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.

Abstract (VO)
Résumé

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.

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.

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

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

@article{CRMATH_2007__345_10_599_0,
     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},
}

TY  - JOUR
AU  - Erik Burman
AU  - Alexandre Ern
AU  - Igor Mozolevski
AU  - Benjamin Stamm
TI  - The symmetric discontinuous Galerkin method does not need stabilization in 1D for polynomial orders $ p⩾2$
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 599
EP  - 602
VL  - 345
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2007.10.028
LA  - en
ID  - CRMATH_2007__345_10_599_0
ER  -

%0 Journal Article
%A Erik Burman
%A Alexandre Ern
%A Igor Mozolevski
%A Benjamin Stamm
%T The symmetric discontinuous Galerkin method does not need stabilization in 1D for polynomial orders $ p⩾2$
%J Comptes Rendus. Mathématique
%D 2007
%P 599-602
%V 345
%N 10
%I Elsevier
%R 10.1016/j.crma.2007.10.028
%G en
%F CRMATH_2007__345_10_599_0

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

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

[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

Jingjing Yin Mixed Discontinuous Galerkin Method for One-dimensional Biharmonic Equations without Penalty Terms, Journal of Physics: Conference Series, Volume 2660 (2023) no. 1, p. 012022 | DOI:10.1088/1742-6596/2660/1/012022
Konstantinos G. Eptaimeros; Constantinos Chr. Koutsoumaris Discontinuous Galerkin FEMs for a gradient beam in static carbon nanotube applications, Mathematical Methods in the Applied Sciences (2021) | DOI:10.1002/mma.7355
K. G. Eptaimeros; C. Chr. Koutsoumaris; G. J. Tsamasphyros Interior penalty discontinuous Galerkin FEMs for a gradient beam and cnts, Applied Numerical Mathematics, Volume 144 (2019), pp. 118-139 | DOI:10.1016/j.apnum.2019.05.020 | Zbl:1451.74188
Hailiang Liu; Peimeng Yin A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems, Journal of Scientific Computing, Volume 77 (2018) no. 1, pp. 467-501 | DOI:10.1007/s10915-018-0756-0 | Zbl:1407.65168
H. Barucq; A. Bendali; M. Fares; V. Mattesi; S. Tordeux A symmetric Trefftz-DG formulation based on a local boundary element method for the solution of the Helmholtz equation, Journal of Computational Physics, Volume 330 (2017), pp. 1069-1092 | DOI:10.1016/j.jcp.2016.09.062 | Zbl:1380.65360
Xiaobing Feng; Thomas Lewis; Michael Neilan Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations, Journal of Computational and Applied Mathematics, Volume 299 (2016), pp. 68-91 | DOI:10.1016/j.cam.2015.10.024 | Zbl:1333.65129
W. Feng; T. L. Lewis; S. M. Wise Discontinuous Galerkin derivative operators with applications to second-order elliptic problems and stability, Mathematical Methods in the Applied Sciences, Volume 38 (2015) no. 18, pp. 5160-5182 | DOI:10.1002/mma.3440 | Zbl:1336.65190
Erik Burman; Benjamin Stamm Bubble stabilized discontinuous Galerkin method for parabolic and elliptic problems, Numerische Mathematik, Volume 116 (2010) no. 2, pp. 213-241 | DOI:10.1007/s00211-010-0304-9 | Zbl:1207.65135
Erik Burman; Benjamin Stamm Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 346 (2008) no. 1-2, pp. 103-106 | DOI:10.1016/j.crma.2007.11.016 | Zbl:1133.65096

Cité par 9 documents. Sources : Crossref, zbMATH

Commentaires - Politique