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

Description

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

Sujets

• Créuse
• Saint Nicaise

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 22
• Langue: English

Extrait

of

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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,[e-mail]

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 ofR 2  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∈L 2 (Ω), let θ∈H 1 (Ω) be the unique solution of 0

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

1