An augmented discontinuous Galerkin method for elliptic problems

Tomás P. Barrios; Rommel Bustinza

doi:10.1016/j.crma.2006.11.003

Numerical Analysis

An augmented discontinuous Galerkin method for elliptic problems
[Une méthode de Galerkin discontinue augmentée pour des problèmes elliptiques]

Présenté par : Philippe G. Ciarlet

Tomás P. Barrios ¹ ; Rommel Bustinza 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

Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 53-58.

Résumé
Abstract

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.

Reçu le : 2006-07-04
Accepté le : 2006-11-02
Publié le : 2006-12-08

DOI : 10.1016/j.crma.2006.11.003

Affiliations des auteurs :

Tomás P. Barrios ¹ ; Rommel Bustinza ²

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     title = {An augmented discontinuous {Galerkin} method for elliptic problems},
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     pages = {53--58},
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Tomás P. Barrios; Rommel Bustinza. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.003/

Bibliographie
Cité par

[1] D.N. Arnold; F. Brezzi; B. Cockburn; L.D. Marini Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, Volume 39 (2001) no. 5, pp. 1749-1779

[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

[3] T.P. Barrios and G.N. Gatica, An augmented mixed finite element method with Lagrange multipliers: a priori and a-posteriori error analyses, Journal of Computational and Applied Mathematics, in press

[4] F. Brezzi; B. Cockburn; L.D. Marini; E. Süli Stabilization mechanisms in discontinuous Galerkin finite element methods, Computer Methods in Applied Mechanics and Engineering, Volume 195 (2006), pp. 3293-3310

[5] R. Bustinza; G.N. Gatica A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions, SIAM Journal on Scientific Computing, Volume 26 (2004) no. 1, pp. 152-177

[6] P. Castillo; B. Cockburn; I. Perugia; D. Schötzau An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM Journal on Numerical Analysis, Volume 38 (2000) no. 5, pp. 1676-1706

[7] G.N. Gatica; F.J. Sayas A note on the local approximation properties of piecewise polynomials with applications to LDG methods, Complex Variables and Elliptic Equations, Volume 51 (2006) no. 2, pp. 109-117

[8] T.J.R. Hughes; A. Masud; J. Wan A stabilized mixed discontinuous Galerkin method for Darcy flow, Computer Methods in Applied Mechanics and Engineering, Volume 195 (2006) no. 25–28, pp. 3347-3381

[9] I. Perugia; D. Schötzau An hp-analysis of the local discontinuous Galerkin method for diffusion problems, Journal of Scientific Computing, Volume 17 (2002), pp. 561-571

[10] J.E. Roberts; J.-M. Thomas Mixed and hybrid methods (P.G. Ciarlet; J.L. Lions, eds.), Finite Element Methods, Part 1, Handbook of Numerical Analysis, vol. II, North-Holland, Amsterdam, 1991

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.

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