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Comptes Rendus Mathématique
Volume 349, n° 19-20
pages 1041-1046 (novembre 2011)
Doi : 10.1016/j.crma.2011.09.005
Received : 11 February 2011 ;  accepted : 8 September 2011
The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods
Lʼeffet de lʼintégration numérique sur la méthode des éléments finis pour des problèmes non-monotones elliptiques, avec application aux méthodes numériques dʼhomogénéisation
 

Assyr Abdulle , Gilles Vilmart
Section de mathématiques, École polytechnique fédérale de Lausanne, station 8, CH-1015 Lausanne, Switzerland 

Abstract

A finite element method with numerical quadrature is considered for the solution of a class of second-order quasilinear elliptic problems of nonmonotone type. Optimal a priori error estimates for the   and the   norms are derived. The uniqueness of the finite element solution is established for a sufficiently fine mesh. Our results permit the analysis of numerical homogenization methods.

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Résumé

On considère des méthodes dʼéléments finis avec intégration numérique par quadrature pour des problèmes elliptiques quasi-linéaires de type non-monotone. Les vitesses de convergence optimales pour les normes   et   sont démontrées ainsi que lʼunicité de la solution numérique pour un maillage suffisamment fin. Ces résultats permettent lʼanalyse multi-échelles de méthodes dʼhomogénéisation numérique.

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© 2011  Published by Elsevier Masson SAS de la part de Académie des sciences.
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