Functional a posteriori estimates for the reaction–diffusion problem

Sergey Repin; Stefan Sauter

doi:10.1016/j.crma.2006.06.024

Numerical Analysis

Functional a posteriori estimates for the reaction–diffusion problem
[Estimations à posteriori de type fonctionnel pour les problèmes de réaction–diffusion]

Présenté par : Philippe G. Ciarlet

Sergey Repin ¹ ; Stefan Sauter ²

¹ V.A. Steklov Institute of Mathematics, Fontanka 27, 191 011, St. Petersburg, Russia
² Institute of Mathematics, Zurich University, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland

Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 349-354.

Résumé
Abstract

Cette Note s'intéresse aux estimations à posteriori de type fonctionnel pour les problèmes de réaction–diffusion. Ces estimations fonctionnelles à posteriori sont obtenues par des méthodes purement fonctionnelles ne faisant en particulier pas appel à des propriétés d'orthogonalité de Galerkine ou des propriétés spéciales des espaces d'approximation. De ce fait elles sont indépendantes des tailles de maillage et fournissent des erreurs fiables pour toute approximation conforme. La généralisation au cas non conforme est également possible. Les estimations établies ici sont efficaces aussi bien dans le cas de coefficients constants que de coefficients oscillant arbitrairement dans certaines parties du domaine. Une telle robustesse est importante dans les applications où certains paramètres peuvent être très grands dans certaines parties et quasi nuls dans d'autres. On montre également que les estimations à posteriori que nous obtenons sont directement calculables et fournissent des estimations optimales.

The Note is concerned with functional type a posteriori estimates for stationary reaction–diffusion problems. Functional a posteriori estimates are derived on purely functional grounds without using any type of the Galerkin orthogonality condition and special properties of approximation spaces. Therefore, they contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. Generalizations to non-conforming approximations are also possible. Estimates derived in the Note are equally efficient for the problems with constant reaction parameter and for those admitting a high variability of it in different parts of the domain. Such a robustness with respect to the reaction parameter is important because in applications the reaction parameter my often be large in one subdomain and almost zero in another one. It is shown that the a posteriori bounds obtained are directly computable and provide sharp error bounds.

Reçu le : 2006-05-26
Accepté le : 2006-06-16
Publié le : 2006-08-08

DOI : 10.1016/j.crma.2006.06.024

Affiliations des auteurs :

Sergey Repin ¹ ; Stefan Sauter ²

¹ V.A. Steklov Institute of Mathematics, Fontanka 27, 191 011, St. Petersburg, Russia
² Institute of Mathematics, Zurich University, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland

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Sergey Repin; Stefan Sauter. Functional a posteriori estimates for the reaction–diffusion problem. Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 349-354. doi : 10.1016/j.crma.2006.06.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.024/

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