A posteriori error estimations of a coupled mixed and standard Galerkin method

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profil-vieg-2012

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Niveau: Supérieur, Licence, Bac+2
A posteriori error estimations of a coupled mixed and standard Galerkin method for second order operators Emmanuel Creuse, Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis LAMAV Institut des Sciences et Techniques de Valenciennes F-59313 - Valenciennes Cedex 9 France Emmanuel.Creuse, June 14, 2006 Abstract In this paper, we consider a discretization method proposed by Wieners and Wohlmuth [26] (see also [16]) for second order operators, which is a coupling between a mixed method in a sub-domain and a standard Galerkin method in the remaining part of the domain. We perform an a posteriori error analysis of residual type of this method, by combining some arguments from a posteriori error analysis of Galerkin methods and mixed methods. The reliability and efficiency of the estimator are proved. Some numerical tests are presented and confirm the theoretical error bounds. 1 Introduction Let us fix a bounded domain ? of R2, with a polygonal boundary. For the sake of simplicity we assume that ? is simply connected. The case of a multiply connected domain can be treated as in [12]. In this paper we consider the following second order problem: For f ? L2(?), let ? ? H10 (?) be the unique solution of div (A??) = ?f in ?, (1) where the matrix A ? L∞(?,R2?2) is supposed to be symmetric and uniformly positive definite.

div ? ?

priori error

crouzeix-raviart property

included into

all elements

universite de valenciennes et du hainaut cambresis

include standard

standard galerkin

Voir

Publié par

profil-vieg-2012

Langue

English

Créuse Saint Nicaise

A posteriori error estimations coupled mixed and standard Galerkin for second order operators

EmmanuelCreus´e,SergeNicaise

method

Universite´deValenciennesetduHainautCambre´sis

LAMAV Institut des Sciences et Techniques de Valenciennes F-59313 - Valenciennes Cedex 9 France Emmanuel.Creuse,Serge.Nicaise@univ-valenciennes.fr

June 14, 2006

Abstract

In this paper, we consider a discretization method proposed by Wieners and Wohlmuth [26] (see also [16]) for second order operators, which is a coupling between a mixed method in a sub-domain and a standard Galerkin method in the remaining part of the domain. We perform an a posteriori error analysis of residual type of this method, by combining some arguments from a posteriori error analysis of Galerkin methods and mixed methods. The reliability and eﬃciency of the estimator are proved. Some numerical tests are presented and conﬁrm the theoretical error bounds.

1 Introduction Let us ﬁx a bounded domain Ω ofR2 For the sake of simplicity, with a polygonal boundary. we assume that Ω is simply connected. The case of a multiply connected domain can be treated as in [12]. In this paper we consider the following second order problem: Forf∈L2(Ω), let θ∈H1(Ω) be the unique solution of 0

div (Arθ) =−fin Ω,(1) where the matrixA∈L∞(Ω,R2×2) is supposed to be symmetric and uniformly positive deﬁnite.

Voir