AN A POSTERIORI ERROR ESTIMATOR FOR THE LAME EQUATION BASED ON H DIV CONFORMING STRESS

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AN A POSTERIORI ERROR ESTIMATOR FOR THE LAME EQUATION BASED ON H(DIV)-CONFORMING STRESS APPROXIMATIONS S. NICAISE?, K. WITOWSKI† , AND B.I. WOHLMUTH† Abstract. We derive a new a posteriori error estimator for the Lame system based on H(div)- conforming elements and equilibrated fluxes. It is shown that the estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established using Argyris elements. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. Key words. equilibrated fluxes, mixed finite elements, a posteriori error estimates, linear elasticity AMS subject classifications. 65N30, 65N15, 65N50 1. Introduction. The finite element methods are commonly used in the nu- merical realization of many problems occurring in engineering applications, like the Laplace equation, the Lame system, the Stokes system, etc.... (see [13, 17]). Adaptive techniques based on a posteriori error estimators have become indispensable tools for such methods. There now exists a large number of publications devoted to the task of analyzing finite element approximations for problems in solid mechanics and obtain- ing locally defined a posteriori error estimates. We refer to the monographs [2, 10, 29] for a good overview on this topic.

  • local postprocessing step

  • error estimator

  • matrix associated

  • square integrable functions

  • higher order

  • lame system

  • norm being

  • elements ?h

  • when using

  • elements


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ANAPOSTERIORIERRORESTIMATORFORTHELAME EQUATION BASED ONH(DIV)-CONFORMING STRESS APPROXIMATIONS S. NICAISE, K. WITOWSKI,ANDB.I. WOHLMUTH
Abstract.WdereposterioiveanewamitsrotareireroresmteysrtfoLahedenobmsaH(div)-conformingelementsandequilibrateduxes.Itisshownthattheestimatorgivesrisetoanupper bound where the constant is one up to higher order terms. The lower bound is also established usingArgyriselements.Thereliabilityandeciencyoftheproposedestimatoriscon rmedbysome numerical tests.
Key words. mixedequilibrated uxes, a posteriori error estimates, linear elements,  nite elasticity
AMS subject classi cations.65N30, 65N15, 65N50
1. Introduction.The nite element methods are commonly used in the nu-merical realization of many problems occurring in engineering applications, like the Laplaceequation,theLamesystem,theStokessystem,etc....(see[13,17]).Adaptive techniques based on a posteriori error estimators have become indispensable tools for such methods. There now exists a large number of publications devoted to the task of analyzing nite element approximations for problems in solid mechanics and obtain-ing locally de ned a posteriori error estimates. We refer to the monographs [2, 10, 29] for a good overview on this topic. Usually upper and lower bounds are proved in order to guarantee the reliability and the eciency of the proposed estimator. Most of the existing approaches involve constants depending on the shape regularity of the elements and/or of the jumps in the coecien ts; but these dependences are often not given. Fortheelasticitysystem,severaldi erentapproachesleadingtovariousesti-mators have been developed (see the review paper [30]). Let us quote the follow-ing methods: Residual type error estimators measure the jump of the discrete ux [7, 8, 30]. Another approach is to solve local subproblems by using higher order elements [7, 9, 11, 12]. Very simple and cheap error estimators are the so-called Zienkiewicz-Zhu estimators based on averaging techniques [1, 2, 32, 33]. Finally we canmentionestimatorsbasedonequilibrateduxesandonthesolutionoflocalNeu-mann boundary value problems [3, 15, 18, 19, 20, 21, 22, 25, 26, 27, 28]. Here we introduce a locally de ned error estimator based onHiv)-(dormiconfgn approximationsforthestressandonequilibrateduxes.Inacertainsenseinthelast described method, we replace the resolution of the local Neumann boundary value problems by explicitH(div)-conforming approximations. That renders our estimator more attractive since no supplementary problems have to be solved. This error esti-mator further yields, up to higher order terms, an upper bound with constant one for the discretization error.
-UineitrsvencleVadetesenneiuaniaHudbretCamLAMAsis,tsti,VnISsictued ences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9 France. email: snicaise@univ-valenciennes.fr -ne afP,tSaitrsrtgatttuI(NAoisninev)SU,ricaNumeulatlSimfopAutetsnitIsandlysidAnaplie waldring 57, 70529 Stuttgart, Germany. email:{htitwskowwoi,muhl}@snaiinu.uts-etrd.ttag This work was supported in part by the Deutsche Forschungsgemeinschaft, SFB 404, B8 1
2
The schedule of the paper is as follows: We recall in Section 2 the boundary value problem and its numerical approximation. Section 3 is devoted to the intro-duction of the estimator and the proofs of the upper and lower bounds. The upper bound directly follows from the construction of the estimator, while the lower bound requires a speci c choice using Argyris elements. In Section 4, we give a practical waytocomputeexplicitlyourestimatorfromtheequilibrateduxes.Finallysome numericaltestsarepresentedinSection5thatcon rmthereliabilityandeciency of our estimator.
2. The boundary value problem of elasticity and notation.In the context of elasticity, vector- and tensor- or matrix-valued functions will be written in boldface form. The scalar product of two tensors or matricesandwill be denoted by:, and is given bya:b=aijbij, the summation convention on repeated indices being invoked. Consider a homogeneous isotropic linear elastic material body which occupies a bounded domain  inR2with Lipschitz boundary . For a prescribed body force f[L2( )]2, the governing equilibrium equation in  reads
 div=f,
where tensor is strain in nitesimal Theis the symmetric Cauchy stress tensor. de ned as a function of the displacementuby(u) :=2(ru+ [ru]>). The dis-placement is assumed to satisfy the homogeneous Dirichlet boundary condition, i.e., u=0on the fourth-order elasticity tensor denoted by . WithC, the constitutive equation reads
=C(u) :=(tr(u))1+ 2(u).
(2.1)
Here,1is the identity tensor, andanderciahamarepmwhs,eretaLehtera constant in view of the assumption of a homogeneous body, and which are assumed positive. We will make use of the spaceL2nnedosde tionfuncelbargetni-erauqfs)o( with the inner product and norm being denoted by (,)0and kk 0, respectively. The spaceH01( ) consists of functions inH1()whiavhchsinhtnouobearndntyihe sense of traces. For the weak or variational formulations we will require the space V:= [H01( )]2is a Hilbert space with inner product (of displacements; this ,)1and norm kk 1de ned in the standard way; that is, (u,v)1:=Pi2=1(ui, vi)1, with the norm being induced by this inner product. To de ne our error estimator, we compute by a local postprocessing step an approximation of the symmetric stress. The stress is inHS(div; ) :={|ji=ij, ijL2( ),div[L2( )]2}with the normk  k0 generated in the standard way by theL2-norm. De ne the bilinear forma(,) by a:VVR, a(u,v) :=RC(u) :(v)dx . Associated with the bilinear form is the energy norm|v|2:=a(v,v),vV. Then the standard form of the weak problem for elasticity takes the following form: given f[L2( )]2, nduVthat satis es
.
a(u,v) = (f,v)0,vV .(2.2) LetThbe a quasi-uniform, shape-regular triangulation of the polygonal domain ˆ WeassumethatallelementsareaneequivalenttothereferencetriangleTwith