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A POSTERIORI ERROR ESTIMATES OF THE STABILIZED CROUZEIX RAVIART FINITE ELEMENT METHOD FOR THE

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A POSTERIORI ERROR ESTIMATES OF THE STABILIZED CROUZEIX-RAVIART FINITE ELEMENT METHOD FOR THE LAME-NAVIER EQUATIONS. M. FARHLOUL ?, S. NICAISE † , AND L. PAQUET † Abstract. We obtain a posteriori error estimates for the (non-locking) stabilized nonconforming method based on the Crouzeix-Raviart element introduced by P. Hansbo and M. G. Larson in [M2AN 37 (2003) 63-72]. A robust (i.e., uniformly with respect to the Lame coefficient) upper bound is proved, while an almost robust lower bound is obtained. Key words. Crouzeix-Raviart element, nonconforming method, stabilized method, nonlocking, a posteriori error estimates. AMS subject classifications. 65N30, 65N15, 65N50 1. Introduction. The finite element methods are widely used for the numeri- cal approximation of many problems occurring in engineering applications, like the Laplace equation, the Lame system, etc.... (see [10, 15]). In practice, adaptive tech- niques based on a posteriori error estimators have become indispensable tools for such methods. Hence there now exists a large number of publications devoted to the analysis of some finite element approximations of problems from solid mechanics and obtaining locally defined a posteriori error estimates. We refer to the monographs [2, 7, 26] for a good overview on this topic.

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A POSTERIORI ERROR ESTIMATES OF THE STABILIZEDCROUZEIX-RAVIART FINITE ELEMENT METHOD FOR THE´LAME-NAVIER EQUATIONS.M. FARHLOUL, S. NICAISE,ANDL. PAQUETAbstract.We obtain a posteriori error estimates for the (non-locking) stabilized nonconformingmethod based on the Crouzeix-Raviart element introduced by P. Hansbo and M. G. Larson in [M2AN37(2003)63-72].Arobust(i.e.,uniformlywithrespecttotheLam´ecoecient)upperboundisproved, while an almost robust lower bound is obtained.Key words.Crouzeix-Raviart element, nonconforming method, stabilized method, nonlocking,a posteriori error estimates.AMS subject classifications.65N30, 65N15, 65N501. Introduction.The finite element methods are widely used for the numeri-cal approximation of many problems occurring in engineering applications, like theLaplaceequation,theLame´system,etc....(see[10,15]).Inpractice,adaptivetech-niques based on a posteriori error estimators have become indispensable tools forsuch methods. Hence there now exists a large number of publications devoted to theanalysis of some finite element approximations of problems from solid mechanics andobtaining locally defined a posteriori error estimates. We refer to the monographs[2, 7, 26] for a good overview on this topic.For the elasticity system in the primal variables, several different approaches havebeen developed: Residual type error estimators [4, 5, 27], methods based on the resolu-tion of local subproblems by using higher order elements [4, 6, 8], averaging techniques(the so-called Zienkiewicz-Zhu estimators) [1, 2, 28, 29] and finally estimators basedon equilibrated fluxes [3, 11, 20, 21, 22, 24]. For methods based on dual variables,like mixed methods, we refer to [12, 13, 14, 9, 19]; note that such methods are usuallylocking free and therefore the obtained estimators are also locking free.Here we analyze a displacement method based on the primal variables introducedby P. Hansbo and M.G. Larson in [18] which is a nonconforming method based on theCrouzeix-Raviart finite element. This method is locking free and very cheap. In [18],the authors derive apriorioptimalerrorestimatesuniformintheLam´eparameterλ(see Theorem 3.1 of [18]). In this paper, we propose an aposteriorierror analysisof this method. Our analysis enters in the family of estimators of residual typesince our error indicator is based on residuals on each triangle and jumps across theinter-element boundaries. We prove reliability of the indicator uniformly inλ(andh), in particular avoiding locking phenomena. The proof is based on the Helmoltzdecomposition of the strain tensor of the error. Efficiency of our indicator follows byusing classical inverse estimates. However, the lower bound of the error (i.e., efficiencyof the indicator) depends onλ; see section 4 for more details.The outline of the paper is as follows: We recall in Section 2 the boundary valueproblem and its numerical approximation. Section 3 is devoted to the introduction ofUniversit´edeMoncton,De´partementdeMathe´matiquesetdeStatistique,Moncton,N.B.,E1A3E9, Canada (mohamed.farhloul@umoncton.ca).Universite´deValenciennesetduHainautCambre´sis,LAMAV,InstitutdesSciences et Techniques de Valenciennes, F-59313 - Valenciennes Cedex 9 France({Serge.Nicaise,Luc.Paquet}@univ-valenciennes.fr).1