The symmetric discontinuous Galerkin method does not need stabilization in 1D for polynomial orders $ p⩾2$

Erik Burman; Alexandre Ern; Igor Mozolevski; Benjamin Stamm

doi:10.1016/j.crma.2007.10.028

Numerical Analysis

The symmetric discontinuous Galerkin method does not need stabilization in 1D for polynomial orders

p ⩾ 2

[La méthode de Galerkine discontinue symétrique est stable en une dimension d'espace pour tout ordre polynômial

p ⩾ 2

]

Présenté par : Philippe G. Ciarlet

Erik Burman ¹ ; Alexandre Ern ² ; Igor Mozolevski ^2,³ ; Benjamin Stamm ¹

¹ IACS/CMCS, Station 8, École polytechnique fédérale de Lausanne, CH 1015, Lausanne, Switzerland
² CERMICS, École des ponts, Université Paris-Est, 6 & 8, avenue Blaise-Pascal, 77455 Marne-la-Vallée cedex 2, France
³ Mathematics Department, Federal University of Santa Catarina, SC, 88040-900, Florianópolis, Brazil

Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 599-602.

Résumé
Abstract

Dans cette Note, nous montrons qu'en une dimension d'espace, la méthode de Galerkine discontinue symétrique pour les problèmes elliptiques d'ordre deux est stable pour tout ordre polynômial $p ⩾ 2$ sans devoir introduire de paramètre de stabilisation. La méthode fournit des ordres de convergence optimaux dans la norme d'énergie (norme $L^{2}$ du gradient brisé plus des termes de saut) et 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 one space dimension, the symmetric discontinuous Galerkin method for second order elliptic problems is stable for polynomial orders $p ⩾ 2$ without using any stabilization parameter. The method yields optimal convergence rates in both the energy norm ( $L^{2}$ -norm of broken gradient plus jump terms) and the $L^{2}$ -norm and can be written in conservative form with fluxes independent of any stabilization parameter.

Reçu le : 2007-05-11
Accepté le : 2007-10-12
Publié le : 2007-11-15

DOI : 10.1016/j.crma.2007.10.028

Affiliations des auteurs :

Erik Burman ¹ ; Alexandre Ern ² ; Igor Mozolevski ^{2, 3} ; Benjamin Stamm ¹

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     author = {Erik Burman and Alexandre Ern and Igor Mozolevski and Benjamin Stamm},
     title = {The symmetric discontinuous {Galerkin} method does not need stabilization in {1D} for polynomial orders $ p\ensuremath{\geqslant}2$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {599--602},
     publisher = {Elsevier},
     volume = {345},
     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.028},
     language = {en},
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Erik Burman; Alexandre Ern; Igor Mozolevski; Benjamin Stamm. The symmetric discontinuous Galerkin method does not need stabilization in 1D for polynomial orders $ p⩾2$. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 599-602. doi : 10.1016/j.crma.2007.10.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.028/

Bibliographie
Cité par

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

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

[4] A. Ern; J.-L. Guermond Discontinuous Galerkin methods for Friedrichs' systems. I. General theory, SIAM J. Numer. Anal., Volume 44 (2006) no. 2, pp. 753-778

[5] 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

[6] 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

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

[8] B. Rivière; M.F. Wheeler; V. Girault 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

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