Free access article

Issue M2AN Volume 35, Number 6, November-December 2001 Page(s) 1079 - 1109 DOI 10.1051/m2an:2001149

DOI: 10.1051/m2an:2001149


M2AN, Vol. 35, N°6, pp. 1079-1109

Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes

Gerd Kunert

TU Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany. (kunert@mathematik.tu-chemnitz.de)

(Received: January 22, 2001. Revised: July 26, 2001. )

Abstract
Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

AMS Subject: 65N15, 65N30, 35B25.

Key words: Error estimator, anisotropic solution, stretched elements, reaction diffusion equation, singularly perturbed problem.


© EDP Sciences, SMAI 2001

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