Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment

Erik Burman; Benjamin Stamm

doi:10.1016/j.crma.2007.11.016

Numerical Analysis

Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment
[Les méthodes de Galerkine discontinue symétrique et non-symétrique stabilisées par des bulles quadratiques]

Présenté par : Philippe G. Ciarlet

Erik Burman ¹ ; Benjamin Stamm ¹

¹ IACS/CMCS, station 8, École polytechnique fédérale de Lausanne, CH 1015, Lausanne, Switzerland

Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 103-106.

Résumé
Abstract

Dans cette Note, nous montrons qu'en deux et trois dimensions d'espace, les méthodes de Galerkine discontinue symétrique ou non-symétrique pour les problèmes elliptiques d'ordre deux sont stable pour l'ordre polynomial $p = 1$ sans devoir introduire de terme de stabilisation si l'espace est enrichi par des bulles quadratiques. La méthode fournit des ordres de convergence optimaux dans la norme d'énergie brisée et, pour la formulation symétrique, dans la norme $L^{2}$ et peut être écrite sous forme conservative avec des flux indépendants de tout paramètre de stabilisation.

In this Note we prove that in two and three space dimensions, the symmetric and non-symmetric discontinuous Galerkin methods for second order elliptic problems are stable when using piecewise linear elements enriched with quadratic bubbles without any penalization of the interelement jumps. The method yields optimal convergence rates in both the broken energy norm and, in the symmetric case, the $L^{2}$ -norm. Moreover the method can be written in conservative form with fluxes independent of any stabilization parameter.

Reçu le : 2007-07-06
Accepté le : 2007-11-22
Publié le : 2007-12-26

DOI : 10.1016/j.crma.2007.11.016

Affiliations des auteurs :

Erik Burman ¹ ; Benjamin Stamm ¹

¹ IACS/CMCS, station 8, École polytechnique fédérale de Lausanne, CH 1015, Lausanne, Switzerland

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     title = {Symmetric and non-symmetric discontinuous {Galerkin} methods stabilized using bubble enrichment},
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     pages = {103--106},
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     language = {en},
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Erik Burman; Benjamin Stamm. Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 103-106. doi : 10.1016/j.crma.2007.11.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.016/

Bibliographie
Cité par

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[3] G. Baker Finite element methods for elliptic equations using nonconforming elements, Math. Comp., Volume 31 (1977), pp. 44-59

[4] F. Brezzi; L.D. Marini Bubble stabilization of discontinuous Galerkin methods (W. Fitzgibbon; R. Hoppe; J. Periaux; O. Pironneau; Y. Vassilevski, eds.), Advances in Numerical Mathematics, Proc. International Conference on the occasion of the 60th birthday of Y.A. Kuznetsov, Institute of Numerical Mathematics of The Russian Academy of Sciences, Moscow, 2006, pp. 25-36

[5] E. Burman; A. Ern; I. Mozolevski; B. Stamm The symmetric discontinuous Galerkin method does not need stabilization in 1d for polynomial orders $p ⩾ 2$ , C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 10, pp. 599-602

[6] E. Burman, B. Stamm, Low order discontinuous Galerkin methods for second order elliptic problems, Technical Report 04-2007, EPFL-IACS, 2007

[7] M.G. Larson; A.J. Niklasson Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case, Numer. Math., Volume 99 (2004), pp. 113-130

[8] M.G. Larson; A.J. Niklasson Analysis of a nonsymmetric discontinuous Galerkin method for elliptic problems: stability and energy error estimates, SIAM J. Numer. Anal., Volume 42 (2004) no. 1, pp. 252-264

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