Numerical Analysis
An augmented discontinuous Galerkin method for elliptic problems
Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 53-58.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.11.003
Barrios, Tomás P. 1; Bustinza, Rommel 2

1 Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile
2 Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
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Barrios, Tomás P.; Bustinza, Rommel. An augmented discontinuous Galerkin method for elliptic problems. Comptes Rendus. Mathématique, Volume 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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[2] T.P. Barrios, R. Bustinza, A priori and a posteriori error analyses of an augmented Galerkin discontinuous formulation, Departamento de Ingeniería Matemática, Universidad de Concepción, Chile, in preparation

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Cited by 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.