no. 1-2
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

Burman, Erik1 ; Stamm, Benjamin1
1 IACS/CMCS, station 8, École polytechnique fédérale de Lausanne, CH 1015, Lausanne, Switzerland
Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 103-106.

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 L2 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 L2-norm. Moreover the method can be written in conservative form with fluxes independent of any stabilization parameter.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.11.016
Burman, Erik 1 ; Stamm, Benjamin 1

1 IACS/CMCS, station 8, École polytechnique fédérale de Lausanne, CH 1015, Lausanne, Switzerland 
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Burman, Erik; Stamm, Benjamin. Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment. Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 103-106. doi : 10.1016/j.crma.2007.11.016. http://www.numdam.org/articles/10.1016/j.crma.2007.11.016/

[1] Arnold, D.N. An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., Volume 19 (1982) no. 4, pp. 742-760

[2] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, L.D. Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2001/2002) no. 5, pp. 1749-1779 (electronic)

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

[4] Brezzi, F.; Marini, L.D. Bubble stabilization of discontinuous Galerkin methods (Fitzgibbon, W.; Hoppe, R.; Periaux, J.; Pironneau, O.; Vassilevski, Y., 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] Burman, E.; Ern, A.; Mozolevski, I.; Stamm, B. 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] Larson, M.G.; Niklasson, A.J. Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case, Numer. Math., Volume 99 (2004), pp. 113-130

[8] Larson, M.G.; Niklasson, A.J. 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

[9] Oden, J.T.; Babuška, I.; Baumann, C. A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., Volume 146 (1998) no. 2, pp. 491-519

[10] Rivière, B.; Wheeler, M.F.; Girault, V. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2001) no. 3, pp. 902-931 (electronic)

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