A posteriori error estimations of a SUPG method for anisotropic diffusion–convection–reaction problems

Thomas Apel; Serge Nicaise

doi:10.1016/j.crma.2007.10.022

Numerical Analysis

A posteriori error estimations of a SUPG method for anisotropic diffusion–convection–reaction problems
[Estimations d'erreur a posteriori d'une méthode SUPG pour des problèmes de diffusion–convection–réaction anisotropes]

Présenté par : Philippe G. Ciarlet

Thomas Apel ¹ ; Serge Nicaise ²

¹ Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany
² Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, ISTV, 59313 Valenciennes cedex 9, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 657-662.

Résumé
Abstract

Cette Note présente un estimateur d'erreur a posteriori du type résiduel pour des problèmes de diffusion–convection–réaction approché par un schéma SUPG sur des triangulations isotropes ou non de $R^{d}$ , $d = 2$ or 3. Cet estimateur est basé sur les sauts des flux et les résidus intérieurs de la solution approchée. Il est construit pour fonctionner sur des maillages anisotropes qui tiennent en compte l'anisotropie éventuelle de la solution. L'équivalence entre la norme de l'erreur et l'estimateur est démontrée.

This Note presents an a posteriori residual error estimator for diffusion–convection–reaction problems approximated by a SUPG scheme on isotropic or anisotropic meshes in $R^{d}$ , $d = 2$ or 3. This estimator is based on the jump of the flux and the interior residual of the approximated solution. It is constructed to work on anisotropic meshes which account for the eventual anisotropic behavior of the solution. The equivalence between the energy norm of the error and the estimator is proved.

Reçu le : 2007-03-08
Accepté le : 2007-10-10
Publié le : 2007-11-09

DOI : 10.1016/j.crma.2007.10.022

Affiliations des auteurs :

Thomas Apel ¹ ; Serge Nicaise ²

¹ Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany
² Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, ISTV, 59313 Valenciennes cedex 9, France

@article{CRMATH_2007__345_11_657_0,
     author = {Thomas Apel and Serge Nicaise},
     title = {A posteriori error estimations of a {SUPG} method for anisotropic diffusion{\textendash}convection{\textendash}reaction problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {657--662},
     publisher = {Elsevier},
     volume = {345},
     number = {11},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.022},
     language = {en},
}

TY  - JOUR
AU  - Thomas Apel
AU  - Serge Nicaise
TI  - A posteriori error estimations of a SUPG method for anisotropic diffusion–convection–reaction problems
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 657
EP  - 662
VL  - 345
IS  - 11
PB  - Elsevier
DO  - 10.1016/j.crma.2007.10.022
LA  - en
ID  - CRMATH_2007__345_11_657_0
ER  -

%0 Journal Article
%A Thomas Apel
%A Serge Nicaise
%T A posteriori error estimations of a SUPG method for anisotropic diffusion–convection–reaction problems
%J Comptes Rendus. Mathématique
%D 2007
%P 657-662
%V 345
%N 11
%I Elsevier
%R 10.1016/j.crma.2007.10.022
%G en
%F CRMATH_2007__345_11_657_0

Thomas Apel; Serge Nicaise. A posteriori error estimations of a SUPG method for anisotropic diffusion–convection–reaction problems. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 657-662. doi : 10.1016/j.crma.2007.10.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.022/

Bibliographie
Cité par

[1] M. Ainsworth; J.T. Oden A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 2000

[2] L. Angermann Balanced a-posteriori error estimates for finite volume type discretizations of convection-dominated elliptic problems, Computing, Volume 55 (1995) no. 4, pp. 305-323

[3] T. Apel Anisotropic Finite Elements: Local Estimates and Applications, Advances in Numerical Mathematics, Teubner, Stuttgart, 1999

[4] P.G. Ciarlet The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978

[5] F. Fierro; A. Veeser A posteriori error estimators, gradient recovery by averaging, and superconvergence, Numer. Math., Volume 103 (2006), pp. 267-298

[6] D. Kay; D. Silvester The reliability of local error estimators for convection–diffusion equations, IMA J. Numer. Anal., Volume 21 (2001) no. 1, pp. 107-122

[7] G. Kunert, A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes, PhD thesis, TU Chemnitz, 1999, Logos, Berlin, 1999

[8] G. Kunert An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes, Numer. Math., Volume 86 (2000), pp. 471-490

[9] G. Kunert A posteriori error estimation for convection dominated problems on anisotropic meshes, Math. Methods Appl. Sci., Volume 26 (2003), pp. 589-617

[10] J. Li Convergence and superconvergence analysis of finite element methods on highly nonuniform anisotropic meshes for singularly perturbed reaction–diffusion problems, Appl. Numer. Math., Volume 36 (2001), pp. 129-154

[11] P. Neittaanmaäki; S. Repin Reliable Methods for Computer Simulation: Error Control and a Posteriori Error Estimates, Studies in Mathematics and its Applications, vol. 33, Elsevier, Amsterdam, 2004

[12] M. Picasso An anisotropic error indicator based on Zienkiewicz–Zhu error estimator: Application to elliptic and parabolic problems, SIAM J. Sci. Comput., Volume 24 (2003) no. 4, pp. 1328-1355

[13] R. Verfürth A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley and Teubner, Chichester and Stuttgart, 1996

[14] R. Verfürth A posteriori error estimators for convection–diffusion equations, Numer. Math., Volume 80 (1998) no. 4, pp. 641-663

Cité par Sources :

Commentaires - Politique