Numerical Analysis
Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids
[Reconstruction du flux et estimations a posteriori pour la méthode de Galerkine discontinue sur des maillages non-coïncidants avec mailles polygonales]

Présenté par : Olivier Pironneau

Ern, Alexandre1 ; Vohralík, Martin2, 3
1 Université Paris-Est, CERMICS, École des Ponts, 6 & 8, avenue Blaise-Pascal, 77455 Marne-la-Vallée cedex 2, France
2 UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
3 CNRS, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 441-444.

Les méthodes de Galerkine discontinues sont bien adaptées pour traiter des maillages non-coïncidants avec mailles polygonales. Nous présentons dans cette Note une reconstruction H(div)-conforme du flux sur de tels maillages pour un problème elliptique. Nous exploitons la propriété de conservativité locale des méthodes de Galerkine discontinues afin de résoudre des problèmes locaux de Neumann approchés par des éléments finis mixtes de Raviart–Thomas–Nédélec. Notre reconstruction peut être utilisée pour une estimation garantie d'erreur a posteriori et également afin d'évaluer une vitesse approchée pour un problème de transport.

Discontinuous Galerkin methods handle very well general polygonal and nonmatching meshes. We present in this Note a H(div)-conforming reconstruction of the flux on such meshes in the setting of an elliptic problem. We exploit the local conservation property of discontinuous Galerkin methods and solve local Neumann problems by means of the Raviart–Thomas–Nédélec mixed finite element method. Our reconstruction can be used in a guaranteed a posteriori error estimate and it is also of independent interest when the approximate flux is to be used subsequently in a transport problem.

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DOI : 10.1016/j.crma.2009.01.017
Ern, Alexandre 1 ; Vohralík, Martin 2, 3

1 Université Paris-Est, CERMICS, École des Ponts, 6 & 8, avenue Blaise-Pascal, 77455 Marne-la-Vallée cedex 2, France
2 UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
3 CNRS, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France 
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Ern, Alexandre; Vohralík, Martin. Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids. Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 441-444. doi : 10.1016/j.crma.2009.01.017. https://www.numdam.org/articles/10.1016/j.crma.2009.01.017/

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