Comptes Rendus
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

Assyr Abdulle1
1 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK
Comptes Rendus. Mathématique, Volume 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.

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

1 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK 
@article{CRMATH_2008__346_1-2_97_0,
     author = {Assyr Abdulle},
     title = {Multiscale method based on discontinuous {Galerkin} methods for homogenization problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {97--102},
     publisher = {Elsevier},
     volume = {346},
     number = {1-2},
     year = {2008},
     doi = {10.1016/j.crma.2007.11.029},
     language = {en},
}
TY  - JOUR
AU  - Assyr Abdulle
TI  - Multiscale method based on discontinuous Galerkin methods for homogenization problems
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 97
EP  - 102
VL  - 346
IS  - 1-2
PB  - Elsevier
DO  - 10.1016/j.crma.2007.11.029
LA  - en
ID  - CRMATH_2008__346_1-2_97_0
ER  - 
%0 Journal Article
%A Assyr Abdulle
%T Multiscale method based on discontinuous Galerkin methods for homogenization problems
%J Comptes Rendus. Mathématique
%D 2008
%P 97-102
%V 346
%N 1-2
%I Elsevier
%R 10.1016/j.crma.2007.11.029
%G en
%F CRMATH_2008__346_1-2_97_0
Assyr Abdulle. Multiscale method based on discontinuous Galerkin methods for homogenization problems. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 97-102. doi : 10.1016/j.crma.2007.11.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.029/

[1] J. Aarnes; B.-O. Heimsund Multiscale discontinuous Galerkin methods for elliptic problems with multiple scales, Lecture Notes Comput. Sci. Eng., vol. 44, Springer-Verlag, 2006, pp. 1-20

[2] A. Abdulle On a-priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM Multiscale Model. Simul., Volume 4 (2005) no. 2, pp. 447-459

[3] A. Abdulle; W. E Finite difference HMM for homogenization problems, J. Comput. Phys., Volume 191 (2003) no. 1, pp. 18-39

[4] A. Abdulle; C. Schwab Heterogeneous multiscale FEM for diffusion problem on rough surfaces, SIAM Multiscale Model. Simul., Volume 3 (2005) no. 1, pp. 195-220

[5] D. Arnold An interior penalty finite element method with discontinuous element, SIAM J. Numer. Anal., Volume 19 (1982), pp. 742-760

[6] D. Arnold; F. Brezzi; B. Cockburn; D. Marini Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 5 (2002), pp. 1749-1779

[7] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978

[8] S. Chen; W. E; C. Shu The heterogeneous multiscale method based on the discontinuous Galerkin method for hyperbolic and parabolic problems, SIAM Multiscale Model. Simul., Volume 3 (2005) no. 4, pp. 871-894

[9] E. De Giorgi; S. Spagnolo Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital., Volume 4 (1973) no. 8, pp. 391-411

[10] W. E; B. Engquist The heterogeneous multi-scale methods, Commun. Math. Sci., Volume 1 (2003) no. 1, pp. 87-132

[11] W. E; P. Ming; P. Zhang Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., Volume 18 (2004) no. 1, pp. 121-156

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

Assyr Abdulle; Achim Nonnenmacher

C. R. Math (2009)

The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods

Assyr Abdulle; Gilles Vilmart

C. R. Math (2011)

A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems

Sven Kaulmann; Mario Ohlberger; Bernard Haasdonk

C. R. Math (2011)