Error estimates for mixed methods

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  • RAIROANALYSE NUMRIQUE

    R. S. FALKJ. E. OSBORNError estimates for mixed methodsRAIRO Analyse numrique, tome 14, no 3 (1980), p. 249-277.

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  • R.A.LR.O. Analyse numrique/Numerical Analysis(vol. 14, n 3, 1980, p, 249 277)

    ERROR ESTIMATES FOR MIXED M E T H O D S (*)

    by R. S. FALK (*) and J. E. OSBORN (2)

    Communicated by P. A. RAVIART

    Abstract. This paper prsents abstract error estimtes for mixed methods for the approximatesolution of elliptic boundary value problems. These estimtes are then applied to obtain quasi-optimalerror estimtes in the usual Sobolev norms for four examples: three mixed methods for the biharmonicproblem and a mixed method for second order elliptic problems.

    Resum. - Dans cet article, on prsente des estimations d'erreur abstraites pour des mthodesmixtes appliques la rsolution approche de problmes aux limites elliptiques. On applique ensuite cesestimations afin d'obtenir des estimations d'erreur quasi-optimales, dans les normes de Sobolevhabituelles, dans quatre exemples : Trois mthodes mixtes pour le problme biharmoniques, et unemthode mixte pour les problmes elliptiques du second ordre.

    1. INTRODUCTION

    In [5] Brezzi studied Ritz-Galerkin approximation of saddle-point problemsarising in connection with Lagrange multipliers. These problems have the form:

    Given f e V' and g e W\ find (u, \|/)e V x W satisfying

    [ ' }

    where Kand Ware real Hubert spaces, and a ( . , .) and fc(., .) are boundedbinear forms on Vx V and VxW9 respectively.

    Given finite dimensional spaces VhczV and WhczW, 0 < h < l , the Ritz-Galerkin approximation (uh, v|/ft) to (u, \j/) is the solution of the followingproblem:

    Find (ufc, v|/A)e VK x Wh satisfying

    a(uh, v) + b(v, **) = (, v), Vi> Vk9 1

    () Received May 1979.The work of the first author was partialiy supported by NSF grant MCS78-02737 and by the

    United States Army under Contract No. DAAG29-75-C-0024, and the second author by NSF GrantMCS78-02851 and by the United States Army under Contract No. DAAG29-75-C-0024.

    (1) Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, U.S.A.(2) Department of Mathematics, University of Maryland, College Park, Maryland 20742, U.S.A.

    R.A.I.R.O. Analyse numrique/Numerical Analysis, 0399-0516/1980/249/$ 5.00AFCET-Bordas-Dunod

  • 250 R. S. FALK, J. E. OSBORN

    The major assumptions in Brezzi's results are

    ^ Y ^ ^Jo\\u\\v, VueZh and Vft, (1.3)veZh \\

    V\\V

    where YO > 0 is a constant independent of h and

    and

    ^ , , 9 ^ ^fcolMU, VcpePF, and Vfe, (1.4)\\V\

    where fco>0 is independent of h.

    Using (1.3) and (1.4) Brezzi proves the following error estimate for theapproximation method determined by (1.2):

    | |-Wfc|k+IK-*/ | | iK^C(iiif | | M - X | | K + inf H^-Till^), VA. (1.5)

    In [1, 2] Babuska studied Ritz-Galerkin approximation of gneraivariationally posed problems. The main resuit of [1, 2], applied to (1.1) and(1.2), is that (1.5) holds provided

    c _ [ f l ( M , P ) + fe(P,\|Q + ft(". < P ) | > T / I I I I , 1 1 , || vS U P HPH -4- II CD II T 0 ( | | | | K + \W\\w), i

    (vt0 is independent of h.

    It is clear from [1, 2, 5] that (1.3) and (1.4) hold if and only if (1.6) holds. (1.3)-(1.4) or, equivalently, (1.6) is referred to as the stabiiity condition for thisapproximation method.

    The results of [1, 2, 5] can be viewed as a strategy for analyzing theseapproximation methods: the approximation method is characterized by certainbilinear forms, norms (spaces), and families of finite dimensional approximatingspaces, and if the method can be shown to be stable with respect to the chosennorms, then the error estimtes in these norms foliow in a simple mannerprovided the bilinear forms are bounded and the approximation properties of Vhand Wh are known in these norms. These results can be used to analyze, forexampie, certain hybrid methods for the biharmonic problem [5, 6] and thestationary Stokes problem [10]. The results of [1, 2] have also been used toanalyze a variety of variationally posed problems that are not of form (1.1).

    R.A.I.R.O. Analyse numrique/Numencal Analysis

  • ERROR ESTIMATES FOR MIXED METHODS 251

    There are other problems of a similar nature, however, where attempts atusing the ideas of [1, 2, 5] were not entirely successful since not all of the abstracthypotheses were satisfied: specifically the Brezzi condition (1.3) or, equivalently,the Babuska condition (1.6), is not satisfied with the usual choice of norms, i. e.,the approximation methods for these problems are not stable with respect to theusual norms. This is the case, for example, in the analysis in [7] of the Hermann-Miyoshi [14, 15, 19] mixed method for the biharmonic problem. In the analysisof this method a natural choice for both || . \\v and |[ . ||^ is the lst order Sobolevnorm; however this method is not stable with respect to this choice. As a resuit ofthis difficulty, the error estimtes obtained in [7] are not quasi-optimal. A similardifficulty arises in the analysis of the Hermann-Johnson [14, 15, 16] and Ciarlet-Raviart [9] mixed methods for the biharmonic problem. In later work of Scholz[23] and Rannacher [21] quasi-optimal error estimtes were obtained for themixed methods of Ciarlet-Raviart and Hermann-Miyoshi, although thesystematic approach of Brezzi and Babuska was abandoned.

    In a forthcoming paper of Babuska, Osiorn, and Pitkranta {3} quasi-optimalerror estimtes for mixed methods for the biharmonic problem are derived by anapplication of the resuit s of Brezzi and Babuska. In this work a new family of(mesh dependent) norms are introduced with respect to which the abovementioned mixed methods (Ciarlet-Raviart, Hermann-Miyoshi, Hermann-Johnson) are stable. Error estimtes in these norms then follow directly from theresults of Brezzi and Babuka, once the approximation properties of thesubspaces Vh and Wh have been determined in these new norms. Error estimtesin the more standard norms are then obtained by using the usual duaiityargument.

    It is the intent of this paper to provide an abstract approach to the analysis ofmixed methods which leads to quasi-optimal error estimtes, uses only standardnorms, and is systematic. We shall assume that existence and uniqueness for thecontinuous (infinit dimensional) problem has been established and develop anabstract framework under which quasi-optimal error estimtes can be derivedfor a variety of examples which do not fit within the convergence theory of Brezziand Babuska using the usual norms.

    Section 2 contains the abstract convergence results of the paper. In section 3we present four examples previously analyzed in the literature and show howerror estimtes can be derived from the theorems in section 2. Three of thesemethods are mixed methods for the biharmonic problem and the fourth is amixed method for a second order problem analyzed by Raviart-Thomas [22, 25].

    It is interesting to note that in this last example the results of Brezzi andBabuska apply with the choice of spaces used by Raviart-Thomas, but fail to

    vol 14, n3, 1980

  • 2 5 2 R. S. FALK, J. E. OSBORN

    yield quasi-optimal error estimtes in all cases due to the way in which thevariables are tied together in the error estimtes. In our analysis the errorestimtes for the two variables are separated and quasi-optimal error estimtesare obtained. For the three mixed methods for the biharmonic problem that areanalyzed in section 3 the results of the present paper and those obtained in [3],using different techniques, are the same. For additional results on mixedmethods see Oden [20]. Finally we note that some basic ideas in the analysis inthis paper are similar to those employed in Scholz [23, 24],

    Throughout this paper, we shall use the Sobolev spaces Wm-p (Q), where Q is aconvex polygon in the plane, m is a nonnegative integer, and 1 g p < oo. On thesespaces we have the seminorms and norms

    and

    When p = 2, we dnote Wm> 2(Q) by Hm(Q) and write

    and

    We will further dnote by W^P{QL) the subspace of WltP(Q) of fonctions thatvanish on F = 3Q and by ifo(O) the subspace of H2(Q) of fonctions that vanishtogether with their normal derivatives on F. For m = 1 and 2 we will also use thespaces H-m(Q) = [H'S(Q)Y [the dual space of if^(fi)] with the norm on H~m(Q)taken to be the usual dual norm. To forther simplify notation we often drop theuse of the subscript Q in the norm when the context is clear.

    2. ABSTRACT RESULTS

    Let V9 W, and H be three real Banach spaces with norms || .\\y, || . ||^, and|| . \\H respectively. We assume VczH with a continuous imbedding. Let a(., .)and b(., .) be continuous bilinear forms on H xH and VxWy respectively:

    (2.1)

    W. (2.2)

    R.A.I.R.O. Analyse numrique/Numerical Analysis

  • ERROR ESTIMATES FOR MIXED METHODS 2 5 3

    We consider the foliowing problem, which we refer to as problem P:

    Given e V1 and g e W\ find (u,ty)eVxW satisfying:

    , (2.3)

    (2.4)

    where (., .) dnotes the pairing between F and V' or W and W'.

    We shall be interested in this problem for a subclass