[Une méthode de Galerkin discontinue augmentée pour des problèmes elliptiques]
Dans cette Note, on se propose d'étudier une méthode de Galerkin discontinue augmentée pour des problèmes elliptiques bidimensionnels avec conditions mixtes à la frontière. L'approche introduit des termes de type Galerkin moindres carrés qui proviennent des équations constitutives et d'équilibre, et qui permettent de chercher les inconnues de flux dans des espaces Raviart–Thomas locaux. L'unicité des solutions est établie sans l'introduction d'opérateurs de relèvement. Une estimation de Céa est établie, qui montre que le taux de convergence de l'erreur, mesuré dans une norme appropriée, est optimal par rapport à la version h. Pour des expériences numériques, cette méthode présente l'avantage d'une réduction des degrés de liberté par rapport aux méthodes classiques de Galerkin discontinues.
In this Note we propose an augmented discontinuous Galerkin method for elliptic linear problems in the plane with mixed boundary conditions. Our approach introduces Galerkin least-squares terms, arising from constitutive and equilibrium equations, which allow us to look for the flux unknown in the local Raviart–Thomas space. The unique solvability is established avoiding the introduction of lifting operators and a Céa estimate is derived, which yields the rate of convergence of error, measured in an appropriate norm, being optimal respect to the h-version. We emphasize that for practical computations, this method reduces the degrees of freedom, with respect to the classical discontinuous Galerkin method.
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@article{CRMATH_2007__344_1_53_0,
author = {Barrios, Tom\'as P. and Bustinza, Rommel},
title = {An augmented discontinuous {Galerkin} method for elliptic problems},
journal = {Comptes Rendus. Math\'ematique},
pages = {53--58},
publisher = {Elsevier},
volume = {344},
number = {1},
year = {2007},
doi = {10.1016/j.crma.2006.11.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2006.11.003/}
}
TY - JOUR AU - Barrios, Tomás P. AU - Bustinza, Rommel TI - An augmented discontinuous Galerkin method for elliptic problems JO - Comptes Rendus. Mathématique PY - 2007 SP - 53 EP - 58 VL - 344 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.11.003/ DO - 10.1016/j.crma.2006.11.003 LA - en ID - CRMATH_2007__344_1_53_0 ER -
%0 Journal Article %A Barrios, Tomás P. %A Bustinza, Rommel %T An augmented discontinuous Galerkin method for elliptic problems %J Comptes Rendus. Mathématique %D 2007 %P 53-58 %V 344 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.11.003/ %R 10.1016/j.crma.2006.11.003 %G en %F CRMATH_2007__344_1_53_0
Barrios, Tomás P.; Bustinza, Rommel. An augmented discontinuous Galerkin method for elliptic problems. Comptes Rendus. Mathématique, Tome 344 (2007) no. 1, pp. 53-58. doi : 10.1016/j.crma.2006.11.003. http://www.numdam.org/articles/10.1016/j.crma.2006.11.003/
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Cité par Sources :
⁎ This research was partially supported by CONICYT-Chile through the FONDECYT project No. 1050842, and the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program.
