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Issue ESAIM: PROC
Volume 27, 2009
CANUM 2008
Page(s) 107 - 121
DOI 10.1051/proc/2009022
Published online 25 June 2009

ESAIM: Proc., 2009, Vol. 27, pp. 107-121
DOI: 10.1051/proc/2009022

A posteriori error analysis for Poisson's equation approximated by XFEM

Patrick Hild1, Vanessa Lleras2 and Yves Renard3

1  Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, U niversité de Franche-Comté, 16 route de Gray, 25030 Besançon, France;
2  Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon, France;
3  Institut Camille Jordan, UMR CNRS 5208, INSA de Lyon, 20 rue Albert Einstein, 69621 Villeurbanne, France;

patrick.hild@univ-fcomte.fr
vanessa.lleras@univ-fcomte.fr
yves.renard@insa-lyon.fr

Published online: 25 June 2009

Abstract
This paper presents and studies a residual a posteriori error estimator for Laplace's equation in two space dimensions approximated by the eXtended Finite Element Method (XFEM). The XFEM allows to perform finite element computations on multi-cracked domains by using meshes of the non-cracked domain. The main idea consists of adding supplementary basis functions of Heaviside type and singular functions in order to take into account the crack geometry and the singularity at the crack tip respectively.


Résumé
Dans ce travail on propose et on étudie un estimateur d'erreur par résidu pour l'équation de Laplace en deux dimensions d'espace discrétisée par la méthode d'éléments finis étendue (XFEM). La XFEM permet de réaliser des simulations par éléments finis sur des domaines multi-fissurés en utilisant des maillages du domaine non fissuré. L'idée principale de la méthode consiste à ajouter des fonctions de base supplémentaires de type Heaviside et des fonctions singulières afin de prendre en compte la géométrie de la fissure et la singularité en pointe de fissure.



© EDP Sciences, ESAIM 2009


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