We derive a nonlinear stabilized Galerkin approximation of the Laplace operator for which we prove a discrete maximum principle on arbitrary meshes and for arbitrary space dimension without resorting to the well-known acute condition or generalizations thereof. We also prove the existence of a discrete solution and discuss the extension of the scheme to convection–diffusion–reaction equations. Finally, we present examples showing that the new scheme cures local minima produced by the standard Galerkin approach while maintaining first-order accuracy in the H1-norm.
Nous introduisons un terme de stabilisation non-linéaire pour lequel nous prouvons un principe du maximum discret pour des approximations de type Galerkin du Laplacien. Le principe du maximum discret est satisfait en dimension quelconque et sans hypothèse particulière sur le maillage. On s'affranchit notamment de la condition bien connue d'acuité ou des généralisations de celle-ci. Nous prouvons également l'existence d'une solution discrète et proposons une extension du schéma aux équations de convection–diffusion–réaction. Enfin, nous présentons des résultats numériques montrant que le schéma élimine bien les minima locaux produits par la méthode de Galerkin standard tout en maintenant une convergence à l'ordre un en norme H1.
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@article{CRMATH_2004__338_8_641_0,
author = {Burman, Erik and Ern, Alexandre},
title = {Discrete maximum principle for {Galerkin} approximations of the {Laplace} operator on arbitrary meshes},
journal = {Comptes Rendus. Math\'ematique},
pages = {641--646},
publisher = {Elsevier},
volume = {338},
number = {8},
year = {2004},
doi = {10.1016/j.crma.2004.02.010},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2004.02.010/}
}
TY - JOUR AU - Burman, Erik AU - Ern, Alexandre TI - Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes JO - Comptes Rendus. Mathématique PY - 2004 SP - 641 EP - 646 VL - 338 IS - 8 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2004.02.010/ DO - 10.1016/j.crma.2004.02.010 LA - en ID - CRMATH_2004__338_8_641_0 ER -
%0 Journal Article %A Burman, Erik %A Ern, Alexandre %T Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes %J Comptes Rendus. Mathématique %D 2004 %P 641-646 %V 338 %N 8 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2004.02.010/ %R 10.1016/j.crma.2004.02.010 %G en %F CRMATH_2004__338_8_641_0
Burman, Erik; Ern, Alexandre. Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes. Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 641-646. doi : 10.1016/j.crma.2004.02.010. http://archive.numdam.org/articles/10.1016/j.crma.2004.02.010/
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