Comptes Rendus
Partial Differential Equations/Numerical Analysis
A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems
Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1233-1238.

Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.

Inspiré par lʼapproche des bases réduites et les méthodes numériques modernes pour des problèmes multi-échelles, nous présentons un nouveau traitement efficace des problèmes hétérogènes multi-échelles. La nouvelle approche repose sur lʼidée de considérer des problèmes hétérogènes multi-échelles comme des équations différentielles partielles paramétrisées, où les paramètres sont des fonctions lisses. Nous construisons alors dans une phase « offline » une base réduite localisée appropriée, utilisée dans une phase « online » pour calculer efficacement des approximations du problème multi-échelle par une méthode Galerkin discontinue sur un maillage grossier. Nous présentons notre nouvelle approche pour des problèmes elliptiques multi-échelles et discutons une estimation dʼerreur à posteriori utilisée lors de la construction de la base réduite localisée. Des expériences numériques sont exposées pour démontrer la efficacité de la nouvelle approche.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.10.024

Sven Kaulmann 1; Mario Ohlberger 2; Bernard Haasdonk 1

1 Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
2 Institute of Computational and Applied Mathematics & Center for Nonlinear Science, University of Münster, Einsteinstr. 62, 48149 Münster, Germany
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     title = {A new local reduced basis discontinuous {Galerkin} approach for heterogeneous multiscale problems},
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Sven Kaulmann; Mario Ohlberger; Bernard Haasdonk. A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1233-1238. doi : 10.1016/j.crma.2011.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.024/

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