Numerical homogenization: survey, new results, and perspectives

Antoine Gloria 1
1 SIMPAF - SImulations and Modeling for PArticles and Fluids
Inria Lille - Nord Europe, LPP - Laboratoire Paul Painlevé - UMR 8524
Abstract : These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar's correctors. Hence the convergence analysis of these methods relies on the H-convergence theory. When one is interested in convergence rates, the story is different. In particular one first needs to make additional structure assumptions on the heterogeneities (say periodicity for instance). In that case, a crucial tool is the spectral interpretation of the corrector equation by Papanicolaou and Varadhan. Spectral analysis does not only allow to obtain convergence rates, but also to devise efficient new approximation methods. For both qualitative and quantitative properties, the development and the analysis of numerical homogenization methods rely on seminal concepts of the homogenization theory. These notes contain some new results.
Type de document :
Article dans une revue
ESAIM: Proceedings, EDP Sciences, 2012, 37, pp.50-116
Liste complète des métadonnées

Littérature citée [57 références]  Voir  Masquer  Télécharger

https://hal.inria.fr/hal-00766743
Contributeur : Antoine Gloria <>
Soumis le : mardi 18 décembre 2012 - 18:04:21
Dernière modification le : mardi 3 juillet 2018 - 11:44:11
Document(s) archivé(s) le : mardi 19 mars 2013 - 03:59:59

Fichier

proc123702.pdf
Fichiers éditeurs autorisés sur une archive ouverte

Identifiants

  • HAL Id : hal-00766743, version 1

Collections

Citation

Antoine Gloria. Numerical homogenization: survey, new results, and perspectives. ESAIM: Proceedings, EDP Sciences, 2012, 37, pp.50-116. 〈hal-00766743〉

Partager

Métriques

Consultations de la notice

359

Téléchargements de fichiers

277