysical flows, plates and shells as anexample. In this paper, we will focus on Neumann boundary condition since they represent a largerrange of physical applications. The periodic boundary conditions can be addressed in a very similarway.

Numerical methods for anisotropic problems have been extensively studied in the literature. Dis-tinct methods have been developed. Domain decomposition techniques using multiple coarse gridcorrections are adapted to the anisotropic equations in [17, 27]. Multigrid methods have been studiedin [16, 32]. For anisotropy aligned with one or two directions, point or plane smoothers are shown tobe very efficient [28]. The hp-finite element method is also known to give good results for singular per-turbation problems [30]. All these methods have in common that they try to discretize the anisotropicPDE as it is written and then to apply purely numerical tricks to circumvent the problems related tolack of accuracy of the discrete solution or to the slow convergence of iterative algorithms. This leadsto methods which are rather difficult to implement.

The approach that we pursue in this paper is entirely different: we reformulate first the originalPDE in such a way that the resulting problem can be efficiently and accurately discretized by straight-forward and easily implementable numerical methods for any anisotropy strength. Our scheme is relatedto the Asymptotic Preserving method introduced in [22]. These techniques are designed to give a precisesolution in the various regimes with no restrictions on the computational meshes and with additionalproperty of converging to the limit solution when ε→0. The derivation of the Asymptotic Preservingmethod requires identification of the limit model. In the case of Singular Perturbation problems, theoriginal problem is reformulated in such a way that the obtained set of equations contain both the limitmodel and the original problem with a continuous transition between them, according to the valuesof ε. This reformulated system of equation sets the foundation of the AP-scheme. These AsymptoticPreserving techniques have been explored in previous studies, for instance quasi-neutral or gyro-fluidlimits [9, 12], as well as anisotropic elliptic problems of the form (1.1) with vector b aligned with a

2


coordinate axis [11, 6].

In this paper, we present a new algorithm which extends the results of [11]. The originality ofthis algorithm consists in the fact, that it is applicable for variable anisotropy directions b, withoutadditional work. The discretization mesh has not to be adapted to the field direction b, but is simply aCartesian grid, whose mesh-size is governed by the desired accuracy, independently on the anisotropystrength ε. All this is possible by a well-adapted mathematical framework (optimally chosen spaces,introduction of Lagrange multipliers). The key idea, as in [11], is to decompose the solution φ intotwo parts: a mean part p which is constant along the field lines and the fluctuation part q consistingof a correction to the mean part needed to recover the full solution. Both parts p and q are solutionsto well-posed problems for any ε> 0. In the limit of ε→0 the AP-reformulation reduces to the socalled Limit model (L-model), whose solution is an acceptable approximation of the P-model solutionfor ε≪1 (see Theorem 2.2). In [11] the Asymptotic Preserving reformulation of the original problemwas obtained in two steps. Firstly, the original problem was integrated along the field lines (z-axis)leading to an ε-independent elliptic problem for the mean part p. Secondly, the mean equation wassubtracted from the original problem and the ε-dependent elliptic problem for the fluctuating partq was obtained. This approach however is not applicable if the field b is arbitrary. In this paper wepresent a new approach. Instead of integrating the original problem along the arbitrary field lines, wechoose to force the mean part p to lie in the Hilbert space of functions constant along the field linesand the fluctuating part q to be orthogonal (in L2 sense) to this space. This is done by a Lagrangemultiplier technique and requires introduction of additional variables thus enlarging the linear systemto be solved. This method allows to treat the arbitrary b field case, regardless of the field topologyand thus eliminates the limitations of the algorithm presented in [11]. We note that an alternativemethod, bypassing the need in Lagrange multipliers, is proposed in [8]. It is based on a reformulationof the original problem as a fourth order equation.

The outline of this paper is the following. Section 2 introduces the original anisotropic ellipticproblem. The original problem will be referred to as the Singular-Perturbation model (P-model). Themathematical framework is introduced and the Asymptotic Preserving reformulation (AP-model) isthen derived. Section 3 is devoted to the numerical implementation of the AP-formulation. Numericalresults are presented for 2D and 3D test cases, for constant and variable fields b. Three methods arecompared (AP-formulation, P-model and L-model) according to their precision for different values of ε.The rigorous numerical analysis of this new algorithm will be the subject of a forthcoming publication.

2. Problem definition

We consider a two or three dimensional anisotropic problem, given on a sufficiently smooth, boundeddomain Ω⊂Rd, d=2,3 with boundary ∂Ω. The direction of the anisotropy is defined by the vectorfield b∈ (C∞(Ω))d, satisfying |b(x)|=1 for all x∈Ω.

Given this vector field b, one can decompose now vectors v∈Rd, gradients ∇φ, with φ(x) a scalarfunction, and divergences ∇·v, with v(x) a vector field, into a part parallel to the anisotropy directionand a part perpendicular to it. These parts are defined as follows:

v|| := (v ·b)b, v⊥ := (Id−b⊗b)v , such that v= v||+v⊥ ,

∇||φ := (b ·∇φ)b, ∇⊥φ := (Id−b⊗b)∇φ, such that ∇φ=∇||φ+∇⊥φ,

∇|| ·v :=∇·v|| , ∇⊥ ·v :=∇·v⊥ , such that ∇·v=∇|| ·v+∇⊥ ·v ,

(2.1)

where we denoted by ⊗ the vector tensor product. With these notations we can now introduce themathematical problem, the so-called Singular Perturbation problem, whose numerical solution is themain concern of this paper.

3


2.1. The Singular Perturbation problem (P-model) We consider the following SingularPerturbation problem

(P )

− 1ε∇‖ ·

(A‖∇‖φε

)−∇⊥ ·(A⊥∇⊥φε)= f in Ω,

1εn‖ ·

(A‖∇‖φε

)+n⊥ ·(A⊥∇⊥φε)=0 on ∂ΩN ,

φε =0 on ∂ΩD ,

(2.2)

where n is the outward normal to Ω and the boundaries are defined by

∂ΩD = x∈∂Ω | b(x) ·n=0, ∂ΩN =∂Ω\∂ΩD. (2.3)

The parameter 0<ε< 1 can be very small and is responsible for the high anisotropy of the problem.The aim is to introduce a numerical scheme, whose computational costs (simulation time and memory),for fixed precision, are independent of ε.We shall assume in the rest of this paper the following hypothesis on the diffusion coefficients and thesource terms

Hypothesis A Let f ∈L2(Ω) and

∂ΩD 6=∅. The diffusion coefficients A‖ ∈L∞(Ω) and A⊥ ∈Md×d(L

∞(Ω)) are supposed to satisfy

0<A0≤A‖(x)≤A1 , f.a.a. x∈Ω, (2.4)

A0||v||2 ≤ vtA⊥(x)v≤A1||v||2 , ∀v∈Rd and f.a.a. x∈Ω. (2.5)

As we intend to use the finite element method for the numerical solution of the P-problem, let us put(2.2) under variational form. For this let V be the Hilbert space

V := φ∈H1(Ω) / φ|∂ΩD=0 , (φ,ψ)V := (∇||φ,∇||ψ)L2 +ε(∇⊥φ,∇⊥ψ)L2 .

Thus, we are seaking for φε ∈V , the solution of

a||(φε,ψ)+εa⊥(φε,ψ)= ε(f,ψ), ∀ψ∈V , (2.6)

where (·, ·) stands for the standard L2 inner product and the continuous bilinear forms a|| :V ×V →Rand a⊥ :V ×V →R are given by

a||(φ,ψ) :=

∫

Ω

A||∇||φ ·∇||ψdx, a⊥(φ,ψ) :=

∫

Ω

(A⊥∇⊥φ) ·∇⊥ψdx. (2.7)

Thanks to Hypothesis A and the Lax-Milgram theorem, problem (2.2) admits a unique solution φε ∈Vfor all fixed ε> 0.

2.2. The Limit problem (L-model) The direct numerical solution of (2.2) may be veryinaccurate for ε≪1. Indeed, when ε tends to zero, the system reduces to

−∇‖ ·(A‖∇‖φ

)=0 in Ω,

n‖ ·(A‖∇‖φ

)=0 on ∂ΩN ,

φ=0 on ∂ΩD.

(2.8)

4


This is an ill-posed problem as it has an infinite number of solutions φ∈G, where

G = φ∈V | ∇‖φ=0 , (2.9)

is the Hilbert space of functions, which are constant along the field lines of b. This shows that thecondition number of the system obtained by discretizing (2.2) tends to ∞ as ε→0 so that its solutionwill suffer from round-off errors.

For this reason, we should approximate (2.2) in the limit ε→0 differently. Supposing that φε →φ0

as ε→0 we identify (at first formally) the problem satisfied by φ0. From the above arguments we knowthat φ0 ∈G. Taking now test functions ψ∈G in (2.6), we obtain

∫

Ω

A⊥∇⊥φε ·∇⊥ψdx=

∫

Ω

fψdx. (2.10)

Passing to the limit ε→0 into this equation yields the variational formulation of the problem satisfiedby φ0 (Limit problem): find φ0 ∈G, the solution of

(L)

∫

Ω

A⊥∇⊥φ0 ·∇⊥ψdx=

∫

Ω

fψdx , ∀ψ∈G , (2.11)

which is a well posed problem. Indeed, the space G ⊂V is a Hilbert space, associated with the innerproduct

(φ,ψ)G := (∇⊥φ,∇⊥ψ)L2 , ∀φ,ψ∈G , (2.12)

and the norm || · ||G is equivalent to the H1 norm. This is due to the Poincare inequality, as

||φ||2L2 ≤C||∇φ||2L2 =C||∇||φ||2L2 +C||∇⊥φ||2L2 =C||∇⊥φ||2L2 , ∀φ∈G .

Hypothesis A and the Lax-Milgram lemma imply the existence and uniqueness of a solution φ0 ∈G ofthe Limit problem (2.11).

Remark 2.1. Let us restrict ourselves for the moment to the simple special case (considered in a pre-vious paper [11]) of the two dimensional domain Ω=(0,Lx)×(0,Lz) in the (x,z) plane with a constantb-field aligned with the Z-axis:

b=

(01

). (2.13)

The functions in the space G are independent of z so that G can be identified to H10 (0,Lx). The limit

problem (2.11) now reads: Find φ0 in H10 (0,Lx) verifying

∫ Lx

0

A⊥(x)∂xφ0(x)∂xψ(x)dx=

∫ Lx

0

f(x)ψ(x)dx, ∀ψ∈H10 (0,Lx),

where A⊥(x)= (1/Lz)∫ Lz

0 A⊥,11(x,z)dz and f(x)= (1/Lz)∫ Lz

0 f(x,z)dz are the mean values of A⊥ andf along the field lines. The limit solution φ0 thus verifies a one-dimensional elliptic equation whosecoefficients are integrated along the anisotropy direction:

−∂x

(A⊥(x)∂xφ

0(x))

= f(x) on (0,Lx),

φ0(0)=φ0(Lx)=0 .(2.14)

5


We see now that φ0(x) is a solution to the one dimensional elliptic problem so that it belongs toH2(0,Lx) provided f ∈L2(Ω). Since φ0 as a function of (x,z) does not depend on z, we have alsoφ0 ∈H2(Ω). This conclusion (φ0 ∈H2(Ω)) remains valid in the case of a cylindrical three dimensionaldomain Ω=Ωxy ×(0,Lz) in the (x,y,z) space with any sufficiently smooth Ωxy in the (x,y) plane and thefield b aligned with the Z-axis, b=(0,0,1)t. Indeed, it is easy to see that φ0 =φ0(x,y) solves in this casean elliptic two dimensional problem in Ωxy similar to (2.14) so that we can apply the standard regularityresults for the elliptic problems. These examples show that it is reasonable to suppose φ0 ∈H2(Ω) alsoin more general geometries of Ω and b. This can be indeed proved under the hypotheses in AppendixA by specifying the (d−1) dimensional elliptic problem for φ0. The proof being rather lengthy andtechnical, we prefer to postpone it to a forthcoming work [13].

2.3. The Asymptotic Preserving approach (AP-model) In this section we introduce theAP-formulation, which is a reformulation of the Singular Perturbation problem (2.2), permitting a“continuous” transition from the (P)-problem (2.2) to the (L)-problem (2.11), as ε→0. For this purpose,each function is decomposed into its mean part along the anisotropy direction (lying in the subspace Gof V) and a fluctuating part (cf. [11]) lying in the L2-orthogonal complement A of G in V , defined by

A := φ∈V |(φ,ψ)=0 , ∀ψ∈G . (2.15)

Note that (·, ·) denote here and elsewhere the inner product of L2(Ω).In what follows, we need the following

Hypothesis B The Hilbert-space V admits the decomposition

V =G⊕⊥A, (2.16)

with G given by (2.9) and A given by (2.15) and where the orthogonality of the direct sum is taken withrespect to the L2-norm. Denoting by P the orthogonal projection on G with respect to the L2 innerproduct:

P :V →G such that (Pφ,ψ)= (φ,ψ) ∀φ∈V , ψ∈G , (2.17)

we shall suppose that this mapping is continuous and that we have the Poincare-Wirtinger inequality

||φ−Pφ||L2(Ω) ≤C||∇||φ||L2(Ω) , ∀φ∈V . (2.18)

Applying the projection P to a function φ is nothing but a weighted average of φ along the anisotropyfield lines of b. The space G is the space of averaged functions (the parallel Gradient of these averagedfunctions being equal to zero), whereas the space A is the space of the fluctuations (the Average of thefluctuations being equal to zero). Note that the decomposition (2.16) is not self evident and it may infact fail on some “pathological” domains Ω. Indeed, although one can always define an L2-orthogonalprojection P φ on the space of functions constant along each field line, for any φ with square-integrable∇||φ, one cannot assure in general that P φ belongs to V for φ∈V since one may lose control of the

perpendicular part of the gradient of Pφ. Fortunately however, Hypothesis B is typically satisfied forthe domains of practical interest. The interested reader is referred to Appendix A for an example of a setof assumptions on Ω and b which entail Hypothesis B and which resume essentially to the requirementfor the field b to intersect ∂ΩN in a uniformly non-tangential manner and for the boundary components∂ΩN and ∂ΩD to be sufficiently smooth.

Let us also define the operator

Q :V →A, Q= I−P . (2.19)

6


Each function φ∈V can be decomposed uniquely as φ=p+q, where p=Pφ∈G and q=Qφ∈A. Usingthis decomposition, we reformulate the Singular-Perturbation problem (2.2). Indeed, replacing φε :=pε +qε in problem (2.2) and taking test functions η∈G and ξ∈A leads to an asymptotic preservingformulation of the original problem: Find (pε,qε)∈G×A such that

a⊥(pε,η)+a⊥(qε,η)= (f,η) ∀η∈G,

a||(qε,ξ)+εa⊥(qε,ξ)+εa⊥(pε,ξ)= ε(f,ξ) ∀ξ∈A.

(2.20)

Contrary to the Singular Perturbation problem (2.2), setting formally ε=0 in (2.20) yields the system

a⊥(p0,η)+a⊥(q0,η) = (f,η), ∀η∈G

a||(q0,ξ) = 0 , ∀ξ∈A,

(2.21)

which has a unique solution (p0,q0)∈G ×A, where p0 is the unique solution of the L-problem (2.11) andq0 ≡ 0. Indeed, taking ξ= q0 as test function in the second equation of (2.21) yields ∇||q0 =0, whichmeans q0 ∈G. But at the same time, q0 ∈A, so that q0 ∈G∩A= 0. Setting then q0 ≡ 0 in the firstequation of (2.21), shows that p0 is the unique solution of the L-problem.Theorem 2.2. For every ε> 0 the Asymptotic Preserving formulation (2.20), under Hypotheses Aand B, admits a unique solution (pε,qε)∈G×A, where φε :=pε +qε is the unique solution in V of theSingular Perturbation model (2.2).These solutions satisfy the bounds

||φε||H1(Ω) ≤C||f ||L2(Ω) , ||qε||H1(Ω) ≤C||f ||L2(Ω) , ||pε||H1(Ω) ≤C||f ||L2(Ω) , (2.22)

with an ε-independent constant C> 0. Moreover, we have

φε →φ0, pε →φ0 and qε →0 in H1(Ω) as ε→0 , (2.23)

where φ0 ∈G is the unique solution of the Limit model (2.11).Proof. The existence and uniqueness of a solution for the P-problem as well as L-problem are

consequences of the Lax-Milgram theorem. The existence and uniqueness of a solution of (2.20) is thenimmediate by construction, remarking that the decomposition φε =pε +qε is unique.The bound ||φε||H1(Ω) ≤C||f ||L2(Ω) is obtained by a standard elliptic argument. Furthermore, pε =Pφε

where P is the L2-orthogonal projector on G, which is a bounded operator in V by (1.4). This impliesthe estimates for pε and qε in (2.22). Since pε ∈G and qε ∈A are bounded, there exist subsequencespεn and qεn that weakly converge for εn →0 to some p0 ∈G and q0 ∈A. Taking ε= εn in (2.20) andpassing to the limit εn →0 we identify (p0,q0) with the unique solution of (2.21), i.e. p0 =φ0 is theunique solution of (2.11) and q0 ≡ 0. Since the limit does not depend on the choice of the subsequence,we have the weak convergence as ε→0, i.e.

pεε→0 p0 in H1(Ω), qεε→0 0 in H1(Ω).

We shall prove now that these convergences are actually strong. Introducing eε =pε −p0, we have

a⊥(eε,η)+a⊥(qε,η)=0, ∀η∈G .

Taking now η= eε in this relation and adding it to the second equation in (2.20), where we put ξ= qε/ε,yields

1

εa||(q

ε,qε)+a⊥(qε +eε,qε +eε)= (f,qε)−a⊥(p0,qε). (2.24)

7


Due to the Poincare-Wirtinger equation (2.18), there exist a constant C> 0 such that

||q||L2(Ω) ≤Ca||(q,q)1/2 , ∀q∈A. (2.25)

In combination with a Young inequality this gives (f,qε)≤||f ||L2||qε||L2 ≤ εC2

2 ||f ||2L2 + 12εa||(q

ǫ,qǫ). Us-ing this in the right hand side of (2.24), we arrive at

1

2εa||(q

ε,qε)+a⊥(qε +eε,qε +eε)≤ εC2

2||f ||2L2 −a⊥(p0,qε).

Noting that qε +eε =φε −p0 and ∇‖eε =0 we can rewrite this last inequality as

1

2εa||(φ

ε −p0,φε −p0)+a⊥(φε −p0,φε −p0)≤ εC2

2||f ||2L2 −a⊥(p0,qε).

Since a⊥(p0,qε)→0 as ε→0 (thanks to the weak convergence qε0) we observe that φε →p0 stronglyin H1(Ω). Reminding again that pε =Pφε and P is bounded in the norm of H1(Ω), we obtain alsopε →Pp0 =p0, which entails qε →0.

Remark 2.3. Let us return to the simple special case discussed in remark 2.1, i.e. Ω=(0,Lx)×(0,Lz)and the b-field given by (2.13). Remind that the space G can be identified in this case with the spaceof functions constant along the Z-axis, which means G := φ∈V / ∂zφ=0. The space A is orthogonal(with respect to the L2-norm) to G and thus contains the functions that have zero mean value along

the Z-axis, i.e. A := φ∈V /∫ Lz

0 φ(x,z)dz=0. Therefore, for φε =pε +qε ∈V, the function pε is themean value of φε in the direction of the field b:

pε =1

Lz

∫ Lz

0

φεdz , (2.26)

and qε is the fluctuating part with zero mean value:

qε =φε − 1

Lz

∫ Lz

0

φεdz. (2.27)

Hypothesis B is thus easily verified. The results obtained in this special case were presented in a previouspaper [11]. In the case of an arbitrary b-field, formula (2.26) is generalized as (1.2) in Appendix A,where the length element along the b-field line is weighted by the infinitesimal cross-sectional area ofthe field tube around the considered b-field-line. This formula can be thus interpreted as a consequenceof the co-area formula. Note that in the special case of a uniform anisotropy direction, the limitproblem can easily be formulated as an elliptic problem depending on the only transverse coordinates(see equation (2.14)). The size of the problem is thus significantly smaller than that of the initial one.This feature still occurs for non-uniform b-fields as long as adapted coordinates and meshes are used.In our case, aligned and transverse coordinates are not at our disposal and the solution of the limitproblem must be searched as a function of the whole set of coordinates.

2.4. Lagrange multiplier space The objective of this work is the numerical solution of system(2.20) and the comparison of the obtained results with those obtained by directly solving the originalproblem (2.2). In a general case, when the field b is not necessarily constant, the discretization ofthe subspaces G and A, is not straightforward, as in the simpler case [11]. In order to overcome thisdifficulty a Lagrange multiplier technique will be used.

8


2.4.1. The A space To avoid the use of the constrained space A, we can remark that A canbe characterized as being the orthogonal complement (in the L2 sense) of the G-space. Thus, insteadof (2.20), the slightly changed system will be solved: find (pε,qε,lε)∈G×V ×G such that

a⊥(pε,η)+a⊥(qε,η)= (f,η) ∀η∈G,

a||(qε,ξ)+εa⊥(qε,ξ)+εa⊥(pε,ξ)+(lε,ξ)= ε(f,ξ) ∀ξ∈V ,

(qε,χ)=0 ∀χ∈G.

(2.28)

The constraint (qε,χ)=0, ∀χ∈G is forcing the solution qε to belong to A, and this property is carriedover to the limit ε→0. We have thus circumvented the difficulty of discretizing A by introducing anew variable and enlarging the linear system.Proposition 2.4. Problems (2.20) and (2.28) are equivalent. Indeed, (pε,qε)∈G×A is the uniquesolution of (2.20) if and only if (pε,qε,lε)∈G×V ×G with lε ≡ 0 is the unique solution of (2.28).

Proof. Let (pε,qε)∈G×A be the unique solution of (2.20). Then, it is immediate to show that(pε,qε,0) solves (2.28). Let now (pε,qε,lε)∈G×V ×G be a solution of (2.28). Then, the last equationof (2.28) implies that qε ∈A. Choosing in the second equation as test function ξ∈G, one gets

εa⊥(qε,ξ)+εa⊥(pε,ξ)+(lε,ξ)= ε(f,ξ), ∀ξ∈G ,

which because of the first equation in (2.28), yields (lε,ξ)=0 for all ξ∈G. Thus lε ≡ 0.

2.4.2. The G space In order to eliminate the problems that arise when dealing with thediscretization of G, the Lagrange multiplier method will again be used. First note that

p∈G ⇔∇||p=0

p∈V⇔

∫

Ω

A||∇||p ·∇||λdx=a||(p,λ)=0, ∀λ∈L

p∈V ,(2.29)

where L is a functional space that should be chosen large enough so that one could find for any p∈V aλ∈L with ∇||λ=∇||p. On the other hand, the space L should be not too large in order to ensure theuniqueness of the Lagrange multipliers in the unconstrained system. A space that satisfies these tworequirements under some quite general assumptions to be detailed later, can be defined as

L := λ∈L2(Ω) /∇||λ∈L2(Ω), λ|∂Ωin=0 , with ∂Ωin := x∈∂Ω / b(x) ·n< 0 . (2.30)

Using the characterization (2.29) of the constrained space G, we shall now reformulate the system(2.28) as follows: Find (pε, λε, qε, lε, µε)∈V ×L×V ×V ×L such that

(AP )

a⊥(pε,η)+a⊥(qε,η)+a||(η,λε)= (f,η) , ∀η∈V ,

a||(pε,κ)=0 , ∀κ∈L,

a||(qε,ξ)+εa⊥(qε,ξ)+εa⊥(pε,ξ)+(lε,ξ)= ε(f,ξ) , ∀ξ∈V ,

(qε,χ)+a||(χ,µε)=0 , ∀χ∈V ,

a||(lε,τ)=0 , ∀τ ∈L.

(2.31)

The advantage of the above formulation, as compared to (2.20), is that we only have to discretize thespaces V and L (at the price of the introduction of three additional variables), which is much easier thanthe discretization of the constrained spaces G and A. More importantly, the dual formulation (2.31)does not require any change of coordinates to express the fact that pε is constant along the b-field lines

9


and that qε averages to zero along these lines. Therefore this formulation is particularly well adaptedto time-dependent b-fields, as it does not require any operation which would have to be reinitiated asb evolves. The system (2.31) will be called the Asymptotic-Preserving formulation in the sequel.

To analyse this Asymptotic-Preserving formulation, we need the following

Hypothesis B’ The trace λ|∂Ωinis well defined for any λ∈V as an element of L2(∂Ωin), with contin-

uous dependence of the trace norm in L2(∂Ωin) on ||λ||V . Moreover, the Hilbert space

V = φ∈L2(Ω) /∇||φ∈L2(Ω) , (φ,ψ)V := (φ,ψ)+(∇||φ,∇||ψ), (2.32)

admits the decomposition

V = G⊕L, (2.33)

where G is given by

G := φ∈V /∇||φ=0 , (2.34)

and L is given by (2.30). The spaces G and G= G ∩V are related in the following way: if g∈G is suchthat

∫∂Ωin

ηgdσ=0 for all η∈G, then g=0.

The decomposition (2.33) is quite natural. It tells simply that any function φ can be decomposed oneach field line as a sum of a function that vanishes at one given point on this line and a constant (whichis therefore the value of φ at this point). Hypothesis B’ will be thus normally satisfied in cases ofpractical interest. For example, we prove in Appendix A that the set of assumptions on the domain Ωand the b-field which can be used to verify Hypothesis B, is also sufficient (but far from necessary) forHypothesis B’. We are now able to show the relation between systems (2.28) and (2.31).Proposition 2.5. Assuming Hypotheses A, B and B’, problem (2.31) admits a unique solution(pε, λε, qε, lε, µε)∈V ×L×V ×V ×L, where (pε,qε,lε)∈G×V ×G is the unique solution of (2.28).

The proof of Proposition 2.5 is based on the following two lemmasLemma 2.6. Assume Hypothesis B’ and let p∈V be such that a||(p,λ)=0, ∀λ∈L. Then p∈G.

Proof. Take any η∈V and decompose η=λ+g with λ∈L and g∈G. We have a||(p,g)=0, hence

a||(p,η)=0 for all η∈V . This entails ∇||p=0, hence p∈G.

Lemma 2.7. Assume Hypothesis B’ and let F ∈V∗ be such that F (η)=0 for all η∈G. Then the problemof finding λ∈L such that

a||(η,λ)=F (η), ∀η∈V , (2.35)

has a unique solution.Proof. Consider the bilinear form b on V ×V

b(u,v)=a||(u,v)+∫

∂Ωin

uvdσ

By Hypothesis B’, this is an inner product on V . Indeed, if b(u,u)=0 then u∈G∩L so that u=0.Riesz representation theorem implies that the problem of finding µ∈V such that

b(η,µ)=F (η), ∀η∈V ,

has a unique solution. We can now decompose µ=λ+g with λ∈L and g∈G. This yields

a||(η,λ)+∫

∂Ωin

ηgdσ=F (η), ∀η∈V ,

10


so that, in particular,∫

∂Ωinηgdσ=0 for all η∈G which implies g=0. We see now that λ is a solution

to (2.35). The uniqueness follows easily.Let us now prove Proposition 2.5.Proof of existence in Proposition 2.5. Take (pε,qε,lε)∈G×V ×G as the unique solution of (2.28).Then, equations 2,3,5 in (2.31) are immediately satisfied. It remains to choose properly the Lagrangemultipliers λε,µε ∈L to satisfy equations 1,4 in (2.31). For this, let us define F1,F2 ∈V∗ by

F1(η) :=1

εa||(q

ε,η), F2(η) :=−(qε,η), ∀η∈V . (2.36)

These functionals are indeed continuous in the norm of V since their definitions do not contain thederivatives in directions perpendicular to b. Since F1(η)=F2(η)=0 for all η∈G, Lemma 2.7 impliesthe existence of λε ∈L and µε ∈L, such that

a||(η,λε)=F1(η), a||(χ,µ

ε)=F2(χ), ∀η,χ∈V . (2.37)

Taking η,χ∈V ⊂V we observe (cf. the second line in (2.28) where lε =0)

a||(η,λε)=

1

εa||(q

ε,η)= (f,η)−a⊥(pε,η)−a⊥(qε,η), ∀η∈V ,

a||(χ,µε)=−(qε,χ), ∀χ∈V ,

which coincides with equations 1,4 in (2.31).Proof of uniqueness in Proposition 2.5. Consider the solution to system (2.31) with f =0. Lemma2.6 implies then that pε,lε ∈G∩V =G and (pε,qε,lε)∈G×V ×G verifies (2.28) with f =0 so that pε =qε = lε =0 by Proposition 2.4. Equations 1,4 in (2.31) now tell us that λε,µε ∈G, but G ∩L= 0, henceλε =µε =0. 2

The presence of 1/ε in the formulas (2.36), (2.37) defining λε indicates at a first sight that λε maytend to ∞ as ε→0 which would be disastrous for an AP numerical method based on (2.31) at verysmall ε. Fortunately λε remains bounded uniformly in ε in the cases of practical interest. It suffices tosuppose that the limit solution φ0 is in H2(Ω) which is a reasonable assumption as discussed in Remark2.1.Proposition 2.8. Assume Hypotheses A, B, B’ and φ0 ∈H2(Ω) where φ0 is the solution to (2.11).Then λε introduced in (2.31) satisfies

||∇||λε||L2 ≤Cmax(||f ||L2 , ||φ0||H2) (2.38)

with a constant C independent of ε.Proof. We will denote all the ε-independent constants by C in this proof. We start from relation

(2.24) in the proof of Theorem 2.2. Dropping the positive term a⊥(qε +eε,qε +eε) it can be rewrittenas

1

εa||(q

ε,qε)≤ (f,qε)−a⊥(φ0,qε).

Since φ0 ∈H2(Ω) we can integrate by parts in the integral defining a⊥(φ0,qε):

−a⊥(φ0,qε)=−∫

Ω

A⊥∇⊥φ0 ·∇⊥q

εdx

=−∫

∂ΩN

(Id−bbt)A⊥∇⊥φ0 ·nqεdσ+

∫

Ω

(∇⊥ ·A⊥∇⊥φ0)qεdx

≤C||φ0||H2

(||qε||L2(∂ΩN )+ ||qε||L2(Ω)

)

11


since ∇φ0 has a trace on ∂Ω and its norm in L2(∂ΩN ) is bounded by C||φ0||H2 . Thus,

1

ε||∇||q

ε||2L2 ≤ C

εa||(q

ε,qε)≤C||f ||L2||qε||L2(Ω)+C||φ0||H2

(||qε||L2(∂ΩN )+ ||qε||L2(Ω)

).

By Poincare-Wirtinger inequality (2.18) (note that Pqε =0) and by Hypothesis B’ we have

max(||qε||L2(Ω), ||qε||L2(∂ΩN ))≤C||∇||qε||L2

so that

1

ε||∇||q

ε||L2 ≤Cmax(||f ||L2 , ||φ0||H2).

This is the same as (2.38) since ∇||λε = 1ε∇||qε according to (2.36) and (2.37).

Remark 2.9. The Limit model (2.11), reformulated using the Lagrange multiplier technique, nowreads: Find (φ0, λ0)∈V ×L such that

(L′)

∫

Ω

A⊥∇⊥φ0 ·∇⊥ψdx+

∫

Ω

A||∇||ψ ·∇||λ0dx=

∫

Ω

fψdx ∀ψ∈V∫

Ω

A||∇||φ0 ·∇||κdx=0 ∀κ∈L.

(2.39)

Problem (2.39) is also well posed assuming Hypotheses A, B, B’ and φ0 ∈H2(Ω). Indeed, the uniquenessof the solution to (2.39) can be proved in exactly the same manner as in the proof of Proposition 2.5above. To prove the existence of a solution, it suffices to take the limit ε→0 in the first two lines of(2.31). Indeed, we know by Theorem 2.2 that pε →φ0, the solution to (2.11), and qε →0 in H1(Ω).Moreover, the family ∇||λε is bounded in the norm of L2(Ω) by Proposition 2.8. We can take thereforea weakly convergence subsequence ∇||λ

εn and identify its limit with ∇||λ0 with some λ0 ∈L (cf.

Lemma 2.7) to see that (φ0,λ0)∈V ×L solves (2.39).

3. Numerical methodThis section concerns the discretization of the Asymptotic Preserving formulation (2.31), based

on a finite element method, and the detailed study of the obtained numerical results. The numericalanalysis of the present scheme is investigated in a forthcoming work [13], in particular we are interestedin the convergence of the scheme, independently of the parameter ε> 0.

Let us denote by Vh ⊂V and Lh ⊂L the finite dimensional approximation spaces, constructed bymeans of appropriate numerical discretizations (see Section 3.1 and Appendix B). We are thus lookingfor a discrete solution (pε

h, λεh, q

εh, l

εh, µ

εh)∈Vh ×Lh ×Vh ×Vh ×Lh of the following system

a⊥(pεh,η)+a⊥(qε

h,η)+a||(η,λεh)= (f,η) , ∀η∈Vh ,

a||(pεh,κ)=0 , ∀κ∈Lh ,

a||(qεh,ξ)+εa⊥(qε

h,ξ)+εa⊥(pεh,ξ)+(lεh,ξ)= ε(f,ξ) , ∀ξ∈Vh ,

(qεh,χ)+a||(χ,µ

εh)=0 , ∀χ∈Vh ,

a||(lεh,τ)=0 , ∀τ ∈Lh .

(3.1)

Our numerical experiments indicate that the spaces Vh and Lh can be always taken of the sametype and on the same mesh. The only difference between these two finite element spaces lies thus in

12


the incorporation of boundary conditions. In general, let Xh denote the complete finite element space(without any restrictions on the boundary) which should be H1 conforming but otherwise arbitrarilychosen. We define then

Vh = vh ∈Xh/vh|∂ΩD =0, (3.2)

Lh = λh ∈Xh/λh|∂Ωin∪∂ΩD =0 . (3.3)

While this choice of Vh is straight forward, the boundary conditions in Lh require special attention.Indeed, nothing in the definition (2.30) of space L on the continuous level indicates that its elementsshould vanish on ∂ΩD. However, this liberty on ∂ΩD is somewhat counter-intuitive. Indeed, theLagrange multiplier λε ∈L serves to impose ∇||pε =0 for some function pε taken from the space V . But,for p∈V the trace on ∂ΩD is zero so that ∇||pε =0 there without the help of a Lagrange multiplier.Of course, this argument is not valid on the continuous level since the trace of functions in L does noteven necessarily exist. However, this may become very important on the finite element level. Indeed,we provide in Appendix B an example of a finite element setting without incorporating λh|∂ΩD =0 intothe definition of Lh, which leads to an ill-posed system (3.1). To avoid this difficulty, we choose Lh asin (3.3) in all our experiments, thus obtaining well-posed problems.

3.1. Discretization Let us present the discretization in a 2D case, the 3D case being a simplegeneralization. The here considered computational domain Ω is a square Ω= [0,1]× [0,1]. All simula-tions are performed on structured meshes. Let us introduce the Cartesian, homogeneous grid

xi = i/Nx , 0≤ i≤Nx , yj = j/Ny , 0≤ j≤Ny, (3.4)

where Nx and Ny are positive even constants, corresponding to the number of discretization intervalsin the x- resp. y-direction. The corresponding mesh-sizes are denoted by hx> 0 resp. hy> 0. Choosinga Q2 finite element method (Q2-FEM), based on the following quadratic base functions

θxi =

(x−xi−2)(x−xi−1)2h2

xx∈ [xi−2,xi],

(xi+2−x)(xi+1−x)2h2

xx∈ [xi,xi+2],

0 else

, θyj =

(y−yj−2)(y−yj−1)2h2

yy∈ [yj−2,yj ],

(yj+2−y)(yj+1−y)2h2

yy∈ [yj ,yj+2],

0 else

(3.5)

for even i,j and

θxi =

(xi+1−x)(x−xi−1)

h2x

x∈ [xi−1,xi+1],

0 else, θyj =

(yj+1−y)(y−yj−1)

h2y

y∈ [yj−1,yj+1],

0 else(3.6)

for odd i,j, we define

Xh := vh =∑

i,j

vij θxi(x)θyj (y) ,

We then search for discrete solutions (pεh, q

εh, l

εh)∈Vh ×Vh ×Vh and (λε

h, µεh)∈Lh ×Lh with Vh and Lh

defined by (3.2) and (3.3). This leads to the inversion of a linear system, the corresponding matrixbeing non-symmetric and given by

A=

A1 A0 A1 0 0A0 0 0 0 0εA1 0 A0+εA1 C 00 0 C 0 A0

0 0 0 A0 0

. (3.7)

13


εAP scheme Limit model Singular Perturbation scheme

L2 error H1 error L2 error H1 error L2 error H1 error

10 7.2×10−6 4.7×10−3 5.0×100 3.51×101 7.2×10−6 4.7×10−3

1 7.3×10−7 4.7×10−4 5.0×10−1 3.51×100 7.3×10−7 4.7×10−4

10−1 1.47×10−7 9.6×10−5 5.0×10−2 3.51×10−1 1.45×10−7 9.4×10−5

10−4 1.28×10−7 8.3×10−5 5.0×10−5 3.61×10−4 1.26×10−7 8.2×10−5

10−6 1.28×10−7 8.3×10−5 5.2×10−7 8.4×10−5 5.9×10−7 8.2×10−5

10−10 1.28×10−7 8.3×10−5 1.28×10−7 8.3×10−5 9.9×10−3 3.12×10−2

10−15 1.28×10−7 8.3×10−5 1.28×10−7 8.3×10−5 7.1×10−1 2.23×100

Table 3.1 – Comparison between the Asymptotic Preserving scheme, the Limit model and theSingular Perturbation model for h=0.005 (200 mesh points in each direction) and constant b:absolute L2-error and H1-error, for different ε-values.

The sub-matrices A0, A1 resp. C correspond to the bilinear forms a||(·, ·), a⊥(·, ·) resp. (·, ·), used in

equations (2.31) and belong to R(Nx+1)(Ny+1)×(Nx+1)(Ny+1). The matrix elements are computed usingthe 2D Gauss quadrature formula, with 3 points in the x and y direction:

∫ 1

−1

∫ 1

−1

f(x,y)=

1∑

i,j=−1

ωiωjf(xi,yj), (3.8)

where x0 = y0 =0, x±1 = y±1 =±√

35 , ω0 =8/9 and ω±1 =5/9, which is exact for polynomials of degree

5.

3.2. Numerical Results

3.2.1. 2D test case, uniform and aligned b-field In this section we compare the numericalresults obtained via the Q2-FEM, by discretizing the Singular Perturbation model (2.2), the Limit model(2.11) and the Asymptotic Preserving reformulation (2.31). In all numerical tests we set A⊥ = Id andA‖ =1. We start with a simple test case, where the analytical solution is known. Let the source termf be given by

f =(4+ε)π2cos(2πx)sin(πy)+π2 sin(πy) (3.9)

and the b field be aligned with the x-axis. Hence, the solution φε of (2.2) and its decompositionφε =pε+qε write

φε =sin(πy)+εcos(2πx)sin(πy), (3.10)

pε =sin(πy) , qε = εcos(2πx)sin(πy) . (3.11)

We denote by φP , φL, φA the numerical solution of the Singular Perturbation model (2.2), theLimit model (2.11) and the Asymptotic Preserving reformulation (2.31) respectively. The comparisonwill be done in the L2-norm as well as the H1-norm. The linear systems obtained after discretizationof the three methods are solved using the same numerical algorithm — LU decomposition implementedin a solver MUMPS[2].

In Figure 3.1 we plotted the absolute errors (in the L2 resp. H1-norms) between the numericalsolutions obtained with one of the three methods and the exact solution, and this, as a function of

14


1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(a) L2 error for a grid with 50×50 points.

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(b) H1 error for a grid with 50×50 points.

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(c) L2 error for a grid with 100×100 points.

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(d) H1 error for a grid with 100×100 points.

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(e) L2 error for a grid with 200×200 points.

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(f) H1 error for a grid with 200×200 points.

Figure 3.1 – Absolute L2 (left column) and H1 (right column) errors between the exact solutionφε and the computed numerical solution φA (AP), φL (L), φP (P) for the test case with constantb. The error is plotted as a function of the parameter ε and for three different mesh-sizes.

the parameter ε and for several mesh-sizes. In Table 3.1, we specified the error values for one fixedgrid and several ε-values. One observes that the Singular Perturbation finite element approximationis accurate only for ε bigger than some critical value εP , the Limit model gives reliable results for ε

15


method # rows # non zero time L2-error H1-error

AP 50×103 1563×103 13.212 s 1.02×10−6 3.34×10−4

L 20×103 469×103 5.227 s 1.14×10−6 3.34×10−4

P 10×103 157×103 3.707 s 1.02×10−6 3.27×10−4

Table 3.2 – Comparison between the Asymptotic Preserving scheme (AP), the Limit model (L)and the Singular Perturbation model (P) for h=0.01 (100 mesh points in each direction) andfixed ε=10−6: matrix size, number of nonzero elements, average computational time and errorin L2 and H1 norms.

smaller than εL, whereas the AP-scheme is accurate independently on ε. The order of convergence forall three methods is three in the L2-norm and two in the H1-norm, which is an optimal result for Q2

finite elements. When designing a robust numerical method one has therefore two options. The firstone is to use an Asymptotic Preserving scheme, which is accurate independently on ε, but requires thesolution of a bigger linear system. The second one is to design a coupling strategy that involves thesolution of the Singular Perturbation formulation and the Limit problem in their respective validitydomains. This is however a very delicate problem, since we observe that the critical values εP and εL

are mesh dependent, namely εP inversely proportional to h and εL proportional to h. Therefore forsmall meshes there may exist a range of ε-values, where neither the Singular Perturbation nor the Limitmodel finite element approximation give accurate results. For our test case, this is even the case formeshes as big as 200×200 points, if one regards the L2-norm. This mesh-size is generally insufficientin the case of real physical applications.

Another interesting aspect with respect to which the three methods must be compared, is thecomputational time and the size of the matrices involved in the linear systems. Table 3.2 shows thatthe Asymptotic Preserving scheme is expensive in computational time and memory requirements, ascompared to the other methods. Indeed, the computational time required to solve the problem is almostfour times bigger than that of the Singular Perturbation scheme. Moreover, the Asymptotic Preservingmethod involves matrices that have five times more rows and ten times more nonzero elements thanthe matrices obtained with the Singular Perturbation approximation. It is however the only schemethat provides the h-convergence regardless of ε. In order to reduce the computational costs, a couplingstrategy for problems with variable ε will be proposed in a forthcoming paper. In sub-domains whereε>εP the Singular Perturbation problem will be solved, in sub-domains where ε<εL the Limit problemwill be solved and only in the remaining part, where neither the Limit nor the Singular Perturbationmodel are valid, the Asymptotic Preserving formulation will be solved.

3.2.2. 2D test case, non-uniform and non-aligned b-field We now focus our attention onthe original feature of the here introduced numerical method, namely its ability to treat nonuniform bfields. In this section we present numerical simulations performed for a variable field b.

First, let us construct a numerical test case. Finding an analytical solution for an arbitrary bpresents a considerable difficulty. We have therefore chosen a different approach. First, we choose alimit solution

φ0 =sin(πy+α(y2−y)cos(πx)

), (3.12)

where α is a numerical constant aimed at controlling the variations of b. For α=0, the limit solutionof the previous section is obtained. The limit solution for α=2 is shown in Figure 3.2. We set α=2in what follows. Since φ0 is a limit solution, it is constant along the b field lines. Therefore we can

16


Figure 3.2 – The limit solution for the test case with variable b.

determine the b field using the following implication

∇‖φ0 =0 ⇒ bx

∂φ0

∂x+by

∂φ0

∂y=0 , (3.13)

which yields for example

b=B

|B| , B=

(α(2y−1)cos(πx)+ππα(y2−y)sin(πx)

). (3.14)

Note that the field B, constructed in this way, satisfies divB=0, which is an important property in theframework of plasma simulation. Furthermore, we have B 6=0 in the computational domain. Now, wechoose φε to be a function that converges, as ε→0, to the limit solution φ0:

φε =sin(πy+α(y2−y)cos(πx)

)+εcos(2πx)sin(πy). (3.15)

Finally, the force term is calculated, using the equation, i.e.

f =−∇⊥ ·(A⊥∇⊥φε)− 1

ε∇‖ ·(A‖∇‖φ

ε).

As in the previous section, we shall compare here the numerical solution of the Singular Perturbationmodel (2.2), the Limit model (2.11) and the Asymptotic Preserving reformulation (2.31), i.e. φP , φL,φA with the exact solution (3.15) . The L2 and H1-errors are reported on Figure 3.3 and Table 3.3.Once again the Asymptotic Preserving scheme proves to be valid for all values of ε, contrary to theother schemes. There is however a difference compared to the constant-b case. For a variable b , thethreshold value εP seems to be independent on the mesh size and is much larger than that of theuniform b test case. This observation limits further the possible choice of coupling strategies, sinceeven for coarse meshes there exists a range of ε-values, where neither the Singular Perturbation nor theLimit model are valid. The coupling strategy, involving all three models, remains however interestingto investigate.

17


εAP scheme Limit model Singular Perturbation scheme

L2 error H1 error L2 error H1 error L2 error H1 error

10 7.2×10−6 4.6×10−3 5.0×100 3.50×101 7.2×10−6 4.6×10−3

1 7.1×10−7 4.6×10−4 5.0×10−1 3.50×100 7.1×10−7 4.6×10−4

10−2 2.05×10−7 1.33×10−4 5.0×10−3 3.50×10−2 2.05×10−7 1.33×10−4

10−4 2.12×10−7 1.38×10−4 5.0×10−5 3.77×10−4 1.74×10−6 1.43×10−4

10−7 2.17×10−7 1.41×10−4 2.22×10−7 1.41×10−4 1.68×10−3 1.26×10−2

10−10 2.17×10−7 1.41×10−4 2.17×10−7 1.41×10−4 3.93×10−1 1.35×100

10−15 2.17×10−7 1.41×10−4 2.17×10−7 1.41×10−4 6.7×10−1 2.32×100

Table 3.3 – Comparison between the Asymptotic preserving scheme, the Limit model and theSingular Perturbation model for h=0.005 (200 mesh points in each direction) and variable b:absolute L2-error and H1-error.

In the next test case we investigate the influence of the variations of the b field on the accuracy ofthe solution. We would like to answer the following question: what is the minimal number of points percharacteristic length of b variations required to obtain an acceptable solution. For this, let us modifythe previous test case. Let b=B/|B|, with

B=

(α(2y−1)cos(mπx)+πmπα(y2−y)sin(mπx)

), (3.16)

m being an integer. The limit solution and φε are chosen to be

φ0 =sin(πy+α(y2−y)cos(mπx)

), (3.17)

φε =sin(πy+α(y2−y)cos(mπx)

)+εcos(2πx)sin(πy). (3.18)

We perform two tests: first, we fix the mesh size and varym to find the minimal period of b for whichthe Asymptotic Preserving method yields still acceptable results. We define a result to be acceptablewhen the relative error is less then 0.01. In the second test m remains fixed and the convergence of thescheme is studied. The results are presented on Figures 3.4 and 3.5.

For ε=1 and 400 mesh points in each direction (h=0.0025) the relative error in the L2-norm,

defined as||φε−φA||L2(Ω)

||φA||L2(Ω), is below 0.01 for all tested values of 1≤m≤ 50. The relative H1-error

||φε−φA||H1(Ω)

||φA||H1(Ω)exceeds the critical value for m> 25. For ε=10−20 the maximal m for which the er-

ror is acceptable in both norms is 20. The minimal number of mesh points per period of b variations is40 in the worst case, in order to obtain an 1% relative error.

Figure 3.5 and Table 3.4 show the convergence of the Asymptotic Preserving scheme with respectto h for m=10 and ε=10−10. We observe that for big values of h the error does not diminish with h.Then, for h< 0.025 the scheme converges at a better rate then 2 for H1-error and 3 for L2-error. Forh< 0.00625 (160 points) the optimal convergence rate in the H1-norm is obtained (which is 32 meshpoints per period of b). The method is super-convergent in the whole tested range for the L2-error.

These results are reassuring, as they prove that the Asymptotic Preserving scheme is precise evenfor strongly varying fields for relatively small mesh sizes, which was not evident. Indeed, the optimalconvergence rate in the H1-norm is obtained for 32 mesh points per b period, and an 1% relative errorfor 40 points. It shows that accurate results can be obtained in more complex simulations, such astokamak plasma for example. The application of the method to bigger scale problems is the subject ofan ongoing work.

18


1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(a) L2 error for a grid with 50×50 points.

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(b) H1 error for a grid with 50×50 points.

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(c) L2 error for a grid with 100×100 points.

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(d) H1 error for a grid with 100×100 points.

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(e) L2 error for a grid with 200×200 points.

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(f) H1 error for a grid with 200×200 points.

Figure 3.3 – Absolute L2 (left column) and H1 (right column) errors between the exact solutionφε and the computed solution φA (AP), φL (L), φP (P) for the test case with variable b. Plottedare the errors as a function of the small parameter ε, for three different meshes.

19


1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

5 10 15 20 25 30 35 40 45 50

L2 errorH1 error

(a) ε=1

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

5 10 15 20 25 30 35 40 45 50

L2 errorH1 error

(b) ε=10−20

Figure 3.4 – Relative L2 and H1 errors between the exact solution φε and the computed solutionφA (AP) for h=0.0025 (400 points in each direction) as a function of m and for ε=1 respectively10−20.

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0.001 0.01 0.1

L2 errorH1 error

Figure 3.5 – Relative L2 and H1 errors between the exact solution φε and the computed solutionφA (AP) for m=10 and ε=10−10 as a function of h.

3.2.3. 3D test case, uniform and aligned b-field Finally, we test our method on a simple3D test case. Let the field b be aligned with the X-axis:

b=

100

. (3.19)

Let Ω= [0,1]× [0,1]× [0,1], and the source term f is such that the solution is given by

φε =sin(πy)sin(πz)+εcos(2πx)sin(πy)sin(πz), (3.20)

pε =sin(πy)sin(πz) , qε = εcos(2πx)sin(πy)sin(πz). (3.21)

Numerical simulations were performed on a 30×30×30 grid. Once again all three methods are com-pared. The L2 and H1-errors are given on Figure 3.6. The numerical results are equivalent with thoseobtained in the 2D test with constant b. Note that it is difficult to perform 3D simulations with more

20


h # points per period L2-error H1-error

0.1 2 4.7×10−1 1.05

0.05 4 5.2×10−1 1.29

0.025 8 1.82×10−1 4.3×10−1

0.0125 16 1.89×10−2 6.4×10−2

0.00625 32 1.41×10−3 1.00×10−2

0.0003125 64 9.3×10−5 2.21×10−3

0.0015625 128 6.1×10−6 5.5×10−4

0.00078125 256 4.6×10−7 1.36×10−4

Table 3.4 – Relative L2 and H1 errors between the exact solution φε and the computed solutionφA (AP) for m=10 and ε=10−10 as a function of h.

refined grids, due to memory requirements on standard desktop equipment as we are doing now. Everyrow in the matrix constructed for the Singular Perturbation model, can contain up to 125 non zeroentries (for Q2 finite elements), while matrices associated with the Asymptotic Preserving reformulationhave rows with up to 375 non zero entries. Furthermore the dimension of the latter is five times bigger.The memory requirements of the direct solver used in our simulations grow rapidly. The remedy couldbe to use an iterative solver with suitable preconditioner. Finding the most efficient method to inversethese matrices is however beyond the scope of this paper. In future work we will address this problemas well as a parallelization of this method.

1e-05

0.0001

0.001

0.01

0.1

1

10

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(a) L2 error for a grid with 30×30×30 points.

0.001

0.01

0.1

1

10

100

1e-15 1e-10 1e-05 1

(P)(L)

(AP)

(b) H1 error for a grid with 30×30×30 points.

Figure 3.6 – Absolute L2 (left column) and H1 (right column) errors between the exact solutionφε and the computed solution φA (AP), φL (L), φP (P) for the 3D test case. The errors areplotted as a function of the anisotropy ratio ε.

4. Conclusions

The asymptotic preserving method presented in this paper is shown to be very efficient for thesolution of highly anisotropic elliptic equations, where the anisotropy direction is given by an arbitrary,but smooth vector field b with non-adapted coordinates and meshes. The results presented here gener-alize the procedure used in [11] and have the important advantage to permit the use of Cartesian grids,

21


independently on the shape of the anisotropy. Moreover, the scheme is equally accurate, independentlyon the anisotropy strength, avoiding thus the use of coupling methods. The numerical study of thisAP-scheme shall be investigated in a forthcoming paper, in particular the ε-independent convergenceresults shall be stated.

Another important related work consists in extending our methods to the case of anisotropy ratiosε, which are variable in Ω from moderate to very small values. This is important, for example, in plasmaphysics simulations as already noted in the introduction. An alternative strategy to the AsymptoticPreserving schemes whould be to couple a standard discretization in subregions with moderate ε witha limit (ε→0) model in subregions with small ε as suggested, for example, in [5, 25]. However, thelimit model is only valid for ε≪1 and cannot be applied for weak anisotropies. Thus, the couplingstrategy requires existence of a range of anisotropy strength where both methods are valid. This israther undesirable since this range may not exist at all, as illustrated by our results in Fig. 3.1.

Appendix A. Decompositions V =G⊕⊥A, V = G ⊕L and related estimates.

We shall show in this Appendix that all the statements in Hypotheses B and B’ can be rigorouslyderived under some assumptions on the domain boundary ∂Ω and on the manner in which it is inter-sected by the field b. We assume essentially that b is tangential to ∂Ω on ∂ΩD and that b penetratesthe remaining part of the boundary ∂ΩN at an angle that stays away from 0 on ∂ΩN . We assume alsothat ∂ΩN consists of two disjoint components for which there exist global and smooth parametrizations.This last assumption can be weakened (existence of an atlas of local smooth parametrizations should besufficient) at the expense of lengthening the proofs. The precise set of our assumptions is the following:

Hypothesis C The boundary of Ω is the union of three components: ∂ΩD where b ·n=0, ∂Ωin

where b ·n≤−α and ∂Ωout where b ·n≥α with some constant α> 0. Moreover, there is a smoothsystem of coordinates ξ1, . . . ,ξd−1 on ∂Ωin meaning that there is a bounded domain Γin ∈Rd−1 anda one-to-one map hin : Γin →Rd such that hin ∈C2(Γin) and ∂Ωin is the image of hin(ξ1, . . . ,ξd−1) as(ξ1, . . . ,ξd−1) goes over Γin. The matrix formed by the vectors (∂hin/∂ξ1, . . . ,∂hin/∂ξ1,n) is invertiblefor all (ξ1, . . . ,ξd−1)∈Γin. Similar assumptions hold also for ∂Ωout (changing Γin to Γout and hin to hout).

Using this hypothesis we can introduce a system of coordinates in Ω such that the field lines ofb coincide with the coordinate lines. To do this consider the initial value problem for a parametrizedordinary differential equation (ODE):

∂X

∂ξd(ξ′,ξd)= b(X(ξ′,ξd)), X(ξ′,0)=hin(ξ′). (1.1)

Here X(ξ′,ξd) is Rd-valued and ξ′ stands for (ξ1, . . . ,ξd−1). For any fixed ξ′ ∈Γin, equation (1.1) shouldbe understood as an ODE for a function of ξd. Its solution X(ξ′,ξd) goes then over the field line of bstarting (as ξd =0) at the point on the inflow boundary ∂Ωin, parametrized by ξ′. This field line hitsthe outflow boundary ∂Ωout somewhere. In other words, for any ξ′ ∈Γin there exists L(ξ′)> 0 suchthat X(ξ′,L(ξ′))∈∂Ωout. The domain of definition of X is thus

D= (ξ′,ξd)∈Rd / ξ′ ∈Γin and 0<ξd<L(ξ′).

Gathering the results on parametrized ODEs, from for instance [34], we conclude that X(ξ′,ξd)=X(ξ1, . . . ,ξd) is a smooth function of all its d parameters, more precisely X ∈C2(D). Evidently, themapX is one-to-one fromD to Ω and thus ξ1, . . . ,ξd provide a system of coordinates for Ω. Moreover thissystem is not degenerate in the sense that the vectors ∂X/∂ξ1, . . . ,∂X/∂ξd are linearly independent ateach point of Ω. Indeed, if this was not the case, then there would exist a non trivial linear combination

22


λ1∂X/∂ξ+ · · ·+λd∂X/∂ξd that would vanish at some point in Ω. But, ODE (1.1) implies

∂

∂ξd

d∑

i=1

λi∂X

∂ξi(ξ′,ξd)=∇b(X(ξ′,ξd)) ·

d∑

i=1

λi∂X

∂ξi(ξ′,ξd)

so that, the unique solution of this ODE, i.e. the linear combination∑d

i=1λi∂X∂ξi

, would vanish on the

whole field line, in particular on the inflow. But this is impossible since ∂X∂ξi

= ∂hin

∂ξi, i=1, . . .,d−1 on the

inflow, while ∂X∂ξd

= b and the vectors(

∂hin

∂ξ1, . . . , ∂hin

∂ξd−1,b)

are linearly independent for all (ξ1, . . . ,ξd−1)∈Γin. We see thus that the Jacobian J =det(∂Xj/∂ξi) does not vanish on Ω so that we can assume thatm<J<M everywhere on Ω with some positive constants m and M (assuming that J is positive doesnot prevent the generality since if J is negative in Ω than one can replace ξ1 by −ξ1). Since X ∈C2(Ω),we have also that J ∈C1(Ω).

One also sees easily that the top of D is given by a smooth function L(ξ′). Indeed, L(ξ′) isdetermined for each ξ′ ∈Γin from the equation X(ξ′,L(ξ′))=hout(η) with some η=(η1, . . . ,ηd−1)∈Γout.We know already that this equation is solvable for ξd =L(ξ′), η1, . . . ,ηd−1 for any ξ′ ∈Γin. To concludethat the solution depends smoothly on ξ′ we can apply the implicit function theorem to the equation

F (ξ′;ξd,η1, . . . ,ηd−1)=X(ξ′,ξd)−hout(η1, . . .,ηd−1)=0.

Indeed, the Rd-valued function F is smooth and the matrix of its partial derivatives with respect toξd,η1, . . . ,ηd−1 is invertible since ∂F/∂ξd = b and ∂F/∂ηi =−∂hout/∂ηi at some point at the outflowand the vectors ∂hout/∂ηi lie in the tangent plane to ∂Ωout while b is nowhere in this plane. We havemoreover that L∈C1(Γin). Indeed, we can prove that all the derivatives of L are bounded. In order todo it, let us remark that the differential of X(ξ′,L(ξ′)) represents a vector in the tangent plane at somepoint on ∂Ωout so that it is perpendicular to the outward normal n. We have thus for any i=1, . . .,d−1

0=n ·(∂X

∂ξi(ξ′,L(ξ′))+

∂X

∂ξd(ξ′,L(ξ′))

∂L

∂ξi(ξ′)

)=n ·

(∂X

∂ξi(ξ′,L(ξ′))+b(ξ′,L(ξ′))

∂L

∂ξi(ξ′)

)

so that

∂L

∂ξi(ξ′)=−

n · ∂X∂ξi

(ξ′,L(ξ′))

n ·b(ξ′,L(ξ′))

and this is bounded since X has bounded partial derivatives and n ·b≥α by the hypothesis. Note alsothat L is strictly positive.

• We can now establish the decomposition V =G⊕⊥A. Take any φ∈V ∩C1(Ω) and introducep∈L2(Ω) by

p(x)=p(ξ′,ξd)=p(ξ′)=

∫ L(ξ′)0

φ(ξ′,t)J(ξ′,t)dt∫ L(ξ′)0 J(ξ′,t)dt

. (1.2)

(from now on we switch back and forth between the Cartesian coordinates x=(x1, . . . ,xd) andthe new ones (ξ′,ξd)= (ξ1, . . . ,ξd)). Evidently, p is constant along each field line. Moreover, p isthe L2-orthogonal projection of φ on the space of such functions. Indeed, if ψ=ψ(ξ′)∈L2(Ω)is any function constant along each field line then

∫

Ω

pψdx=

∫

D

pψJdξ=

∫

Γin

p(ξ′)ψ(ξ′)∫ L(ξ′)

0

J(ξ′,ξd)dξddξ′

=

∫

Γin

∫ L(ξ′)

0

φ(ξ′,ξd)ψ(ξ′)J(ξ′,ξd)dξddξ′ =∫

Ω

φψdx.

23


Let us prove that p∈V , i.e. that its derivatives are square integrable. The change of variablet=L(ξ′)s yields the function

p(ξ′)=

∫ 1

0φ(ξ′,L(ξ′)s)J(ξ′,L(ξ′)s)ds∫ 1

0 J(ξ′,L(ξ′)s)ds.

Now we have ∂p/∂ξd =0 and for all ∂p/∂ξi,i=1,...,d−1 denoting a=a(ξ′)= (∫ 1

0J(ξ′,L(ξ′)s)ds)−1,

φ=φ(ξ′,L(ξ′)s) and same for J we obtain

∂p

∂ξi=

∂a

∂ξi

∫ 1

0

φJds+a

∫ 1

0

∂φ

∂ξiJds+a

∫ 1

0

∂φ

∂ξd

∂L

∂ξisJ ds

+a

∫ 1

0

φ∂J

∂ξids+a

∫ 1

0

φ∂J

∂ξd

∂L

∂ξisds

(1.3)

Using all the previous bounds on the functions L and J and skipping the details of somewhattedious calculations, we arrive at

∫

Ω

(∂p

∂ξi

)2

dx=

∫

Γin

∫ L(ξ′)

0

(∂p

∂ξi

)2

Jdξddξ′

≤C

∫

Γin

∫ L(ξ′)

0

(φ2+

(∂φ

∂ξi

)2

+

(∂φ

∂ξd

)2)Jdξddξ

′

implying

∥∥∥∥∂p

∂ξi

∥∥∥∥2

L2(Ω)

≤C

(‖φ‖2L2(Ω)+

∥∥∥∥∂φ

∂ξi

∥∥∥∥2

L2(Ω)

+

∥∥∥∥∂φ

∂ξd

∥∥∥∥2

L2(Ω)

)≤C ‖φ‖2H1(Ω) .

Thus p∈H1(Ω), hence p∈G and q=φ−p∈A. Since the dependence of p on φ is continuousin the norm of H1(Ω), a density argument shows that the decomposition φ=p+q with p∈Gand q∈A exists for any φ∈V .

• Let us now introduce the operator P as the L2-orthogonal projector on G, that means

P :V →G , φ∈V 7−→Pφ∈G given by (1.2).

Then, the estimates in the preceding paragraph show that the operator P is continuous in thenorm of H1(Ω):

||∇⊥(Pφ)||L2(Ω) ≤C||∇φ||L2(Ω) , ∀φ∈V (1.4)

• We have also the following Poincare-Wirtinger inequality:

||φ−Pφ||L2(Ω) ≤C||∇||φ||L2(Ω) , ∀φ∈V . (1.5)

To prove this, it is sufficient to establish that ||q||L2(Ω) ≤C||∇||q||L2(Ω) for all q∈A. We observethat

||q||2L2(Ω) =

∫

Γin

∫ L(ξ′)

0

q2(ξ′,ξd)J(ξ′,ξd)dξddξ′

and

||∇||φ||2L2(Ω) =

∫

Γin

∫ L(ξ′)

0

(∂q

∂ξd

)2

(ξ′,ξd)J(ξ′,ξd)dξddξ′ .

24


The requirement q∈A is equivalent to

∫ L(ξ′)

0

q(ξ′,ξd)J(ξ′,ξd)dξd =0 f.a.a. ξ′ ∈Γin . (1.6)

We have thus to prove for every ξ′

∫ L(ξ′)

0

q2(ξ′,ξd)J(ξ′,ξd)dξd ≤C2

∫ L(ξ′)

0

(∂q

∂ξd

)2

(ξ′,ξd)J(ξ′,ξd)dξd

provided (1.6). Fixing any ξ′, making the change of integration variable ξd =L(ξ′)t and intro-ducing the functions u(t)= q(ξ′,L(ξ′)t)J(ξ′,L(ξ′)t) and J(t)=J(ξ′,L(ξ′)t), we rewrite the lastinequality as

∫ 1

0

u2(t)

J(t)dt≤ C2

L2(ξ′)

∫ 1

0

(u′(t)J(t)

− u(t)

J2(t)J ′(t)

)2

J(t)dt. (1.7)

Since∫ 1

0 u(t)dt=0 we have by the standard Poincare inequality

∫ 1

0

u2(t)dt≤C2P

∫ 1

0

(u′(t))2dt. (1.8)

• Let us turn to the verification of Hypothesis B’. Take any u∈V. We want to prove that one candecompose u=p+q with p∈G and q∈L and the trace of u on ∂Ωin (denoted g) is in L2(∂Ωin).In the ξ-coordinates we can write a surface element of ∂Ωin as dσ=S(ξ′)dξ′ with a function Ssmoothly depending on ξ′. We see now that for u suffuciently smooth

||g||2L2(∂Ωin)=

∫

Γin

g2(ξ′)S(ξ′)dξ′

≤C

∫ 1

0

∫

Γin

[u2(ξ′,L(ξ′)s)+

1

L(ξ′)

(∂u

∂ξd

)2

(ξ′,L(ξ′)s)

]S(ξ′)dξ′ds

(by a one-dimensional trace inequlity)

≤C||u||2V

By density, the trace g is thus defined for any u∈V with ||g||L2(∂Ωin) ≤C||u||V . Taking p=

p(ξ′)= g(ξ′) we observe by a similar calculation that ||p||L2(Ω) ≤C||u||V so that p∈G. Bydefinition q=u−p∈L.

Appendix B. On the choice of the finite element space Lh.Let Ω be the rectangle (0,Lx)×(0,Ly) and the anisotropy direction be constant and aligned with

the y-axis: b=(0,1). Let us use the Qk finite elements on a Cartesian grid, i.e. take some basis functionθxi(x), i=0, . . .,Nx and θyj (y), j=0, . . .,Ny and define the complete finite element space Xh (withoutany restrictions on the boundary) as spanθxi(x)θyj (y) 0≤ i≤Nx, 0≤ j≤Ny. The following subspaceis then used for the approximation of the unknowns p,q,l∈V

Vh = vh ∈Xh/vh|∂ΩD =0.

We want to prove that taking for the approximation of λ,µ∈L the space Lh under the form

Lh = λh ∈Xh/λh|∂Ωin =0 , (2.1)

25


leads to an ill posed problem (3.1).

Claim There exist λh ∈Lh, λh 6=0, such that a||(λh,ph)=0 for all ph ∈Vh. In fact there are exactly2Ny linearly independent functions having this property.

Remark B.1. In the continuous case, the equation

a||(p,λ)=0 , ∀p∈V ,

implies λ=0 by density arguments. These density arguments are lost when discretizing the spaces Vresp. L.Proof of the Claim. We can suppose that the basis functions θij(x,y) := θxi(x)θyj (y) are enumeratedso that θij(0,y)=0 for all i≥ 1 and θ0j(0,y) 6=0. Hence for all ph =

∑pijθij ∈Vh, the coefficients satisfy

p0j =0 since the part of the boundary x=0 is in ∂ΩD. Let M =(mik)0≤i,k≤Nx be the mass matrixin the x-direction: mik =

∫θxi(x)θxk

(x)dx. This matrix is invertible, hence there is a vector a∈RNx+1

that solves Ma= e with e∈RNx+1, e=(1,0, . . .,0)t. Take any fixed integer j, 1≤ j≤Ny and defineλh ∈Lh as λh =

∑aiθij . Then, for all ph =

∑pklθkl ∈Vh we have

a||(λh,ph)=∑

i,k,l

aipkl

∫

Ω

∂θij

∂y

∂θkl

∂ydxdy

=∑

i,k,l

aipkl

∫ Lx

0

θxi(x)θxk(x)dx

∫ Ly

0

θ′yj(y)θ′yl

(y)dy

=∑

k,l

δk0pkl

∫ Ly

0

θ′yj(y)θ′yl

(y)dy=0.

As we can do this for all (i,j), i=0, 1≤ j≤Ny and in the same manner for all (i,j), i=Nx, 1≤ j≤Ny,there are 2Ny linearly independent functions with the property a||(λh,ph)=0 for all ph ∈Vh.

We see now that the system (3.1) with zero right hand side f =0 possesses non-zero solutions(pε

h, λεh, q

εh, l

εh, µ

εh)= (0,λj

h,0,0,0) where λjh is any of the functions constructed in the preceding para-

graph. It means that (3.1) is ill posed, i.e. the corresponding matrix is singular.

Acknowledgement. This work has been supported by the Marie Curie Actions of the Euro-pean Commission under the contract DEASE (MEST-CT-2005-021122), by the French ’Commissariata l’Energie Atomique (CEA)’ under contracts ELMAG (CEA-Cesta 4600156289), and GYRO-AP(Euratom-CEA V 3629.001), by the Agence Nationale de la Recherche (ANR) under contract IODISEE(ANR-09-COSI-007-02), by the ’Fondation Sciences et Technologies pour l’Aeronautique et l’Espace(STAE)’ under contract PLASMAX (RTRA-STAE/2007/PF/002) and by the Scientific Council of theUniversite Paul Sabatier, under contract MOSITER. Support from the French magnetic fusion pro-gramme ’federation de recherche sur la fusion par confinement magnetique’ is also acknowledged. Theauthors would like to express their gratitude to G. Gallice and C. Tessieras from CEA-Cesta for bring-ing their attention to this problem, to G. Falchetto, X. Garbet and M. Ottaviani from CEA-Cadarache,for their constant support to this research programme.

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28


AN ANISOTROPIC ERROR ESTIMATOR FOR THE CRANK-NICOLSON METHOD : APPLICATION TO

A PARABOLIC PROBLEM

par

ALEXEI LOZINSKI, MARCO PICASSO AND VIRABOUTH PRACHITTHAM

paru dans

SIAM Journal of Scientific Computing

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. SCI. COMPUT. c© 2009 Society for Industrial and Applied MathematicsVol. 31, No. 4, pp. 2757–2783

AN ANISOTROPIC ERROR ESTIMATOR FOR THECRANK–NICOLSON METHOD: APPLICATION TO A PARABOLIC

PROBLEM∗

ALEXEI LOZINSKI† , MARCO PICASSO‡ , AND VIRABOUTH PRACHITTHAM‡

Abstract. In this paper we derive two a posteriori upper bounds for the heat equation. Acontinuous, piecewise linear finite element discretization in space and the Crank–Nicolson methodfor the time discretization are used. The error due to the space discretization is derived usinganisotropic interpolation estimates and a postprocessing procedure. The error due to the timediscretization is obtained using two different continuous, piecewise quadratic time reconstructions.The first reconstruction is developed following G. Akrivis, C. Makridakis, and R. H. Nochetto [Math.Comp., 75 (2006), pp. 511–531], while the second one is new. Moreover, in the case of isotropicmeshes only, upper and lower bounds are provided as in [R. Verfurth, Calcolo, 40 (2003), pp. 195–212]. An adaptive algorithm is developed. Numerical studies are reported for several test cases andshow that the second error estimator is more efficient than the first one. In particular, the seconderror indicator is of optimal order with respect to both the mesh size and the time step when usingour adaptive algorithm.

Key words. anisotropic finite elements, a posterori error estimates, Crank–Nicolson

AMS subject classifications. 65M15, 65M50, 65M60

DOI. 10.1137/080715135

1. Introduction. A posteriori error analysis is at the base of refinement/coarsening procedures in mesh adaptivity techniques. An impressive amount of workis available for a huge number of problems, as evidenced by the reviews [AO00, Ver96,BR03, BS01, Han05] and the references therein. The goal of a posteriori error esti-mates is to derive an easily computable bound of the error in order to ensure globalcontrol of the solution. The theory is particularly well developed in the case of ellipticproblems; see, for instance, [AO93, AN98, Ver94, BV00]. In the case of parabolic prob-lems, most papers deal with the Euler implicit time discretization (see, for instance,[Pic98, BBM05, CF04]), or higher order discontinuous Galerkin methods [EJ91, EJ95].However, little attention was paid to the popular Crank–Nicolson method for parabolicproblems. In [Ver03], the θ-scheme was considered for the time discretization of theheat equation and a posteriori bounds were derived applying standard energy tech-niques for a continuous, piecewise linear approximation in time. However, when con-sidering the case θ = 1/2, which corresponds to the Crank–Nicolson scheme, the errorestimator corresponding to the energy norm was of suboptimal order with respect tothe time step. In order to restore the appropriate second order of convergence, itwas suggested in [AMN06] to work with a continuous, piecewise quadratic polynomialfunction in time rather than a piecewise linear one. The so-called Crank–Nicolsonreconstruction was then introduced for a semidiscrete time discretization of a general

∗Received by the editors February 5, 2008; accepted for publication (in revised form) March 30,2009; published electronically June 25, 2009.

http://www.siam.org/journals/sisc/31-4/71513.html†Universite Paul Sabatier, Institut de Mathematiques de Toulouse, 118 route de Narbonne, 31062

Toulouse Cedex 9, France ([email protected]).‡Institut d’Analyse et Calcul Scientifique, EPFL-SB-IACS-ASN, Ecole Polytechnique Federale de

Lausanne, CH-1015 Lausanne, Switzerland ([email protected], [email protected]).The work of the third author was supported by the Swiss National Science Foundation.

2757



2758 A. LOZINSKI, M. PICASSO, AND V. PRACHITTHAM

parabolic problem. This approach was generalized in [AMN07] to Runge–Kutta andGalerkin methods.

In the present paper, we are interested in the fully discrete situation, taking theexample of the linear heat equation ∂u/∂t − Δu = f discretized in space by con-tinuous piecewise linear finite elements and in time by the Crank–Nicolson method.We extend the results of [AMN06] to the fully discrete case by introducing piecewisequadratic (in time) reconstructions of the numerical solution and using them to ob-tain a posteriori error estimates. We consider two reconstructions. The first one is adirect transposition of the reconstruction from [AMN06] to our fully discrete setting,whereas the second one is new. It is based on a finite difference approximation of∂2u/∂t2 rather than an approximation of ∂f/∂t + Δ(∂u/∂t) and thus formally re-sembles the reconstruction from [AMN07]. A posteriori upper bounds are derived forboth reconstructions and are used to construct an adaptive algorithm for both thetime step and the space step. Numerical experiments show that the first error esti-mator does not lead to an optimal rate of convergence with respect to the time stepwhen using our adaptive algorithm, especially in situations when the error is mainlydue to the space discretization. On the other side, the second error estimator alwaysprovides a fair representation of the true error.

Another feature of our work is to use finite elements on anisotropic triangulations,that is to say, meshes having triangles with possibly large aspect ratio. The goal is toreduce, as much as possible, the number of degrees of freedom required to reach a givenlevel of accuracy. For this reason, anisotropic a posteriori error estimates have recentlyreceived much attention [Kun00, Kun02, KN03, KV00, AL96, Ape99, Sie96, BJP04],with the goal to derive error estimates for which the involved constants do not dependon the triangle’s aspect ratio. Here we pursue the anisotropic approach developedin [Pic98, Pic03, Pic06], which are, in turn, based on the anisotropic interpolationestimates derived in [FP03, FP01]. The results in this paper are thus presented inthe anisotropic framework, and the numerical tests are done on examples that leadnaturally to highly stretched meshes. Note, however, that our anisotropic a posteriorierror bounds can also be translated in the isotropic setting, i.e., when the triangulationsatisfies the minimum angle condition.

The outline of the paper is as follows. In the next section, we present the modelproblem and its time and space discretization. We will then introduce in section 3some definitions and notations relative to the mesh anisotropy. The a posteriori errorestimates for the two reconstructions are presented in section 4. Section 5 is devotedto the description of an adaptive algorithm in space and time. A numerical study iscarried out for several test cases in section 6.

2. The heat equation and its discretization. Consider a polygonal domainΩ of R2 with boundary ∂Ω. Given a final time T > 0, a function f : Ω × (0, T ) → R,and an initial condition u0 : Ω → R, we consider the following problem: find u :Ω × (0, T ) → R such that

(2.1)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∂u

∂t− Δu = f in Ω × (0, T ),

u = 0 on ∂Ω × (0, T ),

u(·, 0) = u0 in Ω.

For the sake of simplicity, homogeneous Dirichlet boundary conditions are consid-ered. However, the analysis can be extended to mixed Dirichlet–Neumann boundary



AN ERROR ESTIMATOR FOR THE CRANK–NICOLSON METHOD 2759

conditions; see Example 6.2 at the end of the paper. We suppose henceforth thatf ∈ L2(0, T ; H−1(Ω)), u0 ∈ L2(Ω), and seek (see, for example, [DL92]) a solutionu ∈ W with

(2.2) W = w ∈ L2(0, T ; H10 (Ω)) and ∂w/∂t ∈ L2(0, T ; H−1(Ω))

such that u(·, 0) = u0 and

(2.3)

⟨∂u

∂t, v

⟩+

∫

Ω

∇u · ∇v dx = 〈f, v〉 ∀v ∈ H10 (Ω) and a.e. t ∈ (0, T ),

where < ·, · > denotes the duality pairing between H−1(Ω) and H10 (Ω). In order to

describe the time discretization corresponding to (2.3), we introduce a partition of theinterval [0, T ] into subintervals In = [tn−1, tn], n = 1, . . . , N , such that 0 = t0 < t1 <· · · < tN = T , and we denote the time steps by τn = tn − tn−1. For any 0 < h < 1, letTh be a conforming triangulation of Ω into triangles K (not necessarily satisfying theminimum angle condition) with diameter hK less than h. We define by Vh the usualfinite element space of continuous, piecewise linear functions on Th:

Vh =vh ∈ C0(Ω); vh|K ∈ P1; ∀K ∈ Th

,

and we set

V 0h = Vh ∩ H1

0 (Ω).

We suppose that f ∈ C0(0, T ; L2(Ω)) and u0 ∈ C0(Ω). We set fn(·) = f(·, tn), andwe compute u0

h = rhu0, where rh is the Lagrange interpolant corresponding to V 0h .

The Crank–Nicolson scheme consists of seeking unh ∈ V 0

h for all n = 1, . . . , N suchthat ∀vh ∈ V 0

h ,

(2.4)∫

Ω

unh − un−1

h

τnvh dx +

1

2

∫

Ω

(∇un

h + ∇un−1h

)· ∇vh dx =

1

2

∫

Ω

(fn + fn−1)vh dx.

Throughout the paper the following notations will be used for n = 1, . . . , N :

∂nuh =un

h − un−1h

τn, u

n−1/2h =

1

2(un

h + un−1h )

and

∂nf =fn − fn−1

τn, fn−1/2 =

1

2(fn + fn−1).

With these notations, we can rewrite (2.4) as

(2.5)

∫

Ω

∂nuh vh dx +

∫

Ω

∇un−1/2h · ∇vh dx =

∫

Ω

fn−1/2vh dx,

for all vh ∈ V 0h . As in [Pic98], we introduce the continuous, piecewise linear approxi-

mation in time defined for all t ∈ In by

uhτ(x, t) =t − tn−1

τnun

h +tn − t

τnun−1

h(2.6)

= un−1/2h + (t − tn−1/2)∂nuh,




where tn−1/2 = (tn + tn−1)/2. So for all vh ∈ V 0h , (2.4) or (2.5) can be rewritten as

∫

Ω

∂nuh vh dx +

∫

Ω

∇uhτ · ∇vh dx =

∫

Ω

fn−1/2vh dx(2.7)

+ (t − tn−1/2)

∫

Ω

∇∂nuh · ∇vh dx.

3. Anisotropic finite elements. In order to describe the mesh anisotropy,we introduce some definitions and properties taken from [FP03, FP01]. Alternativenotations can be found using the framework of [Kun00, KV00]. For any triangle K

of Th, we consider TK : K → K, the affine application that maps the triangle K intoK. Let MK ∈ R2×2 and tK ∈ R2 be the matrix and the vector defining such a map,and we have

x = TK(x) = MK x + tK ∀x ∈ R2.

Since MK is invertible, it admits a singular value decomposition

MK = RTKΛKPK ,

where RK and PK are orthogonal and ΛK is diagonal with positive entries. In thefollowing we set

(3.1) ΛK =

(λ1,K 0

0 λ2,K

)and RK =

(rT1,K

rT2,K

)

with the choice λ1,K ≥ λ2,K . We refer to section 2 of [Pic06] for examples of such atransformation.

Let us now recall some results on interpolation on anisotropic meshes proved in[FP01, FP03]. The classical minimum angle condition is not required in this context.However, for each vertex, the number of neighboring vertices should be bounded fromabove, uniformly with respect to the mesh size h. There is another restriction onthe mesh (see [Pic03] for a rigorous definition and illustrations) that prevents, looselyspeaking, the stretching directions r1,K , r2,K from changing too abruptly between theadjacent triangles of the mesh. In the rest of this paper we suppose that the familyTh meets the above-mentioned restrictions. In practice, the BL2D anisotropic meshgenerator [BL96] that we have used meets these restrictions.

Proposition 3.1 ([FP01, FP03]). Let Ih : H10 (Ω) → V 0

h be the Clement inter-polation operator [Cle75]. There is a constant C independent of the mesh size andaspect ratio such that, for any v ∈ H1(Ω) and any K ∈ Th, we have

(3.2) ‖v − Ihv‖L2(K) + λ2,K‖∇(v − Ihv)‖L2(K) + λ1/22,K‖v − Ihv‖L2(∂K) ≤ CωK(v).

Here ωK(v) is defined by

ω2K(v) = λ2

1,K

(rT1,KGK(v)r1,K

)+ λ2

2,K

(rT2,KGK(v)r2,K

),

λi,K and ri,K are given by (3.1), and GK(v) is the following 2 × 2 matrix:

GK(v) =∑

T∈ΔK

⎛⎜⎜⎜⎝

∫

T

(∂v

∂x1

)2

dx

∫

T

∂v

∂x1

∂v

∂x2dx

∫

T

∂v

∂x1

∂v

∂x2dx

∫

T

(∂v

∂x2

)2

dx

⎞⎟⎟⎟⎠ ,

where K represents the set of triangles of Th having a common vertex with K.




Remark 3.2. The reader should note that similar interpolation error estimatescan be found in [Kun00, KV00]. For instance, estimate (3.2) is similar to estimates(11) and (12) in [Kun00] with different notations (λ1,Kr1,K plays the role of p1 in[Kun00]).

4. A posteriori error estimates.

4.1. Piecewise quadratic reconstructions of the numerical solution. Itwas observed in [AMN06] that a direct use of uhτ in an a posteriori error analysiswould lead to a suboptimal estimate. It was proposed there to work rather witha piecewise quadratic time reconstruction of the numerical solution. We now recallbriefly the construction of [AMN06] made for a Crank–Nicolson discretization of anabstract evolutionary equation

∂u

∂t+ Au = f(t)

posed in a Hilbert space V , where A is a positive definite, self-adjoint, linear operatoron V with the dense domain in V . We discretize the last equation in time as

(4.1)un − un−1

τn+ A

un + un−1

2=

f(tn) + f(tn−1)

2

and consider the linear interpolation

uτ (t) = un−1 + (t − tn−1)un − un−1

τn, t ∈ In

and the quadratic one

(4.2) uτ (t) = uτ (t)+1

2(t−tn−1)(t−tn)

(f(tn) − f(tn−1)

τn− A

un − un−1

τn

), t ∈ In;

see (3.5) in [AMN06]. The latter reconstruction allows an a posteriori error estimateto be obtained for (4.1), which is of optimal second order.

We return now to the heat equation (2.3) and its discretization (2.4) or equiv-alently (2.5). Comparing (2.5) and (4.1), we can interpret the operator A in (2.5)as the finite-dimensional approximation of −Δ in (2.5). This analogy allows us tointroduce the following quadratic reconstruction:

(4.3) uhτ (x, t) = uhτ (x, t) +1

2(t − tn−1)(t − tn)wn

h , t ∈ In, 1 ≤ n ≤ N,

where uhτ is defined in (2.6) and wnh ∈ V 0

h is defined by

(4.4)

∫

Ω

wnhvh dx =

∫

Ω

(∂nf vh − ∇∂nuh · ∇vh) dx ∀vh ∈ V 0h .

This reconstruction is the analog of (4.2) in the case of the heat equation discretizedin both space and time. We will refer to uhτ as the two-point reconstruction since itinvolves only un

h and un−1h . We will use it to construct an upper bound for the error

analogous to that of [AMN06]. However, considering the numerical results obtainedwith the time error estimator based on uhτ , we cannot conclude that it gets theoptimal rate of convergence with respect to the time step when using our adaptive




algorithm. That is why we propose an alternative quadratic reconstruction in thenext paragraph.

As a motivation, we observe that wnh in (4.3) is formally an approximation of

∂f/∂t + Δ∂u/∂t in the time slab In, and the latter is equal to ∂2u/∂t2 by (2.1). Itseems natural then to try to replace wn

h in (4.3) by a finite difference approximationof ∂2u/∂t2. We introduce thus

(4.5) uhτ (x, t) = uhτ (x, t) +1

2(t − tn−1)(t − tn)∂2

nuh, t ∈ In, 2 ≤ n ≤ N,

where

(4.6) ∂2nuh =

unh − un−1

h

τn− un−1

h − un−2h

τn−1

(τn + τn−1)/2.

Note that uhτ is again continuous, piecewise quadratic in time. We will refer to it asthe three-point quadratic reconstruction since it involves un

h, un−1h , and un−2

h . Notethat uhτ , uhτ , and uhτ coincide at all times t1, . . . , tN .

Remark 4.1. The reconstruction uhτ restricted to the time interval In is the uniquequadratic polynomial, which coincides with un−2

h , un−1h , un

h at time tn−2, tn−1, tn,respectively. Indeed, denoting the latter by Pn(x, t), we observe that it is equal touhτ at t = tn−1 and t = tn and uhτ is linear in time on In. Hence, necessarily

Pn(x, t) = uhτ (x, t) + Cn(x)(t − tn−1)(t − tn)

for some Cn ∈ V 0h . Then we find Cn deriving Pn twice in t and taking into account

that Pn(·, tn−2) = un−2h . We thus find that Cn = ∂2

nuh/2 and recover (4.5).Remark 4.2. The requirement that wn

h vanish on the boundary can seem incon-sistent with the nature of the finite element problem for it, which does not containany derivatives and thus does not require boundary conditions. We note, however,that wn

h can be interpreted as an approximation of ∂f/∂t + Δ∂u/∂t = ∂2u/∂t2. Asu is equal to zero on ∂Ω, its second derivative in time is also equal to zero there.Consequently, it is reasonable to define wn

h as an element of V 0h .

4.2. A posteriori error estimates on anisotropic meshes. We will nowpresent the two error indicators based on uhτ and uhτ and used subsequently in ouradaptive algorithm. In both cases, a standard energy technique is used that leads tocombined error indicators in space and time. The estimator in space is similar to theone considered in [Pic03, Pic06]. In what follows we keep the notations of sections 2and 4.1 and set e = u − uhτ , e = u − uhτ , and e = u − uhτ . We announce now ourmain results.

Theorem 4.3. Let f be the linear interpolant of f defined by

f(·, t) =t − tn−1

τnfn +

tn − t

τnfn−1, t ∈ In, 1 ≤ n ≤ N,

and suppose that the mesh is such that there exists a constant c independent of thetime step, mesh size, and aspect ratio such that

(4.7) λ21,K

(rT1,KGK(e)r1,K

)≤ cλ2

2,K

(rT2,KGK(e)r2,K

)∀K ∈ Th.




Then there is a constant C independent of the time step, mesh size, and aspect ratiosuch that

∫ T

0

‖∇e‖2L2(Ω) dt + ‖e(·, T )‖2L2(Ω) ≤ ‖e(·, 0)‖2L2(Ω)

+ C

N∑

n=1

∑

K∈Th

∫ tn

tn−1

(‖f − ∂nuh + Δuhτ‖L2(K)

+1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)ωK(e) dt

+

∫ tn

tn−1

∥∥∥f − f∥∥∥2

L2(K)dt +

τ5n

120‖∇wn

h‖2L2(K) +λ22,Kτ3

n

12‖wn

h‖2L2(K)

.

Here [·] denotes the jump of the bracketed quantity across an internal edge, [·] = 0 foran edge on the boundary ∂Ω, and n is the unit edge normal (in arbitrary direction).

Theorem 4.4. Let

(4.8) f = fn−1/2 + (t − tn−1/2)fn − fn−2

τn + τn−1, t ∈ In, 2 ≤ n ≤ N,

and suppose that the mesh is such that there exists a constant c independent of thetime step, mesh size, and aspect ratio such that

(4.9) λ21,K

(rT1,KGK(e )r1,K

)≤ cλ2

2,K

(rT2,KGK(e )r2,K

)∀K ∈ Th.

Then there is a constant C independent of the time step, mesh size, and aspect ratiosuch that

∫ T

t1‖∇e‖2L2(Ω) dt + ‖e(·, T )‖2L2(Ω) ≤

∥∥e(·, t1)∥∥2

L2(Ω)

+ C

N∑

n=2

∑

K∈Th

∫ tn

tn−1

(‖f − ∂nuh + Δuhτ‖L2(K)

+1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)ωK(e) dt +

∫ tn

tn−1

∥∥∥f − f∥∥∥2

L2(K)dt

+τ2

n−1τ3n

48+

τ5n

120

∥∥∇∂2nuh

∥∥2L2(K)

+λ22,Kτ3

n

12

∥∥∂2nuh

∥∥2L2(K)

.

Remark 4.5. The bounds in Theorems 4.3 and 4.4 are not traditional a posteriorierror estimates since they involve ωK(e) or ωK(e) and hence the gradient of theexact solution u. A way to approximate it by a computable quantity was proposedin [Pic03, Pic06]. The resulting error estimator was proved very efficient for severalproblems, in particular, the Poisson equation and the Euler discretization of the heatequation. We will also apply this technique in constructing our error estimators andthe adaptive algorithm for the Crank–Nicolson scheme in this paper.

Remark 4.6. As we will see in section 5, our anisotropic adaptive algorithm ensuresassumptions (4.7) and (4.9) to be fulfilled with c = 1. A similar assumption has beenmade in [Pic06] in order to prove a lower bound in the framework of the Laplaceproblem.




We will prove here only Theorem 4.4 because the proof of Theorem 4.3 can followthe same lines with even less technicalities. We first need the following result.

Proposition 4.7. Let uhτ be defined by (4.5) and f by (4.8), then we have

∫

Ω

∂uhτ

∂tvh dx+

∫

Ω

∇uhτ · ∇vh dx =

∫

Ω

fvh dx+(t−tn−1/2)τn−1

2

∫

Ω

∇∂2nuh · ∇vh dx,

for all vh ∈ V 0h and for any t ∈ In, 2 ≤ n ≤ N .

Proof. We choose any 2 ≤ n ≤ N and t ∈ In in this proof. We have from (4.5)that

(4.10)∂uhτ

∂t= ∂nuh + (t − tn−1/2)∂2

nuh.

Then, using (2.7), we have for all vh ∈ V 0h

∫

Ω

∂uhτ

∂tvh dx +

∫

Ω

∇uhτ · ∇vh dx

(4.11)

=

∫

Ω

fn−1/2 vh dx + (t − tn−1/2)

∫

Ω

∂2

nuh vh + ∇∂nuh · ∇vh

dx.

We invoke now (2.4) at the time tn−1

∫

Ω

un−1h − un−2

h

τn−1vh dx +

∫

Ω

∇(un−1h + un−2

h )

2· ∇vh dx =

∫

Ω

fn−1 + fn−2

2vh dx,

and we subtract it from (2.4) to obtain for all vh ∈ V 0h

∫

Ω

∂2

nuh vh + ∇(

unh − un−2

h

τn + τn−1

)· ∇vh

dx =

1

τn + τn−1

∫

Ω

(fn − fn−2)vh dx.

Thus

∫

Ω

∂2

nuh vh + ∇∂nuh · ∇vh

dx =

1

τn + τn−1

∫

Ω

(fn − fn−2)vh dx(4.12)

+τn−1

2

∫

Ω

∇∂2nuh · ∇vh dx.

It suffices now to insert (4.12) into (4.11) to obtain the result.




Proof of Theorem 4.4. We choose any 2 ≤ n ≤ N and t ∈ In. Using (2.7), (4.10),and Proposition 4.7, we obtain for all v ∈ H1

0 (Ω) and all vh ∈ V 0h∫

Ω

∂e

∂tv dx +

∫

Ω

∇e · ∇v dx

=

∫

Ω

(f − ∂nuh) v dx −∫

Ω

∇uhτ · ∇v dx − (t − tn−1/2)

∫

Ω

∂2nuh v dx

=

∫

Ω

(f − ∂nuh) (v − vh) dx −∫

Ω

∇uhτ · ∇(v − vh) dx

+

∫

Ω

(f − ∂uhτ

∂t

)vh dx −

∫

Ω

∇uhτ · ∇vh dx − (t − tn−1/2)

∫

Ω

∂2nuh(v − vh)dx

=

∫

Ω

(f − ∂nuh) (v − vh) dx −∫

Ω

∇uhτ · ∇(v − vh) dx

− (t − tn−1/2)τn−1

2

∫

Ω

∇∂2nuh · ∇vh dx +

∫

Ω

(f − f) vh dx

− (t − tn−1/2)

∫

Ω

∂2nuh (v − vh) dx.

Then taking vh = Ihv and integrating by parts on each triangle K, we obtain

∫

Ω

∂e

∂tv dx +

∫

Ω

∇e · ∇v dx =

(4.13)

∑

K∈Th

∫

K

(f − ∂nuh + Δuhτ ) (v − Ihv) dx − 1

2

∫

∂K

[∇uhτ · n] (v − Ihv) dx

− (t − tn−1/2)τn−1

2

∫

K

∇∂2nuh · ∇Ihv dx +

∫

K

(f − f) Ihv dx

− (t − tn−1/2)

∫

K

∂2nuh (v − Ihv) dx

.

Taking v = e, rewriting ∇e · ∇e = 12 |∇e|2 + 1

2 |∇e|2 − 12 |∇e − ∇e|2 with

(4.14) ‖∇(e− e)‖2L2(Ω) = ‖∇(uhτ −uhτ )‖2L2(Ω) =1

4(t− tn−1)2(t− tn)2‖∇∂2

nuh‖2L2(Ω),

and using the Cauchy–Schwarz, Young, and Poincare inequalities together with Propo-sition 3.1, we have

1

2

d

dt

∫

Ω

|e|2 dx +1

2

∫

Ω

|∇e|2 dx +1

2

∫

Ω

|∇e|2 dx

≤∑

K∈Th

C1

[(‖f − ∂nuh + Δuhτ‖L2(K) +

1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)ωK(e)

+ |t − tn−1/2|∥∥∂2

nuh

∥∥L2(K)

ωK(e)

]

+

p

8(τn−1)

2(t − tn−1/2)2 +1

8(t − tn−1)2(t − tn)2

∥∥∇∂2nuh

∥∥2L2(K)

+p

2

∥∥∥f − f∥∥∥2

L2(K)+

1 + C22

2p‖∇Ihe‖2L2(K)

,




where C1 is the constant of Proposition 3.1, C2 is the constant in the Poincare in-equality, and p > 0 is arbitrary. Error equidistribution inequality (4.9) combined withProposition 3.1 implies that

(4.15) ωK(e) ≤ C3λ2,K ‖∇e‖L2(K) and ‖∇Ihe‖L2(K) ≤ C4 ‖∇e‖L2(K) .

We have then

1

2

d

dt

∫

Ω

|e|2 dx +1

2

∫

Ω

|∇e|2 dx +1

2

∫

Ω

|∇e|2 dx

(4.16)

≤∑

K∈Th

C1

[(‖f − ∂nuh + Δuhτ‖L2(K) +

1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)ωK(e)

+p

2λ22,K(t − tn−1/2)2

∥∥∂2nuh

∥∥2L2(K)

]+

C1C23

2p‖∇e‖2L2(K)

+p

8τ2n−1(t − tn−1/2)2 +

1

8(t − tn−1)2(t − tn)2

∥∥∇∂2nuh

∥∥2L2(K)

+p

2

∥∥∥f − f∥∥∥2

L2(K)+

1 + C22

2p‖∇Ihe‖2L2(K)

.

Finally, use the second inequality of (4.15) in (4.16), choose p = C1C23 + C2

4 (1 + C22 ),

and integrate (4.16) between t = tn−1 and t = tn to obtain

∫ tn

tn−1

‖∇e‖2L2(Ω) dt + ‖e(·, tn)‖2L2(Ω) ≤∥∥e(·, tn−1)

∥∥2L2(Ω)

+ C∑

K∈Th

∫ tn

tn−1

(‖f − ∂nuh + Δuhτ‖L2(K) +

1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)ωK(e) dt

+

∫ tn

tn−1

∥∥∥f − f∥∥∥2

L2(K)dt +

τ2n−1τ

3n

48+

τ5n

120

∥∥∇∂2nuh

∥∥2L2(K)

+λ22,Kτ3

n

12

∥∥∂2nuh

∥∥2L2(K)

,

where C = max(1, 2C1, pC1, p). Summing up this inequality on n and noting thate(tn) = e(tn) ∀n leads to the final result.

4.3. A posteriori error indicators. As already noted in the previous section,the upper bounds derived in Theorems 4.3 and 4.4 are not traditional a posteriori errorestimates since they involve ωK(e) and ωK(e) and hence the exact unknown solutionu. Therefore, following [Pic03, Pic06], we introduce some Zienkiewicz–Zhu postpro-cessing in order to estimate these unknown quantities. From the theoretical viewpoint,on general unstructured meshes, the Zienkiewicz–Zhu error estimator [ZZ87, ZZ92] isproved to be equivalent to the true error; see, for instance, [Rod94, Car04] for isotropicmeshes and [KN03] for anisotropic meshes. The asymptotic exactness of this estimatorcan be justified whenever superconvergence occurs, that is, on parallel [Rod94, AO00]or mildly structured meshes [XZ04]. Zienkiewicz–Zhu postprocessing for anisotropicmeshes was used in [KN03] to approximate the so-called matching function and in




[Pic03, Pic06] to build triangulations with large aspect ratio. The numerical exper-iments showed that the Zienkiewicz–Zhu error estimator performs surprisingly well,even on general unstructured anisotropic meshes. Here, we follow this approach anduse this postprocessing method to derive our space error indicator. The interestedreader should note that alternative gradient recovery methods based on least squarefitting have been proposed in [NZ04, ZN05] and have proved to be even more robustthan Zienkiewicz–Zhu postprocessing.

Our Zienkiewicz–Zhu error estimator is defined to be the difference between ∇uhτ

and an approximate L2(Ω) projection onto Vh:

ηZZ(uhτ ) =

(ηZZ1 (uhτ )

ηZZ2 (uhτ )

)=

⎛⎜⎜⎝

(I − Πh)

(∂uhτ

∂x1

)

(I − Πh)

(∂uhτ

∂x2

)

⎞⎟⎟⎠ ,

where Πh(∇uhτ ) ∈ Vh is defined by its values at each vertex P as

Πh(∇uhτ )(P ) =

⎛⎜⎜⎝

Πh

(∂uhτ

∂x1

)(P )

Πh

(∂uhτ

∂x2

)(P )

⎞⎟⎟⎠ =

1∑

K∈ThP∈K

|K|

⎛⎜⎜⎜⎜⎜⎝

∑

K∈ThP∈K

|K|(

∂uhτ

∂x1

)

|K

∑

K∈ThP∈K

|K|(

∂uhτ

∂x2

)

|K

⎞⎟⎟⎟⎟⎟⎠

.

Then, our error indicator is obtained by replacing GK(e) in ωK(e) and GK(e) inωK(e) by GK(uhτ ) defined for any vh ∈ Vh by

GK(vh) =

⎛⎜⎜⎝

∫

K

(ηZZ1 (vh)

)2dx

∫

K

ηZZ1 (vh)ηZZ

2 (vh) dx

∫

K

ηZZ1 (vh)ηZZ

2 (vh) dx

∫

K

(ηZZ2 (vh)

)2dx

⎞⎟⎟⎠ .

We define the anisotropic space error estimator ηA as

ηA =

(N∑

n=1

∑

K∈Th

(ηA

K,n(uhτ ))2)1/2

,

where the contributions ηAK,n are defined on each triangle K of Th and each time

interval In as

(ηA

K,n(uhτ ))2

=

∫ tn

tn−1


1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)

×(

λ21,K

(rT1,KGK(uhτ )r1,K

)+ λ2

2,K

(rT2,KGK(uhτ )r2,K

))1/2

dt.

(4.17)

We introduce now two time error estimators: ηT corresponding to the two-pointreconstruction uhτ (cf. Theorem 4.3) and ηT corresponding to the three-point recon-struction uhτ (cf. Theorem 4.4) defined, respectively, by

ηT =

(N∑

n=1

(ηT

n (uhτ ))2)1/2

and ηT =

(N∑

n=2

(ηT

n (uhτ ))2)1/2

.




The contributions ηTn and ηT

n are computed on each time interval In, n ≥ 1, via

(4.18)(ηT

n (uhτ ))2

=

∫ tn

tn−1

∥∥∥f − f∥∥∥2

L2(Ω)dt +

τ5n

120‖∇wn

h‖2L2(Ω) +∑

K∈Th

λ22,Kτ3

n

12‖wn

h‖2L2(K) ,

and for n ≥ 2,

(4.19)

(ηT

n (uhτ ))2

=

∫ tn

tn−1

∥∥∥f − f∥∥∥2

L2(Ω)dt +

τ2n−1τ

3n

48+

τ5n

120

∥∥∇∂2

nuh

∥∥2L2(Ω)

+∑

K∈Th

λ22,Kτ3

n

12

∥∥∂2nuh

∥∥2L2(K)

.

In our implementation, all the integrals between tn−1 and tn are approximated by themidpoint rule.

In order to measure the quality of our estimators, the estimated error is comparedto the true error introducing the so-called effectivity index, ei. Thus, we define thefollowing effectivity indices in both space

eiA =ηA

‖e‖L2(0,T ;H1(Ω))

and time

eiT

=ηT

‖e‖L2(0,T ;H1(Ω)), ei

T=

ηT

‖e‖L2(t1,T ;H1(Ω)), eiT =

ηT

‖e‖L2(0,T ;H1(Ω)).

We will also check the behavior of the Zienkiewicz–Zhu error estimator. We thusintroduce the corresponding global estimator and the effectivity index

ηZZ =

(N∑

n=1

∑

K∈Th

∫ tn

tn−1

∫

K

|ηZZ(uhτ )|2 dx dt

)1/2

and eiZZ =ηZZ

‖e‖L2(0,T ;H1(Ω)).

4.4. A posteriori error estimates on isotropic meshes. We deviate nowfrom the framework of anisotropic meshes to the more traditional case of isotropicones; i.e., we consider in this subsection a family of triangulations Th satisfying theminimum angle condition, so that λ1,K λ2,K hK ≤ h for all K ∈ Th. A much morecomplete theory of a posteriori error estimation can be constructed for the Crank–Nicolson method in this case. Indeed, the error indicator in space can be rewrittenin a closed form (no longer involving the unknown exact solution). Moreover, upperand lower bounds of optimal order can be proved for the space-time error estimatorprovided the error is measured with respect to the quadratic reconstructions uhτ oruhτ , and we choose a slightly stronger norm than before, namely,

|| · ||2W = || · ||2L2(0,T ;H10 (Ω)) +

∥∥∥∥∂·∂t

∥∥∥∥2

L2(0,T ;H−1(Ω))

.

Note that this is the natural norm of the space W introduced in (2.2), and we have||u||L∞(0,T ;L2(Ω)) ≤ ||u||W for all u ∈ W . Such a norm was used in [Ver03] to obtainupper and lower bounds for the θ-scheme. More precisely, it was proved that the error




||u − uhτ ||W was bounded above and below by some error estimator being of orderO(h + τ), even for the Crank–Nicolson scheme θ = 1/2. Here, we can prove that theerrors ||e||W = ||u − uhτ ||W or ||e||W = ||u− uhτ ||W are bounded above and below byan error estimator which is of optimal order O(h + τ2).

Let us outline this theory in the case of the three-point reconstruction uhτ , keepingin mind that similar statements can be made about the two-point reconstruction uhτ .Consider the isotropic space error indicator corresponding to (4.17),

(ηiso

K,n(uhτ ))2

=

∫ tn

tn−1

(h2

K ‖f − ∂nuh + Δuhτ‖2L2(K) + hK ‖[∇uhτ · n]‖2L2(∂K)

)dt,

and the time error indicator (4.19), in which we keep only the second order finitedifference in time of the approximated solution:

(ηT,iso

n (uhτ ))2

=τ2

n−1τ3n

48+

τ5n

120

∥∥∇∂2nuh

∥∥2L2(Ω)

.

Then we have the global upper bound of the error

(4.20) ||e||2L2(t1,T ;H10 (Ω)) +

∥∥∥∥∂e

∂t

∥∥∥∥2

L2(t1,T ;H−1(Ω))

≤ ||e(·, t1)||2L2(Ω)

+ CN∑

n=2

∑

K∈Th

(ηisoK,n)2+(ηT,iso

n )2+

∫ tn

tn−1

∥∥∥f − f∥∥∥2

H−1(Ω)dt+

∑

K∈Th

h2Kτ3

n

12

∥∥∂2nuh

∥∥2L2(K)

and the local in time lower bound for all n ≥ 2

(4.21)

1

3(ηT,iso

n )2 ≤ ||e||2L2(tn−1,tn;H10 (Ω)) +

∥∥∥∥∂e

∂t

∥∥∥∥2

L2(tn−1,tn;H−1(Ω))

+

∫ tn

tn−1

∥∥∥f − f∥∥∥2

H−1(Ω)dt.

We no longer have to assume the error equidistribution (4.9), but the constant C nowdepends on the mesh aspect ratio. The last term in (4.20) is dominated by (ηT,iso

n )2

provided h = O(τ2), and thus it can be neglected in practice.The proof of (4.20) is essentially the same as that of Theorem 4.4. Indeed, using

standard estimates for the Clement interpolation Ih on isotropic meshes in (4.13), wearrive at

∫

Ω

∂e

∂tv dx +

∫

Ω

∇e · ∇v dx

(4.22)

≤

C∑

K∈Th

(h2

K ‖f − ∂nuh + Δuhτ‖2L2(K) + hK ‖[∇uhτ · n]‖2L2(∂K)

)

+ (t − tn−1/2)2τ2n−1

4||∇∂2

nuh||2L2(Ω) + ||f − f ||2H−1(Ω)

+ (t − tn−1/2)2∑

K∈Th

h2K ||∂2

nuh||2L2(K)

12

× ||v||H1(Ω), ∀v ∈ H10 (Ω), t ∈ [tn−1, tn].




Table 4.1Convergence results using uniform isotropic meshes and constant time steps, case (a).

h τ(∫ T

0

∫Ω |∇e|2

)1/2eiZZ eiA ei

Tei

T

0.0125 0.025 0.17 0.10 0.25 17.23 14.050.00625 0.025 0.17 0.05 0.125 17.35 14.150.003125 0.025 0.17 0.024 0.06 17.39 14.170.0125 0.0125 0.047 0.44 1.09 16.25 13.220.00625 0.0125 0.043 0.24 0.60 17.85 14.530.003125 0.0125 0.042 0.12 0.30 18.42 14.92

Setting v = e, integrating in time and noting (4.14) yields estimate (4.20) for||e||2

L2(t1,T ;H10 (Ω))

. In a similar manner, we can take v in (4.22) such that

∫

Ω

∂e

∂tv =

∥∥∥∥∂e

∂t

∥∥∥∥H−1(Ω)

||v||H1(Ω)

and simplify by ||v||H1(Ω). We thus obtain estimate (4.20) for ‖∂e/∂t‖L2(t1,T ;H−1(Ω)).On the other hand, the identity in Proposition 4.7 can be rewritten as

1

2

(−(t − tn−1/2)τn−1 − (t − tn−1)(t − tn)

)∫

Ω

∇∂2nuh · ∇vh dx

=

∫

Ω

∂e

∂tvh dx +

∫

Ω

∇e · ∇vh dx −∫

Ω

(f − f)vh dx.

This is valid for any vh ∈ V 0h . In particular, taking vh = ∂2

nuh and integrating fromtn−1 to tn we obtain the lower bound (4.21).

4.5. A numerical study of the error estimators with uniform time stepsand mesh size. We study here the effectivity indices corresponding to the two errorestimators ηT and ηT on several test cases for which the error comes from either thespace discretization or the time discretization, or from both of them. We consider theproblem (2.1) with Ω = (0, 1) × (0, 1), the unit square, T = 1, and the exact solutionu given by

case (a) u(x, y, t) = sin(15πt) sin(πx) sin(πy),

case (b) u(x, y, t) = sin(πt/2) sin(10πx) sin(10πy),

case (c) u(x, y, t) = sin(πt) sin(πx) sin(πy).

Note that in case (a) the error should be mainly due to the time discretization, whilein case (b) it should be mainly due to space discretization. Case (c) provides anexample in which the error comes from both time and space discretization. Thenumerical results are reported in Tables 4.1–4.3. Isotropic meshes and constant timesteps are used in all the experiments of this section.

Referring to Table 4.1, we observe that, the computed error in the test case (a)is mainly due to the time discretization. Indeed, for a given time step, the error doesnot depend on the space step h, and for constant h, the error is divided by four whenthe time step τ is divided by two. Moreover the two time error estimators ηT and ηT

behave as the true error.Referring to Table 4.2, case (b), the error is now mainly due to space discretiza-

tion. We observe that for constant h, the error does not depend on the time step τ ,




Table 4.2Convergence results using uniform isotropic meshes and constant time steps, case (b).

h τ(∫ T

0

∫Ω |∇e|2

)1/2eiZZ eiA ei

Tei

T

0.00625 0.05 1.11 1.01 2.47 0.48 0.610.00625 0.025 1.11 1.01 2.47 0.12 0.160.00625 0.0125 1.11 1.01 2.47 0.03 0.040.003125 0.05 0.56 1.00 2.46 0.96 1.210.003125 0.025 0.56 1.00 2.46 0.24 0.310.003125 0.0125 0.56 1.00 2.46 0.06 0.078

Table 4.3Convergence results using uniform isotropic meshes and constant time steps, case (c).

h τ(∫ T

0

∫Ω |∇e|2

)1/2eiZZ eiA ei

Tei

T

0.025 0.05 0.044 1.00 2.46 0.52 0.420.00625 0.025 0.011 1.00 2.45 0.51 0.430.0015625 0.0125 0.0028 1.00 2.45 0.51 0.490.025 0.0125 0.044 1.00 2.46 0.032 0.0270.00625 0.00625 0.011 1.00 2.45 0.032 0.0340.0015625 0.003125 0.0028 1.00 2.45 0.032 0.067

that the space effectivity index stays close to 2.5, and that the Zienkiewicz–Zhu errorestimator is asymptotically exact. Thus, when the error is mainly due to the spacediscretization we can see that the space error estimator ηA behaves as the true error.

Referring to Table 4.3, case (c), the error comes now from both time and spacediscretizations. We observe that the error in the L2(0, T ; H1(Ω)) norm is O (h +τ2), that the space error estimator and the two time error estimators, ηT and ηT ,are equivalent to the true error, and that the Zienkiewicz–Zhu error estimator isasymptotically exact. Thus, using uniform time steps and mesh size, we observe thatthe two time error estimators, ηT and ηT , provide a good representation of the trueerror.

5. Adaptive algorithm. We now propose a time and space adaptive finiteelement algorithm. We will describe this algorithm while using (4.17) and (4.19).Since the time error estimator needs a solution un−2

h , we do not change the first timestep. For n ≥ 2, the idea is to build successive triangulations T n

h with possibly largeaspect ratio and to choose appropriate time steps τn so that the relative estimatederror in space and time in the L2(0, T ; H1(Ω)) norm is close to a preset toleranceTOL, for example,

(5.1) 0.875 TOL ≤((ηA)2 + (ηT )2

)1/2

(∫ T

0

∫

Ω

|∇uhτ |2 dx dt

)1/2≤ 1.125 TOL.

In doing so, we are beyond the scope of the theory developed in the previous sectionsince the mesh Th was not allowed to vary in time there. A more rigorous adaptiveprocedure would have to include the error due to the interpolations from T n−1

h to T nh .

We do not attempt to develop such a theory here and conjecture that the interpolationerror can be neglected provided the total number of remeshings does not depend onthe prescribed tolerance TOL, which will be the case in the numerical experiments.




Thus, in order to satisfy (5.1) we require that, for all n ≥ 1, the error indicatorin space is such that

0.8752 TOL2

2

∫ tn

tn−1

∫

Ω

|∇uhτ |2 dx dt ≤∑

K∈Th

(ηA

K,n(uhτ ))2

≤ 1.1252 TOL2

2

∫ tn

tn−1

∫

Ω

|∇uhτ |2 dx dt,(5.2)

and, for all n ≥ 2, the error indicator in time is such that

0.8752 TOL2

2

∫ tn

tn−1

∫

Ω

|∇uhτ |2 dx dt ≤∑

K∈Th

(ηT

K,n(uhτ ))2

≤ 1.1252 TOL2

2

∫ tn

tn−1

∫

Ω

|∇uhτ |2 dx dt.(5.3)

Note that we do not take into account the error in time corresponding to the firsttime step.

If (5.2) is not satisfied, the BL2D anisotropic mesh generator [BL96] is invoked toconstruct another mesh based on the space error indicator ηA. The P1 interpolationbetween the previous mesh T n−1

h and the new mesh T nh is carried out by the BL2D

mesh generator. Thus, BL2D provides us an interpolated solution rnh(un−1

h ) of un−1h

on the new mesh, where rnh is the Lagrange interpolant operator on T n

h . Then, aftereach remeshing, we seek un

h ∈ V 0h such that ∀vh ∈ V 0

h ,

(5.4)∫

Ω

unh − rn

h(un−1h )

τnvh dx+

1

2

∫

Ω

(∇un

h + ∇rnh(un−1

h ))

· ∇vh dx =1

2

∫

Ω

(fn + fn−1)vh dx.

Since BL2D requires the data to be given at the mesh vertices rather than triangles,we calculate at each vertex P the anisotropic error estimator defined as

ηAi,P,n (uhτ) =

⎛⎜⎝

∑

K∈ThP∈K

(ηA

i,K,n(uhτ ))4⎞⎟⎠

1/4

,

where ηAi,K,n is the local error estimator on triangle K in the direction ri,K defined as

(ηA

i,K,n(uhτ ))2

=

∫ tn

tn−1


1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)

×(

λ2i,K

(rTi,KGK(uhτ )ri,K

))1/2

dt

and a gradient matrix GP,n such that

GP,n(uhτ ) =∑

K∈ThP∈K

((ηA1,K,n(uhτ )

)4+(ηA2,K,n(uhτ )

)4)GK(uhτ ).

The eigenvectors of matrix GP provide the desired direction of the anisotropy atvertex P , and the directional error estimators ηA

i,P,n are used to prescribe the new




Set T 0h , u0

h, n = 1, t = 0 InitializationsDo while t < T Time loop

t := t + τn Increment the current time step

Compute unh on mesh T n−1

hDo for all triangles K of T n−1

hCompute r1,K , r2,K , λ1,K , λ2,K Directions and amplitudes of stretching

Compute GK Approximated error gradient matrixCompute ηT

K,n Time error estimator

Compute ηAK,n Space error estimator

End DoIf (5.2) and (5.3) are satisfied The mesh and the time step are correct

T nh := T n−1

h Same meshn := n + 1

ElseIf (5.2) is not satisfied Mesh adaptation

Do for all vertices P

Compute GP,n, ηA1,P,n, ηA

2,P,n Averaged values on vertices

Set directions of anisotropy r1,P and

r2,P to eigenvectors of GP

If ηA1,P,n is too small (resp. too large) Coarsening (resp. refinement)

the mesh size in the first direction in the direction r1,P .of anisotropy should beincreased (resp. decreased)

If ηA2,P,n is too small (resp. too large) Coarsening (resp. refinement)

the mesh size in the second direction in the direction r2,P

of anisotropy should beincreased (resp. decreased)

End DoBuild new anisotropic mesh T n

h using BL2DIf (5.3) is not satisfied Time adaptation

If∑

K∈Th

(ηT

K,n(uhτ ))2

is too small

(resp. too large) τn should beincreased (resp. decreased)

T n−1h := T n

ht := t − τn

End IfEnd Do

Fig. 5.1. Adaptive algorithm.

stretching amplitudes. In practice, the error is equidistributed in the two directionsof maximum and minimum stretching. This ensures conditions (4.7) and (4.9) to befulfilled with c = 1. Once an acceptable mesh has been constructed, we check (5.3)and modify the current time step if necessary. Our adaptive algorithm is summarizedin Figure 5.1. More technical details are available in [Pic03] in a similar context. Notethat we use the same algorithm with (4.17) and (4.18).

In order to monitor the mesh anisotropy, we define the maximum aspect ratioand the mean aspect ratio respectively defined by

ar = maxK∈Th

λ1,K

λ2,Kand ar =

∑

K∈Th

λ1,K

λ2,K

∑

K∈Th

1.

6. A numerical study of the adaptive algorithm. We apply here the adap-tive algorithm described in Figure 5.1 to two test cases exhibiting increasing levels of




Table 6.1Example 6.1. Convergence results using uniform isotropic meshes and constant time steps.

h τ(∫ T

0

∫Ω

|∇e|2)1/2

eiZZ eiA eiT

eiT

0.025 0.05 0.22 0.99 2.50 17.51 14.320.00625 0.025 0.055 0.99 2.50 18.15 14.790.0015625 0.0125 0.014 1.00 2.52 18.28 14.930.025 0.0125 0.22 1.04 2.59 1.15 0.940.00625 0.00625 0.056 1.00 2.52 1.14 0.930.0015625 0.003125 0.014 1.00 2.52 1.14 0.93

anisotropy. For the first test case we start all our adaptive simulations on an isotropic10 × 10 mesh and for the second test case with an anisotropic 100 × 2 mesh. More-over, we choose the time step τ1 = 0.05. We monitor the absolute error εabs in theL2(0, T ; H1(Ω)) norm, the relative error εrel in the same norm, the number of nodesnbn, maximum, and mean aspect ratio, all computed for the mesh at the final timeT . We also report the number of time steps nbτ needed to reach the final time andthe number of remeshings nbm occurred.

Example 6.1. Set Ω = (0, 1) × (0, 1), the unit square, T = 1, and choose u0 and fso that the solution u of (2.1) is given by

(6.1) u(x, y, t) = e−100r2(x,y,t),

where

r2(x, y, t) = (x − 0.3 − 0.4 β(t))2 + (y − 0.3 − 0.4 β(t))2

and

β(t) = 0.5 + 0.5 tanh

(t − 0.5

0.2

).

Thus, u is a Gaussian function, whose center moves from point (0.3, 0.3) at t = 0 topoint (0.7, 0.7) at t = 1 with a transport velocity 0.4β′(t)(1, 1)T peaking at t = 0.5.Before starting our adaptive algorithm, we first want to study the effectivity indicescorresponding to the two time error indicators ηT and ηT when using uniform timesteps and mesh size. In order to reduce the error due to the first time step, we replace∇u0

h, in (2.4), by rh∇u0. We have reported in Table 6.1 the results obtained whenh = 10τ2 and h = 160τ2. We can observe that the error in the L2(0, T ; H1(Ω))norm is O (h + τ2), that the space error estimator and the two time error estimators,ηT and ηT , are equivalent to the true error as their effectivity indices tend to aconstant value, and that the Zienkiewicz–Zhu error estimator is asymptotically exact.

Moreover, when h = 10τ2, the time effectivity indices eiT

and eiT

are similar tothose of Table 4.1. On the other hand, when h = 160τ2 the error due to the time

discretization is divided by 16 so as the time effectivity indices eiT

and eiT. In Figure

6.1, the values of wnh and ∂2

nuh are compared to ∂2u/∂t2 at time t = 0.5 along theaxis y = x and with h = τ = 0.00625. We observe that ∂2

nuh provides a smootherapproximation of ∂2u/∂t2 than wn

h . Indeed, we can notice slight oscillations whenapproaching ∂2u/∂t2 with wn

h .We now use the adaptive algorithm with first the three-point time error esti-

mator (4.19) and the anisotropic space error estimator (4.17). We have reported in




-500

-400

-300

-200

-100

0

100

200

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4-500

-400

-300

-200

-100

0

100

200

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Fig. 6.1. Example 6.1. ∂2nuh (solid line, left) and and wn

h (solid line, right) compared to∂2u/∂t2 (dotted line) at time t = 0.5 and with h = τ = 0.00625.

Table 6.2Example 6.1. True error and effectivity indices of the adapted solution at final time T = 1.

The three-point time error estimator (4.19) is used. Top: solving (2.4) if T n−1h = T n

h and (5.4) if

T n−1h = T n

h . Bottom: solving (6.2) if T n−1h = T n

h and (6.3) if T n−1h = T n

h .

TOL εrel εabs eiZZ eiA eiT

nbn nbτ nbm ar ar

0.25 0.085 0.15 0.88 2.11 2.07 854 48 18 2.41 8.340.125 0.043 0.76 0.88 2.12 2.02 3232 67 22 2.53 10.590.0625 0.022 0.038 0.88 2.11 2.02 12610 96 27 2.78 18.490.03125 0.011 0.019 0.87 2.09 1.95 46253 136 31 2.78 18.58

0.25 0.078 0.134 0.99 2.37 2.30 853 48 18 2.42 9.840.125 0.039 0.066 0.99 2.38 2.28 3189 67 23 2.54 10.470.0625 0.02 0.034 1.00 2.38 2.23 11714 96 27 2.79 14.820.03125 0.01 0.017 1.00 2.38 2.24 48062 135 30 2.92 18.72

Table 6.2 some numerical results with several values of tolerance TOL. We observein Table 6.2 (top) that the error is divided by two each time the tolerance TOLis, and that both the time error indicator ηT and the space error indicator ηA areof optimal order. Indeed, we note that ηT is of optimal order as the number oftime steps, nbτ , is approximatively multiplied by

√2 when TOL is divided by two

and that ηA is also of optimal order as the number of nodes, nbn, at final time, isapproximatively multiplied by four when TOL is divided by two. Moreover, both thetime and space effectivity indices tend to a constant value, which shows that ηT andηA are equivalent to the true error. However, we remark that the Zienkiewicz–Zhuerror estimator is not asymptotically exact. We have a value of eiZZ around 0.88,whereas we expect this value to be close to one. This discrepancy can be attributed tothe interpolation error between two successive meshes that is not taken into account inour theoretical estimates. Therefore, in order to recover its asymptotical convergencewe decide to replace, if T n−1

h is identical to T nh , ∇un−1

h , in (2.4), by its Z-Z recovery,Πn

h

(∇(un−1

h ))

and if T n−1h is different from T n

h , ∇rnh(un−1

h ), in (5.4), by its Z-Z

recovery, Πnh

(∇rn

h(un−1h )

). Thus, if T n−1

h and T nh are identical, we seek un

h ∈ V 0h such

that ∀vh ∈ V 0h ,

(6.2)∫

Ω

unh − un−1

h

τnvh dx+

1

2

∫

Ω

(∇un

h + Πnh

(∇(un−1

h )))

· ∇vh dx =1

2

∫

Ω

(fn + fn−1)vh dx,




and if T n−1h and T n

h are different, we seek unh ∈ V 0

h such that ∀vh ∈ V 0h ,

(6.3)

∫

Ω

unh − rn

h(un−1h )

τnvh dx +

1

2

∫

Ω

(∇un

h + Πnh

(∇rn

h(un−1h )

))· ∇vh dx

=1

2

∫

Ω

(fn + fn−1)vh dx.

We have reported in Table 6.2 (bottom) the corresponding results. We now observethat the Zienkiewicz–Zhu error estimator is asymptotically exact as its effectivityindex is close to one. All the previous observations remain unchanged, especially thegood behavior of both the time and space error indicators. We have reported in Figure6.2 the corresponding numerical simulation when the tolerance TOL = 0.125.

In Figure 6.3, we plot the evolution of the number of nodes and time step sizecorresponding to the numerical simulation presented in Figure 6.2. We observe thatthe number of nodes remains almost constant whereas the time step size decreasesuntil t = 0.5 and increases until the final time T = 1. The evolution of the time stepsize thus fits the transport velocity. The three-point time error estimator ηT seemsthen to be a good approximation of the true error. Finally, we plot in Figure 6.4 thevalue of ∂2

nuh when solving (2.4) if T n−1h = T n


h and whensolving (6.2) if T n−1

h = T nh and (6.3) if T n−1

h = T nh compared to ∂2u/∂t2 at time

t = 0.5 along the axis y = x and when TOL = 0.125. We note a better approximationof ∂2u/∂t2 when solving (6.2) if T n−1


h = T nh .

We now use the time error estimator (4.18) instead of (4.19) in the adaptivealgorithm, and we report in Table 6.3 (top) some numerical results with several valuesof the tolerance TOL. The results obtained show that the good behavior observed inthe case of constant time and space steps is not preserved when using our adaptivealgorithm. Indeed, we can see that the time error indicator ηT is not of optimal orderof convergence since the number of time steps, nbτ , is not multiplied by

√2 when

we divide TOL by two. For a tolerance TOL = 0.125, the number of the time stepswas 67 when using the time error indicator ηT , whereas it becomes 1860 for the timeerror indicator ηT . This significant difference shows that ηT tends to dramaticallyover-predict the true error. In Figure 6.5 (left) we plot the evolution of the timestep size when TOL = 0.125. We can see that the time step evolution is completelydifferent and shows an irregular progression with very small time steps around a valueof 0.005.

In order to reduce this overprediction, we decide to solve (6.2) if T n−1h = T n

h and(6.3) if T n−1

h = T nh . We have reported in Table 6.3 (bottom) the numerical results for

several values of TOL. We see that the number of time steps, nbτ , is not multiplied by√2 when we divide TOL by two so the optimal order of convergence is not recovered.

We plot in Figure 6.5 (right) the evolution of the time step size when TOL = 0.125.We note the same irregular profile, but its general aspect seems to be closer to theone presented in Figure 6.3. Indeed, neglecting these oscillations we roughly have thesame evolution as in Figure 6.3. We suspect that this irregular profile is due to theinterpolation error after each remeshing. Thus, we introduce the relative two-pointtime error estimator η n

rel and the relative error εnrel defined, respectively, by

(6.4) η nrel =

∑

K∈Th

(ηT

K,n(uhτ ))2

∫ tn

tn−1

∫

Ω

|∇uhτ |2 dx dt

and εnrel =

∫ tn

tn−1

∫

Ω

|∇e|2 dx dt

∫ tn

tn−1

∫

Ω

|∇uhτ |2 dx dt

,




Fig. 6.2. Example 6.1. Adapted meshes and solutions obtained with the time error estimator(4.19) and TOL = 0.125 when solving (6.2) if T n−1


h = T nh . From left to

right: time t = 0, 0.05, 0.5, and 1 (151, 2674, 3333, and 3189 nodes, respectively).

and we plot in Figure 6.6 the evolution of these two quantities for a tolerance TOL =0.125 when solving (6.2) if T n−1


h = T nh . We can see that

each remeshing results in a jump of the two-point time error estimator that causesthe irregular evolution. In Figure 6.7 we plot wn

h compared to ∂2u/∂t2 at time t = 0.5along the axis y = x and when TOL = 0.125. Two implementations of our adaptivealgorithm are reported here: first when solving (2.4) if T n−1


h =T n

h , and second when solving (6.2) if T n−1h = T n


h . We can seethat wn

h does not at all approach ∂2u/∂t2 if the Zienkiewicz–Zhu recovery method isnot used. The huge oscillations of wn

h observed in this case can explain why the timeerror indicator ηT extensively overpredicts the true error. A much better behavior




0

500

1000

1500

2000

2500

3000

3500

4000

0 0.2 0.4 0.6 0.8 1

num

ber

ofnodes

time

10−5

10−4

10−3

10−2

10−1

0 0.2 0.4 0.6 0.8 1

tim

est

ep

time

Fig. 6.3. Example 6.1. Number of nodes (left) and time step size (right) with respect to timet when solving (6.2) if T n−1


h = T nh . The time error estimator is given by

(4.19) and TOL = 0.125.

-500

-400

-300

-200

-100

0

100

200

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4-500

-400

-300

-200

-100

0

100

200

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Fig. 6.4. Example 6.1. ∂2nuh (solid line) compared to ∂2u/∂t2 (dotted line) at time t = 0.5 and

when TOL = 0.125. Left: We solve (2.4) if T n−1h = T n


h . Right: We solve

(6.2) if T n−1h = T n


h .

Table 6.3Example 6.1. True error and effectivity indices of the adapted solution at final time T = 1.

The two-point time error estimator (4.18) is used. Top: Solving (2.4) if T n−1h = T n

h and (5.4) if

T n−1h = T n

h . Bottom: Solving (6.2) if T n−1h = T n


h .


nbn nbτ nbm ar ar

0.25 0.074 0.128 1.00 2.39 2.25 885 620 15 2.38 9.520.125 0.038 0.066 1.00 2.40 2.14 2871 1860 19 2.70 16.450.0625 0.019 0.033 1.00 2.40 2.20 11924 4882 22 2.71 12.680.03125 0.01 0.017 1.00 2.39 2.20 48038 23851 26 2.95 19.51

0.25 0.078 0.133 1.00 2.40 2.28 852 71 15 2.44 10.660.125 0.038 0.066 1.00 2.41 2.20 3125 127 18 2.67 12.530.0625 0.019 0.033 1.00 2.41 2.17 12347 260 23 2.89 16.860.03125 0.0096 0.0166 1.00 2.40 2.18 47513 1695 25 3.01 20.57

of wnh is observed if we solve (6.2) when T n−1

h = T nh and (6.3) when T n−1

h = T nh .

However, this approximation also suffers from important spurious oscillations, and itis still not good enough to recover the optimal order of convergence of the time errorestimator ηT .




10−5

10−4

10−3

10−2

10−1

0 0.2 0.4 0.6 0.8 1

tim

est

ep

time

10−5

10−4

10−3

10−2

10−1

0 0.2 0.4 0.6 0.8 1

tim

est

ep

time

Fig. 6.5. Example 6.1. Time step size with respect to time t when solving (2.4) if T n−1h = T n

h

and (5.4) if T n−1h = T n

h (left) and when solving (6.2) if T n−1h = T n


h (right).The time error estimator is given by (4.18) and TOL = 0.125.

10−5

10−4

10−3

10−2

10−1

100

101

102

103

0 0.2 0.4 0.6 0.8 1

time

new meshη n

relnrel

Fig. 6.6. Example 6.1. Evolution of η nrel and εn

rel when solving (6.2) if T n−1h = T n

h and (6.3)

if T n−1h = T n

h . The time error estimator is given by (4.18) and TOL = 0.125.

-2000

-1000

0

1000

2000

3000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

-2000

-1000

0

1000

2000

3000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Fig. 6.7. Example 6.1. wnh (solid line) compared to ∂2u/∂t2 (dotted line) at time t = 0.5 and

when TOL = 0.125. Left: We solve (2.4) if T n−1h = T n


h . Right: We solve

(6.2) if T n−1h = T n


h .




We now consider another test case with more anisotropic finite elements. Thisexample illustrates the situation where the solution varies in only one direction andthus leads to very stretched meshes. We do all the computations by replacing ∇(un−1

h )and ∇rn

h(un−1h ) in (2.4) and in (5.4), , respectively, by its Zienkiewicz–Zhu recovery,

respectively, Πnh

(∇(un−1

h ))

and Πnh

(∇rn

h(un−1h )

).

Example 6.2. Set Ω = (0, 1) × (0, 1), the unit square, T = 1.5, and choose u0 andf so that the solution u of (2.1) is given by

(6.5) u(x, y, t) = 0.5 + 0.5 tanh

(x − 0.2 − 0.3 β2(t)

0.05

),

where

β2(t) =

⎧⎪⎪⎨⎪⎪⎩

0.5 + 0.5 tanh

(t − 0.4

0.05

)if t ≤ 0.75,

1.5 + 0.5 tanh

(t − 1.1

0.05

)if t > 0.75.

Note that we consider here a problem with mixed Dirichlet–Neumann boundary con-dition. Indeed, we impose u = 0 along the left side of Ω, u = 1 along the right side,and impose homogeneous Neumann boundary conditions along the top and bottomsides. Thus, the solution u exhibits an internal layer moving with a velocity 0.3β′

2(t)having sharp peaks centered at times t = 0.4 and t = 1.1. We apply the adaptivealgorithm with the three-point time error estimator ηT . Adapted meshes are shownin Figure 6.8 at time t = 0, 0.05, 1, and 1.5 and with a tolerance TOL = 0.03125. Weplot in Figure 6.9 (left) the corresponding time step size evolution. We see that thetime step size follows the velocity profile of the solution. Thus, for high velocity theadaptive algorithm chooses to use small time step and vice-versa. In Table 6.4 (top)we report numerical experiments with several values of the tolerance TOL. We ob-serve that ηZZ is asymptotically exact and that the space anisotropic error estimatorηA and the three-point time error estimator ηT are equivalent to the true error. Thenumber of time steps is approximatively multiplied by

√2 when TOL is divided by

two so that the optimal order is recovered.We now use (4.18) instead of (4.19) in the adaptive algorithm. Since the Dirichlet

boundary conditions are prescribed only on the part of the boundary in this example,the two-point time error estimator, namely, the definition of wn

h (4.4), should bechanged. We require here wn

h to vanish only along the left and right sides of Ω, andwe impose homogeneous Neumann boundary conditions along the top and bottomsides. The evolution of the time step size is reported in Figure 6.9 (right) for thesame simulation as in Figure 6.8. A chaotic evolution is observed. Indeed, we seemany perturbations, of which the most important of them appear during the twopeaks of acceleration and deceleration. We can also note that between the two peaks,when the velocity is very small, the time step size stays approximatively constantaround a value of 0.02 whereas we expect an increase. For solutions exhibiting highaspect ratio, the behavior of the two-point time error estimator seems then not to bea really good representation of the true error. In Table 6.4 (bottom) we report somenumerical results with several values of tolerance. We see that the two-point timeerror estimator is not of optimal order as the number of time iterations grows whenthe tolerance is divided by two.




Fig. 6.8. Example 6.2. Adapted meshes obtained with the time error estimator (4.19) and thetolerance TOL = 0.03125. From left to right: time t = 0, 0.05, 1, and 1.5 (320, 1042, 1015, and 892nodes, respectively).

10−4

10−3

10−2

10−1

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4

tim

est

ep

time

10−4

10−3

10−2

10−1

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4

tim

est

ep

time

Fig. 6.9. Example 6.2. Time step size with respect to time t with TOL = 0.03125. Left: Thetime error estimator is given by (4.19). Right: The time error estimator is given by (4.18).

Table 6.4Example 6.2. True error and effectivity indices of the adapted solution at final time T = 1.5.

Top: The time error estimator (4.19) is used. Bottom: The time error estimator (4.18) is used.


nbn nbτ nbm ar ar

0.125 0.03 0.095 0.99 2.87 2.68 155 142 84 63 2310.0625 0.015 0.046 0.99 2.89 2.91 348 201 52 108 5190.03125 0.007 0.022 0.99 2.96 2.99 892 285 52 165 7050.015625 0.004 0.012 1.00 2.88 2.71 4408 401 40 118 847

0.125 0.03 0.09 1.01 2.93 2.85 163 187 81 69 3050.0625 0.015 0.046 1.00 2.98 2.83 345 301 57 109 4790.03125 0.0075 0.023 1.00 2.95 2.96 892 909 52 169 9480.015625 0.0037 0.011 1.00 2.89 2.95 6054 10147 38 86 904




7. Conclusion. Numerical experiments show that ηT is a good representationof the true error even with solutions exhibiting high aspect ratio. Indeed, the ex-pected second order of convergence with respect to the time step has been recoveredexperimentally for several test cases. However, for the time error estimator ηT , wedid not manage to recover experimentally the optimal second order of convergence.The difficulty in approximating ∇wn

h involved in ηT seems to be the major prob-lem of this approach. We have extended these results to other problems such as thetime-dependent (convection dominated) convection diffusion [PP08] and simulationof electroosmotic flows in microfluidic devices [PPG09].

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SPARSE TENSOR-PRODUCT FOKKER-PLANCK-BASED METHODS FOR NONLINEAR

BEAD-SPRING CHAIN MODELS OF DILUTE POLYMER SOLUTIONS

par

PASCAL DELAUNAY, ALEXEI LOZINSKI, AND ROBERT G. OWENS

paru dans

CRM Proceedings and Lecture Notes

Centre de Recherches MathématiquesCRM Proceedings and Lecture Notes

Sparse tensor-product Fokker-Planck-based methodsfor nonlinear bead-spring chain models of dilute

polymer solutionsPascal Delaunay, Alexei Lozinski, and Robert G. Owens

Abstract. In this paper we present a spectral method for the solution of a high-dimensional Fokker-Planck equation describing the evolution of the congurationprobability density function for a dilute solution of idealized polymer chains. Thenumerical method draws on ideas found in the literature on sparse grids and al-lows solutions to be obtained at much lower cost, for comparable accuracy, thanusing a full tensor-product of one-dimensional basis functions. Numerical resultsfor homogeneous shear ow serve to illustrate the attractive features of the newmethod.

Keywords: Fokker-Planck equation, Brownian dynamics, bead-spring chains, sparsetensor product.

1. IntroductionThe dimensionless equation of conservation of linear momentum and of mass for

an incompressible polymeric solution may be written as

ReDv

Dt− β∇2v +∇p = ∇ · τ ,(1)

∇ · v = 0,(2)where v denotes the velocity, p the pressure, Re a Reynolds number and β is asolvent to total viscosity ratio. The tensor τ accounts for the contribution of thedissolved polymer to the Cauchy stress tensor and has traditionally been determinedas the solution to a closed-form dierential or integral equation. In acknowledgementof the unphysical (and, therefore, undesirable) assumptions that may be implied inmodelling the elastic stress tensor of a complex uid as the solution to a closed-form equation a more recent trend, in the so-called micro-macro approach [13],has been the computation of τ from a coarse-grained kinetic theory model for thepolymer conformations. Although most papers in the micro-macro rheological liter-ature have been concerned with the solution of stochastic dierential equations (see,for example, [12, 14, 19]), there is growing representation by advocates of numericalmethods specially adapted to the solution of an equivalent Fokker-Planck equation[5, 6, 7, 8, 9, 17, 23, 27].

Our own recent work has been directed towards developing Fokker-Planck-basedhigh-order methods for both dilute [4, 15] and concentrated [16] polymer solutionsthat are at the same time cheaper and more accurate than stochastic techniques.

c©0000 (copyright holder)

1


2 PASCAL DELAUNAY, ALEXEI LOZINSKI, AND ROBERT G. OWENS

The number of conguration space dimensions in all our papers to date has been low(≤ 3), however, and the cost advantage of Fokker-Planck-based methods is quicklylost once high-dimensional simulations are attempted. This is because of the so-calledcurse of dimension: standard full tensor-product bases for the probability densityfunction (pdf) ψ lead to computational eort that can grow exponentially with totaldimension. Higher-order conguration spaces are encountered in bead-spring chainmodels of polymer solutions, for example.

Our aim in this paper is to present a suitably inexpensive numerical methodfor the determination of the joint probability density function ψ for the internalcongurations of a bead-spring chain that Fokker-Planck-based numerical methodswill become competitive with stochastic techniques for (at least some) congurationdimensions exceeding 3. Recently, Venkiteswaran and Junk [24, 25] have employeda quasi-Monte Carlo method [18] for the simulation of the Fokker-Planck equationfor a dilute solution of polymers modelled by a bead-spring chain and undergoingsimple shear ow. Monte Carlo methods are a well-known approach for overcomingthe curse of dimension in high dimensional systems since the error estimate in termsof the number of nodes M (say) behaves like

√M , independently of the dimension.

Although the provable error estimate of Venkiteswaran and Junk [25] was no betterthan for a standard Monte Carlo method, their new method manifested less variancein the results and, for xed accuracy, the authors achieved an improvement in thecomputational time over the simple Monte Carlo method. Another way of attemptingto overcome the curse of dimension in the present context is by using sparse tensor-product spaces [22, 28] in order to represent ψ. In this class of methods a high-dimensional basis is derived from a one-dimensional basis by forming tensor productsof univariate formulae whose indices lie in a simplex. An extensive and recent reviewof sparse grid methods may be found in the survey paper of Bungartz and Griebel[3]. The advantages of using a sparse grid approach are seen in the cost savings madewhen compared with the use of traditional full tensor-product bases for the dependentvariables, without compromising on the order of accuracy. For example, whereas thenumber of degrees of freedom on a full tensor product nite element mesh of meshsize h in d dimensions grows like O(h−d), that associated with a sparse tensor productonly grows like O(h−1| log h|d−1). At the same time, the order of accuracy in the H1

norm of both methods is O(hp) for nite elements of degree p ≥ 1 and sucientlysmooth approximated functions [26].

In Section 2 we begin by explaining how the multi-dimensional Fokker-Planckequation is derived from the bead-spring micro-mechanical model. The eigenfunctionsof the Fokker-Planck operator for a single dumbbell are computed using a Galerkinspectral method in conguration space and we then employ both full and sparsetensor products of these eigenfunctions to represent the conguration pdf ψ for thefull chain. Exactly how this is done is described in Section 3. An implicit stochasticalgorithm is detailed in Section 4, and in the same section we explain how the elasticstress contribution to the total Cauchy stress is calculated on the basis of the solutionsto the stochastic and equivalent Fokker-Planck equations. The advantages of using asparse tensor product basis for ψ rather than the usual full tensor product for suchfunctions is showcased in Section 5 by comparing the costs of the two approachesagainst that of the Brownian dynamics simulation for simple shear ow at variousshear rates.


SPARSE TENSOR-PRODUCT FOKKER-PLANCK-BASED METHODS 3

2. Micromechanical model and governing equationsThe micromechanical model used to represent a single linear polymer chain in this

paper consists of (d+1) beads, each having the same mass m and joined consecutivelyby d nonlinear massless springs, as shown in Fig. 1. The equations of motion for thebead having position vector rj (j = 2, . . . , d) may be written

(3) md2rjdt2

= ζ

(v(rj)−

drjdt

)+ F (rj+1 − rj)− F (rj − rj−1) +Bj,

whereas for the beads at the ends the chain, (3) simplies to

md2r1dt2

= ζ

(v(r1)−

dr1dt

)+ F (r2 − r1) +B1,(4)

md2rd+1

dt2= ζ

(v(rd+1)−

drd+1

dt

)− F (rd+1 − rd) +Bd+1.(5)

In Eqns. (3)-(5) ζ is a friction coecient, v(rj) is the velocity of the solvent atthe point with position vector rj, F (rj+1 − rj) = Hf(| rj+1 − rj |)(rj+1 − rj) (forsome spring constant H and spring law f) is the force acting on the jth bead due totension in the spring of the jth segment and Bj denotes the Brownian forces actingon the jth bead. The acceleration terms in Eqns. (3)-(5) may be neglected (see [10]

(q )F

O

r

r r

r1

23

4

q

q

q1 2

3

x0

−

1

1

F (q )

Figure 1. Schematic of a bead-spring chain of d = 3 segments

for a justication) and no account is taken of any segment-segment or bead-beadinteraction apart from that which exists through the tension in the segment springs.In the present paper for simplicity, the chains are supposed to live on a plane and thevectors rj, Bj etc. are hence two-dimensional. Note that simulations of bead-springchains evolving in the 3-dimensional physical space should be feasible by the approachanalogous to that described below (cf. [4] for the case of a single dumbbell model).



The Brownian forces are approximated by a Wiener process:(6) Bjdt =

√2kBTζdWj,

where kB is Boltzmann's constant, T is the absolute temperature and Wj(t) =(Wj,1(t),Wj,2(t)) is a Gaussian process in 2 dimensions with(7) 〈Wj,k(t)〉 = 0, 〈Wj,k(t)Wj′,k′(t

′)〉 = δjj′δkk′ min (t, t′).

Introducing the jth connector vector(8) qj = rj+1 − rj, j = 1, . . . , d,

and the position vector of the centre of mass

(9) rc =1

d+ 1

d+1∑

j=1

rj,

we see that the equations of motion (3)-(5) may be rewritten in the form

dqj(t) =

[v(rj+1)− v(rj)−

1

ζ

d∑

k=1

AjkF (qk)

]dt+

√4kBT

ζdW q

j (t),(10)

drc(t) =1

d+ 1

d+1∑

j=1

v(rj)dt+

√2kBT

ζ(d+ 1)dW c(t),(11)

where W qj (t) =

Wj+1−Wj√2

and W c(t) = 1√d+1

∑d+1j=1 Wj(t) both have the same distri-

bution as Wj, and A is the Rouse matrix

A =

2 −1 (0)

−1. . . . . .. . . . . . −1

(0) −1 2

.

Suppose that the ow is homogeneous (i.e., ∇v is a constant matrix so that weare permitted to write v(rj+1) − v(rj) = ∇v · qj). Then, after scaling qk withl0 =

√kBTH

it may be shown (see Theorem 3.30 in Öttinger [19], for example) thatthe Fokker-Planck equation corresponding to (10)-(11) and describing the evolutionof the conguration pdf ψ is(12)∂ψ

∂t= Ld

FP (ψ) =d∑

k=1

Lkψ− 1

4λ

d∑

k=2

(Nk−1x Mk

x+Nk−1y Mk

y )ψ−1

4λ

d−1∑

k=1

(Nk+1x Mk

x+Nk+1y Mk

y )ψ.

where

Lk = ∇k · (−∇v · qk +1

2λF (qk) +

1

2λ∇k), M

kx =

∂

∂qk,x, Mk

y =∂

∂qk,y,

Nkx = Fx(qk) +

∂

∂qk,xand Nk

y = Fy(qk) +∂

∂qk,y.(13)

In (13) λ = ζ/4H is a conguration relaxation time,

(14) qk =

(qk,xqk,y

)and ∇k =

∂∂qk,x

∂∂qk,y

.



In the present paper the spring force law employed for each of the segments is thatof the FENE model [2]

(15) F (q) =q

1− |q|2b

,

where√b = qmax/l0 and qmax is the maximum extensibility of the segment spring.

3. A high-order sparse grid Fokker-Planck-based methodIn this section we describe how we generate numerical approximations to the

leading eigenfunctions of the Fokker-Planck operator L1FP and then employ a sparse

tensor product of these eigenfunctions to represent the conguration pdf ψ for thefull chain.

3.1. Construction of an eigenbasis. The connector vector q for a single FENEdumbbell (d = 1) lives in a disk of radius

√b and we may write the two cartesian

components (qx, qy) of q in terms of planar polar coordinates(16) qx = ρ cos θ, qy = ρ sin θ,

with ρ ∈ [0,√b] and θ ∈ [0, 2π]. The rst step that we take towards solving the

Fokker-Planck equation (12) numerically is the construction of an eigenbasis consist-ing of solutions ψ to the eigenproblem1

(17) L1FP ψ = λiψ.

We compute the leading N (say) eigenfunctions corresponding to the eigenvaluesλ1, . . . , λN having real parts that are smallest in absolute value: 0 ≥ Re(λ1) ≥Re(λ2) ≥ . . . ≥ Re(λN) ≥ . . . This is done using a Fourier-Legendre spectral Galerkinmethod in a manner to be described below.

We denote the even and odd2 parts of an eigenfunction ψ satisfying (17) by ψ0

and ψ1. Then(18) ψ = ψ0 + ψ1,

where ψ0 and ψ1 are unique and dened by

ψ0(ρ, θ) =ψ(ρ, θ) + ψ(ρ, θ + π)

2,

ψ1(ρ, θ) =ψ(ρ, θ)− ψ(ρ, θ + π)

2.(19)

We note that since the spaces of even and odd functions are invariant under theFokker-Planck operator L1

FP at equilibrium, both ψ0 and ψ1 are themselves eigen-functions of this operator, belonging to the same eigenspace as ψ.

Let us introduce the change of dependent variables ψ −→ α for the eigenproblem(17) where

(20) ψ0 =

(1− ρ2

b

)s

α0, ψ1 = ρ

(1− ρ2

b

)s

α1,

1The eigenvalues λi are not to be confused with the relaxation time λ in (12).2A function f(t, ρ, θ) will be called even if f(t, ρ, θ) = f(t, ρ, θ + π) and odd if f(t, ρ, θ) =

−f(t, ρ, θ + π)



α0 and α1 are, respectively, the even and odd parts of α, and s is a real parametergreater than unity. The motivation for the change of variables from ψ to α is rstly tocircumvent the numerical problems created by the unboundedness of F as ρ2 −→ b inthe term F (q)ψ of L(1)

FP (see Eqn. (12) when d = 1). Secondly, the factors multiplyingα0 and α1 in (20) ensure that ψ = 0 on ρ =

√b. Finally, if ψ is even we see from (20)

that ∂ψ/∂ρ = 0 identically at ρ = 0 and, if odd, that ψ = 0 at ρ = 0.Using the shorthand notation

(21) ψ0a(ρ) = ρa

(1− ρ2

b

)s

, a = 0, 1,

we see that we may write(22) ψ = ψ0

0α0 + ψ01α1,

and that therefore there exists an invertible operator R(1) (say) such that α = R(1)ψ

and dened for any function ψ of (t, q) having even and odd parts ψ0 and ψ1 as(23) R(1)ψ =

(ψ00

)−1ψ0 +

(ψ01

)−1ψ1.

The original eigenproblem (17) may now be written in terms of α as(24) L1

FP α = λiα,

where(25) L1

FP = R(1)L1FP

(R(1)

)−1,

or, in terms of the action of L1FP on the even and odd parts of α,

(26) L1FP αa =

(ψ0a

)−1 L1FP

(ψ0aαa

), a = 0, 1.

3.1.1. A spectral Galerkin method. In this paper α0 and α1 will be representedin terms of even and odd Fourier-Legendre series in [−1, 1]× [0, 2π] after performingthe change of variable ρ to η, where

ρ2 = b

(1 + η

2

).

This yields

(27) α =1∑

i=0

NR∑

k=0

2NF∑

l=i

αiklhk(η)Θil(θ),

where(28) Θil(θ) = (1− i) cos lθ + i sin lθ,

and hk(η)0≤k≤NRare Lagrange interpolating polynomials based on the (NR + 1)

Gauss-Legendre points in (−1, 1). From the representation (27) of α it may be seenthat the total number of Fourier modes used is 1 + 2NF + 2NF = 4NF + 1.

An algebraic system of equations corresponding to (24) is obtained using a Galerkinspectral method in which test functions of the form hm(η), hm(η)Θodd

il or hm(η)Θevenil

(0 ≤ m ≤ NR, 1 ≤ i ≤ 2, 1 ≤ l ≤ NF ) are employed. Integrals with respect to η areevaluated using the standard Gauss quadrature rule based on the Gauss-Legendrepoints, and integrals with respect to θ are evaluated analytically. The eigenvaluesand associated eigenvectors of the resulting (NR + 1)(4NF + 1)× (NR + 1)(4NF + 1)matrix are computed by rst reducing the matrix to upper Hessenberg form and thenusing a hybrid double-shifted LR-QR algorithm (see Haag and Watkins [11]). All of



this is done by a call to the standard Intel library routine EVLRG. We shall denotethe eigenfunctions of the single segment operator L1

FP that are computed in this wayby ϕk (k = 1, . . . , (NR + 1)(4NF + 1)) in the sequel.

3.2. A sparse tensor product spectral method for ψ(t, q1, . . . , qd). As ageneralisation of the (unique) decomposition (18) of the function ψ into a sum of itseven and odd parts it may be easily shown that a unique representation exists of anyfunction ψ of the independent variables t and q1, . . . , qd in terms of 2d new functionsψ

(d)p

(29) ψ(t, q1, . . . , qd) =1∑

p1,...,pd=0

ψ(d)p (t, ρ1, θ1, . . . , ρd, θd),

where p = (p1, . . . , pd) is a parity vector, the component pi being equal to zero if ψ(d)p

is even in the variable qi and pi equalling 1 if ψ(d)p is odd in qi, i.e. for k = 1, . . . , d

ψ(d)p (t, ρ1, θ1, . . . , ρk, θk + π, . . . , ρd, θd) = (−1)pkψ(d)

p (t, ρ1, θ1, . . . , ρk, θk, . . . , ρd, θd).

As in the change of variable (22) we now write down the solution ψ to the fullFokker-Planck equation (12) in terms of new functions α(d)

p :

ψ(t, q1, . . . , qd) =1∑

p1,...,pd=0

(d∏

j=1

ρpjj (1− ρ2

b)s

)α(d)p (t, ρ1, θ1, . . . , ρd, θd),

=1∑

p1,...,pd=0

ψ0pα

(d)p (t, ρ1, θ1, . . . , ρd, θd)(30)

where

(31) ψ0p(ρ1, . . . , ρd) =

d∏

j=1

ρpjj

(1− ρ2j

b

)s

=d∏

j=1

ψ0pj(ρj),

ψ0pj(ρj) being as dened in (21).In just the same way as we introduced the change of variables α = α0 + α1 in

Section 3.1 we now write down the new variable

(32) α =1∑

p1,...,pd=0

α(d)p (t, ρ1, θ1, . . . , ρd, θd),

in terms of which we will attempt to express the Fokker-Planck equation (12). Thisis easily done by observing that there exists an invertible operator R(d) such that(33) α = R(d)ψ,

where, given any function ψ of t and q1, . . . , qd written (uniquely) in the form (29)

(34) R(d)ψ =1∑

p1,...,pd=0

(ψ0p

)−1ψ(d)p (t, ρ1, θ1, . . . , ρd, θd).

Eqn. (12) therefore becomes

(35) ∂α

∂t= Ld

FP (α)



where(36) Ld

FP = R(d)LdFP

(R(d)

)−1.

The transformed Fokker-Planck equation may be written in full as(37)∂α

∂t= Ld

FPα =d∑

k=1

Lkα− 1

4λ

d∑

k=2

(Nk−1x Mk

x+Nk−1y Mk

y )α−1

4λ

d−1∑

k=1

(Nk+1x Mk

x+Nk+1y Mk

y )α,

where(38) Lk = R(d)Lk

(R(d)

)−1, Nk−1

x Mkx = R(d)Nk−1

x Mkx

(R(d)

)−1, etc.

Equation (37) is solved subject to an initial condition consisting of the equilibriumsolution. In terms of the pdf ψ this is

(39) ψ = ψ(eq) =

(b+ 2

2πb

)d d∏

k=1

(1− ρ2k

b

)b/2

,

which is equivalent (see (32)) to

(40) α = α(eq) = α(d)0 =

(b+ 2

2πb

)d d∏

k=1

(1− ρ2k

b

)b/2−s

.

3.2.1. A sparse grid spectral Galerkin method. We will now utilise the eigenfunc-tions ϕk described in Section 3.1 to generate a basis for the variable α (see Eqn.(32)). Dene a sequence of spaces V0, V1, V2, . . . such that(41) V0 = ∅,and(42) Vl = span ϕ1, . . . , ϕNl

,where Nl = min(2l − 1, (NR + 1)(4NF + 1)), l = 1, . . . , L and L is the integer denedsuch that, given NR and NF , 2L−1 − 1 < (NR + 1)(4NF + 1) ≤ 2L − 1.

Then the spaces V0, V1, V2, . . . are nested:V0 ⊂ V1 ⊂ V2 ⊂ . . .

and we may dene dierence spacesW1,W2,W3, . . . expressing the hierarchical excessby Wl = Vl \ Vl−1.

Letting i = (i1, . . . , id) be a d−dimensional multi-index we introduce the productfunction

(43) Φi(η1, θ1, . . . , ηd, θd) =d∏

j=1

ϕij(ηj, θj).

From the manner in which the eigenproblem (24) is solved numerically, each eigen-function ϕij turns out to be either odd or even (and even if they weren't we couldconstruct such a set). We seek to approximate α at level l in one of two ways. Therst is a full tensor product (FTP)

(44) α ≈∑

i:1≤‖i‖∞≤Nl

αi(t)Φi(η1, θ1, . . . , ηd, θd) ∈ Vl⊗

· · ·⊗

Vl,



and the second a sparse tensor product (STP) [22, 28](45)α ≈

∑

k:d≤‖k‖1≤d+l−1

∑

i∈Ik,dαi(t)Φi(η1, θ1, . . . , ηd, θd) ∈

⊕

k:d≤‖k‖1≤d+l−1

Wk1

⊗· · ·⊗

Wkd ,

where the index set Ik,d in (45) is dened to consist of those i such that ϕij ∈Wkj :(46) Ik,d = i ∈ Rd : 2kj−1 ≤ ij ≤ min(2kj − 1, (NR + 1)(4NF + 1)).As an example of the formulae (44) and (45) we take d = 2 and l = 2. In this case,the FTP representation (44) becomes

α ≈3∑

i1=1

3∑

i2=1

αi1,i2(t)ϕi1ϕi2 ,

whereas (45) would now readα ≈ α1,1ϕ1ϕ1 + α1,2ϕ1ϕ2 + α2,1ϕ2ϕ1 + α1,3ϕ1ϕ3 + α3,1ϕ3ϕ1.

Let us introduce a dual space V ∗l given by

V ∗l = spanf1, . . . , fNl

,where each function fi (i = 1, 2, . . . , Nl) in this dual basis is dened as(47) fi(v) = 〈ϕ∗

i , v〉1, ∀v ∈ Vl.

In (47) 〈·, ·〉1 is an inner product on Vl dened by

〈f, g〉1 =∫ 2π

θ=0

∫ √b

ρ=0

f(ρ, θ)g(ρ, θ) ρ dρdθ,

=b

4

∫ 2π

θ=0

∫ 1

η=−1

f(ρ(η), θ)g(ρ(η), θ)dηdθ,(48)

and the functions ϕ∗i Nl

i=0 ∈ Vl, chosen so that fi(ϕj) = δij. For this to be the case,we express ϕ∗

i as a linear combination of the basis functions ϕj:

(49) ϕ∗i =

Nl∑

j=1

aijϕj,

and see that

(50) 〈ϕ∗i , ϕk〉1 =

Nl∑

j=1

aij〈ϕj, ϕk〉1,

so that the Nl × Nl matrix A = (aij) should be chosen as the inverse of the Grammatrix G having (i, j)th component(51) gij = 〈ϕi, ϕj〉1.An algebraic set of discrete equations corresponding to (37) is found by using aGalerkin method as follows: The d−dimensional operator

∂

∂t− Ld

FP ,

is applied to either the full (44) or sparse (45) tensor product representation of α.Then the product is taken with test functions of the form(52) Φ∗

j = Πdl=1ϕ

∗jl,



(with j drawn from the same index set as in (44) or (45)) to get, for all suitable j,

(53) 〈∂α∂t

− LdFPα,Φ

∗j〉d = 0,

where 〈·, ·〉d generalises the inner product introduced in (48) to an integral computedover the domain in R2d: |qj| <

√b, j = 1, . . . , d.

Writing α in its form (44) or (45), (53) may be expressed as

(54)∑

i

dαi(t)

dt〈Φi,Φ

∗j〉d − αi(t)〈Ld

FPΦi,Φ∗j〉d = 0,

where the index set for i and j depends on whether an STP or FTP representationfor α is used. Using the orthogonality properties of the bases ϕ∗

i and ϕj (that is,by construction, fi(ϕj) = δij) we then arrive at

dαj(t)

dt−

d∑

k=1

Nl∑

ik=1

αj1...jk−1ikjk+1...jd〈Lkϕik , ϕ∗jk〉1

− 1

4λ

d∑

k=2

Nl∑

ik−1=1

Nl∑

ik=1

αj1...jk−2ik−1ikjk+1...jd

×[〈Nk−1

x ϕik−1, ϕ∗

jk−1〉1〈Mk

x ϕik , ϕ∗jk〉1

+ 〈Nk−1y ϕik−1

, ϕ∗jk−1

〉1〈Mky ϕik , ϕ

∗jk〉1]

− 1

4λ

d−1∑

k=1

Nl∑

ik=1

Nl∑

ik+1=1

αj1...jk−1ikik+1jk+2...jd

×[〈Mk

x ϕik , ϕ∗jk〉1〈Nk+1

x ϕik+1, ϕ∗

jk+1〉1

+ 〈Mky ϕik , ϕ

∗jk〉1〈Nk+1

y ϕik+1, ϕ∗

jk+1〉1]= 0.(55)

Discretization in time is eected via an implicit scheme where, denoting the vectorof unknowns by α, the partial time derivative is approximated by a simple forwarddierence

(56) ∂α

∂t≈ αn+1 −αn

∆t,

and α is evaluated at time step (n+1) in the terms in (37) corresponding to∑

k Lkα

and at time level n elsewhere. All of this leads to the fully discrete scheme

αn+1 −∆td∑

k=1

Lkαn+1 =αn − ∆t

4λ

d∑

k=2

(Nk−1x Mk

x + Nk−1y Mk

y )αn

− ∆t

4λ

d−1∑

k=1

(Nk+1x Mk

x + Nk+1y Mk

y )αn,(57)

where an overbar on a quantity indicates that it is a matrix. The coecient matrixof αn+1 in the scheme (57) is approximated (to O(∆t)2) in our code by the matrixequivalent of the operator

⊗dk=1

(I −∆tLk

).



4. Stochastic algorithm and stress calculatorIn homogeneous polymeric ows the probability density function ψ will be in-

dependent of the centre of mass of a chain rc. We therefore neglect the stochasticequation (11) for the centre of mass of a polymer chain (it just moves with the lo-cal velocity) and, scaling the position vector qj of the jth connector vector withl0 =

√kBTH

, see that its equation of motion (10) becomes

(58) dqj =

[∇v · qj −

1

2λ

qj

1− q2j/b

]dt+

[1

4λF (qj+1) +

1

4λF (qj−1)

]dt+

√1

λdW q

j .

Eqn. (58) is solved subject to an initial condition consisting of a random sampleq1, · · · , qd drawn from the equilibrium distribution (39). Since all the equations(58) (j = 1, . . . , d) are coupled we advance the calculation through one time step∆t by performing one sweep from q1 to qd of an iterative method whereby, for agiven index `j' the vectors q1, . . . , qj−1 are assumed known at t = (n + 1)∆t andqj+1, . . . , qd at time t = n∆t. This leads to the scheme

qn+1j,x +

4t2λ

qn+1j,x

1− q2/b= qnj,x +

(∇v · qn

j

)x∆t

+[Fx(q

nj+1) + Fx(q

n+1j−1 )

]∆t+

√∆t

λWj,x = rnj,x,(59)

qn+1j,y +

∆t

2λ

qn+1j,y

1− q2/b= qny +

(∇v · qn

j

)y∆t

+[Fy(q

nj+1) + Fy(q

n+1j−1 )

]∆t+

√∆t

λWj,y = rnj,y,(60)

whereq2 = (qn+1

j,x )2 + (qn+1j,y )2

and

(61) Wj =1√∆t

(W qj (t+∆t)−W q

j (t)),

which implies that each component Wj ∼ N(0, 1). >From the previous equations

(62) qn+1j,x =

rnj,x

1 + 4t2λ

11−q2/b

and qn+1j,y =

rnj,y

1 + 4t2λ

11−q2/b

,

and therefore,

qn+1j

(1 +

4t2λ

1

1− q2/b

)= rn

j .

Taking absolute values in the above and setting r =√(

rnj,x)2

+(rnj,y)2 we get the

cubic equation

q3 − rq2 − bq

(1 +

∆t

2λ

)+ br = 0.

This equation has one real root q between 0 and√b. Once this has been computed

qj,x and qj,y may be found from Eqn. (62).



The elastic extra stress tensor τ appearing in (1) may be written in the form

(63) τ =d∑

j=1

τ j,

where τ j is the contribution of the jth segment in all the polymer chains of thesolution to the total elastic extra stress. In the case of a homogeneous ow the stressdepends only upon time. The Kramers expression for τ j at time n∆t now reads

(64) τ j(n∆t) ≈(1− β)

We

(b+ 4)

b

[I + 〈qn

j

⊗F (qn

j )〉],

where 〈·〉 denotes an average taken over all M chains in the solution and We denotesa Weissenberg number, dened as We = λγ; a dimensionless shear rate. In terms ofthe joint probability density function ψ, τ j(n∆t) is written as

τ j(n∆t) =(1− β)

We

(b+ 4)

b

[I +

∫

|q1|<√b

. . .

∫

|qd|<√b

qj

⊗F (qj)

×ψ(n∆t, q1, . . . , qd)dq1 . . . dqd] ,(65)and the convergence of (64) as M −→ ∞ to the evaluation of (65) with the exact ψis assured by the strong law of large numbers.

5. Example: Steady simple shear owIn this section we present results for a homogeneous shear ow v = (vx, vy) = (y, 0)

using both the FTP (44) and STP (45) representations for the transformed distrib-ution function α. The stochastic method (59)-(64) was used with M = 10000 and20000 chains and the tensile elastic stress τxx computed with M = 20000 consideredto be a reference solution, against which the Fokker-Planck-based calculations of τxxcould be compared. To this end, for the Weissenberg numbers considered, a timestep ∆t = 0.01 was used and an average value for τxx over the time interval [7, 10]computed. Averages of τxx computed over later time intervals showed no signicantvariation and τxx was considered, as a consequence, to have reached steady state.

The CPU time in seconds on a Pentium IV machine, as well as the relative errorfor τxx committed by the STP and FTP methods for diering values of NR and NF

and at two dierent Weissenberg numbers (We = 1 and 3) are shown in Figs. 2and 3. Unless otherwise stated, the level of approximation l in the STP methodwas taken equal to the maximum value L. At both Weissenberg numbers the STPmethod is considerably cheaper than an FTP method computed with the same level ofdiscretisation for the eigenbasis. At the same time, at the lower Weissenberg number,the relative error with the STP method is lower than with the corresponding FTPmethod and the STP method may be seen to be competitive with the stochasticapproach for longer chains than its FTP counterpart. At the larger Weissenbergnumber, the FTP calculations are more accurate than those performed with the STPrepresentation and the deterioration of the accuracy advantage of the STP methodfor small chains (d ≤ 3) may be attributable to the use of a Galerkin formulation.FTP calculations were not considered feasible for d > 3.

A point of interest in Fig. 3 is that by taking a ner mesh for the eigenfunctioncalculation (NR = 12, NF = 8) but choosing the level of approximation l = 7 < L wecan get a signicant improvement in accuracy over the FTP results from a coarsermesh (NR = 5, NF = 3) at comparable CPU cost.



d

CP

Utim

e(s

ec)

2 3 4 5 6

10-1

100

101

102

103

S:10000S:20000STP5/2STP5/3FTP5/2FTP5/3

d

DO

F

2 3 4 5 6

103

104

105

STP5/2STP5/3FTP5/2FTP5/3

(a) (b)

d

Rel

ativ

eer

ror

(%)

2 3 4 5 6

1

2

3

45678

STP5/2STP5/3FTP5/2FTP5/3

(c)

Figure 2. We = 1. (a) CPU time (s), (b) number of degrees offreedom, and (c) percentage relative error for τxx computed using (65).FTP i/j: full tensor product (44), NR = i, NF = j. STP i/j: sparsetensor product (45), NR = i, NF = j. Also shown are the CPU timeconsumed by the stochastic method (59)-(64) with M = 10, 000 and20, 000 realizations.

6. ConclusionsIn this paper we have presented a new spectral Galerkin method for the solution

of a high-dimensional Fokker-Planck equation arising in the modelling of dilute solu-tions of long idealized polymers. The method is inspired by ideas in the sparse gridliterature and at modest levels of uid memory and shear rate (We ≈ 1) is demonstra-bly superior in accuracy and cost to a traditional full tensor product representationfor the polymer conguration distribution function. The method extends the attrac-tiveness of Fokker-Planck-based methods (as a competitive alternative to stochastic



d

CP

Utim

e(s

ec)

2 3 4 5

100

101

102

103

S:10000S:20000STP5/3STP12/6(L7)FTP5/3

d

DO

F

2 3 4 5 6

103

104

105

STP5/3STP12/6(L7)FTP5/3

(a) (b)

d

Rel

ativ

eer

ror

(%)

2 3 4 5

100

101

STP5/3STP12/6(L7)FTP5/3

(c)

Figure 3. We = 3. (a) CPU time (s), (b) number of degrees offreedom, and (c) percentage relative error for τxx computed using (65).FTP 5/3: full tensor product (44), NR = 5, NF = 3. STP 5/3: sparsetensor product (45), NR = 5, NF = 3. STP 12/8(L7): sparse tensorproduct (45), NR = 12, NF = 8, l = 7. Also shown are the CPU timeconsumed by the stochastic method (59)-(64) with M = 10, 000 and20, 000 realizations.

simulations) to longer polymer chains than was previously possible, although ulti-mately in shear ows at suciently high Weissenberg numbers (strong ows, highlyelastic liquids) it would seem probable that well constructed stochastic methods willhave the advantage in terms of cost and accuracy.



AcknowledgmentsThe second author was supported for the duration of this project by CTI project

6437.1 IWS-IW

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CUST, Génie Mathématique et Modélisation, Université Blaise Pascal II, Clermont-Ferrand, France.

Chaire d'Analyse et Simulations Numériques, Section de Mathématiques, EcolePolytechnique Fédérale de Lausanne, CH 1015 Lausanne, Switzerland

Current address: Department of Mathematics, University of Houston, Houston, TX 77204-3008,USA

Département de Mathematiques et de Statistique, Université de Montréal, CP6128, succursale centre-ville, Montréal QC H3C 3J7, Canada


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