no. 1-2
Numerical Analysis
Multiscale method based on discontinuous Galerkin methods for homogenization problems
[Schéma multi-échelles basé sur une méthode de Galerkin discontinue pour des problèmes d'homogénéisation]

Présenté par : Philippe G. Ciarlet

Abdulle, Assyr1
1 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK
Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 97-102.

Nous proposons une méthode multi-échelles combinant un schéma macroscopique et des schémas microscopiques pour la résolution numérique d'équations elliptiques avec des coefficients fortement oscillants. Le schéma macroscopique, basé sur un macro-maillage, a pour objectif de fournir une approximation du problème effectif (homogénéisé). Les paramètres de ce schéma, a priori inconnus, sont obtenus pendant l'assemblage du problème effectif, à l'aide de schémas microscopiques mis en oeuvre sur des micro-cellules contenues dans le macro-maillage. Dans cette Note, nous expliquons comment ce couplage peut-être réalisé avec un schéma macroscopique basé sur une méthode de Galerkin discontinue. Nous montrons que les flux locaux nécessaire à la mise en oeuvre d'une telle méthode peuvent être construits à l'aide des solutions disponibles dans les cellules microscopiques. Une analyse d'erreur globale des schémas couplés est présentée.

We propose a multiscale method for elliptic problems with highly oscillating coefficients based on a coupling of macro and micro methods in the framework of the heterogeneous multiscale method. The macro method, defined on a macroscopic triangulation, aims at recovering the effective (homogenized) solution of an unknown macro model. The unspecified data of this model are computed by micro methods on sampling domains during the macro assembly process. In this Note, we show how to construct such a coupling with a discontinuous macro finite element space. We show that the flux information needed in this formulation in order to impose weak interelement continuity can be recovered from the known micro calculations on the sampling domains. A fully discrete analysis is presented.

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Accepté le :
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DOI : 10.1016/j.crma.2007.11.029
Abdulle, Assyr 1

1 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK 
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Abdulle, Assyr. Multiscale method based on discontinuous Galerkin methods for homogenization problems. Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 97-102. doi : 10.1016/j.crma.2007.11.029. http://www.numdam.org/articles/10.1016/j.crma.2007.11.029/

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