Numerical Analysis
Multiscale method based on discontinuous Galerkin methods for homogenization problems
Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 97-102.

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.

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.

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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, Volume 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/

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

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

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

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

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

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

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

[8] Chen, S.; E, W.; Shu, C. 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] De Giorgi, E.; Spagnolo, S. 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] E, W.; Engquist, B. The heterogeneous multi-scale methods, Commun. Math. Sci., Volume 1 (2003) no. 1, pp. 87-132

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

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