[Approximation par la méthode de Galerkine discontinue avec un principe variationnel discret pour un Laplacien non-linéaire]
On analyse une méthode de Galerkine discontinue afin d'approcher le problème modèle du Laplacien non-linéaire. La propriété essentielle du schéma proposé est que celui-ci jouit d'un principe variationnel discret. On prouve la convergence des approximations discrètes vers la solution exacte.
A discontinuous Galerkin method is analyzed to approximate the nonlinear Laplacian model problem. The salient feature of the proposed scheme is that it is endowed with a discrete variational principle. The convergence of the discrete approximations to the exact solution is proven.
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@article{CRMATH_2008__346_17-18_1013_0,
author = {Burman, Erik and Ern, Alexandre},
title = {Discontinuous {Galerkin} approximation with discrete variational principle for the nonlinear {Laplacian}},
journal = {Comptes Rendus. Math\'ematique},
pages = {1013--1016},
publisher = {Elsevier},
volume = {346},
number = {17-18},
year = {2008},
doi = {10.1016/j.crma.2008.07.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.07.005/}
}
TY - JOUR AU - Burman, Erik AU - Ern, Alexandre TI - Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian JO - Comptes Rendus. Mathématique PY - 2008 SP - 1013 EP - 1016 VL - 346 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.07.005/ DO - 10.1016/j.crma.2008.07.005 LA - en ID - CRMATH_2008__346_17-18_1013_0 ER -
%0 Journal Article %A Burman, Erik %A Ern, Alexandre %T Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian %J Comptes Rendus. Mathématique %D 2008 %P 1013-1016 %V 346 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.07.005/ %R 10.1016/j.crma.2008.07.005 %G en %F CRMATH_2008__346_17-18_1013_0
Burman, Erik; Ern, Alexandre. Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian. Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 1013-1016. doi : 10.1016/j.crma.2008.07.005. http://www.numdam.org/articles/10.1016/j.crma.2008.07.005/
[1] Finite element approximation of the p-Laplacian, Math. Comp., Volume 61 (1993) no. 204, pp. 523-537
[2] A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions, SIAM J. Sci. Comput., Volume 26 (2004) no. 1, pp. 152-177 (electronic)
[3] Finite element error estimates for nonlinear elliptic equations of monotone type, Numer. Math., Volume 54 (1989) no. 4, pp. 373-393
[4] The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002
[5] D.A. Di Pietro, A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations, 2008, submitted for publication, Hal document 00278925
[6] Discontinuous Galerkin methods for non-linear elasticity, Internat. J. Numer. Methods Engrg., Volume 67 (2006) no. 9, pp. 1204-1243
[7] Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité, d'une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle (RAIRO Analyse Numérique), Volume 9 (1975) no. R-2, pp. 41-76
[8] Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian, Numer. Math., Volume 89 (2001) no. 2, pp. 341-378
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