An accurate $ \mathbf{H}(\mathrm{div})$ flux reconstruction for discontinuous Galerkin approximations of elliptic problems

Alexandre Ern; Serge Nicaise; Martin Vohralík

doi:10.1016/j.crma.2007.10.036

Numerical Analysis

An accurate

H (div)

flux reconstruction for discontinuous Galerkin approximations of elliptic problems

Presented by: Olivier Pironneau

Alexandre Ern ¹; Serge Nicaise ²; Martin Vohralík ³

¹ CERMICS, École des ponts, Université Paris-Est, 6 & 8 avenue B. Pascal, 77455 Marne-la-Vallée cedex 2, France
² LAMAV, Université de Valenciennes and CNRS, 59313 Valenciennes cedex, France
³ LJLL, Université Pierre et Marie Curie (Paris 6), B.C. 187, 4, place Jussieu, 75252 Paris cedex 5, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 709-712.

Abstract
Résumé

We introduce a new $H (div)$ flux reconstruction for discontinuous Galerkin approximations of elliptic problems. The reconstructed flux is computed elementwise and its divergence equals the $L^{2}$ -orthogonal projection of the source term onto the discrete space. Moreover, the energy-norm of the error in the flux is bounded by the discrete energy-norm of the error in the primal variable, independently of diffusion heterogeneities.

On introduit une nouvelle reconstruction dans $H (div)$ du flux pour des approximations par la méthode de Galerkine discontinue de problèmes elliptiques. Le flux reconstruit est calculé localement sur chaque maille et sa divergence est égale à la projection $L^{2}$ -orthogonale du terme source sur l'espace discret. De plus, l'erreur en norme d'énergie sur le flux est bornée par l'erreur en norme d'énergie discrète sur la variable primale, indépendamment des hétérogénéités dans la diffusion.

Received: 2007-10-03
Accepted: 2007-10-12
Published online: 2007-11-19

DOI: 10.1016/j.crma.2007.10.036

Author's affiliations:

Alexandre Ern ¹; Serge Nicaise ²; Martin Vohralík ³

¹ CERMICS, École des ponts, Université Paris-Est, 6 & 8 avenue B. Pascal, 77455 Marne-la-Vallée cedex 2, France
² LAMAV, Université de Valenciennes and CNRS, 59313 Valenciennes cedex, France
³ LJLL, Université Pierre et Marie Curie (Paris 6), B.C. 187, 4, place Jussieu, 75252 Paris cedex 5, France

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     author = {Alexandre Ern and Serge Nicaise and Martin Vohral{\'\i}k},
     title = {An accurate $ \mathbf{H}(\mathrm{div})$ flux reconstruction for discontinuous {Galerkin} approximations of elliptic problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {709--712},
     publisher = {Elsevier},
     volume = {345},
     number = {12},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.036},
     language = {en},
}

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Alexandre Ern; Serge Nicaise; Martin Vohralík. An accurate $ \mathbf{H}(\mathrm{div})$ flux reconstruction for discontinuous Galerkin approximations of elliptic problems. Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 709-712. doi : 10.1016/j.crma.2007.10.036. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.036/

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