Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids

Alexandre Ern; Martin Vohralík

doi:10.1016/j.crma.2009.01.017

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

Alexandre Ern ¹ ; Martin Vohralík ^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

Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 441-444.

Résumé
Abstract

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.

Reçu le : 2008-12-12
Accepté le : 2009-01-16
Publié le : 2009-03-04

DOI : 10.1016/j.crma.2009.01.017

Affiliations des auteurs :

Alexandre Ern ¹ ; Martin Vohralík ^{2, 3}

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     author = {Alexandre Ern and Martin Vohral{\'\i}k},
     title = {Flux reconstruction and a posteriori error estimation for discontinuous {Galerkin} methods on general nonmatching grids},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {441--444},
     publisher = {Elsevier},
     volume = {347},
     number = {7-8},
     year = {2009},
     doi = {10.1016/j.crma.2009.01.017},
     language = {en},
}

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%A Alexandre Ern
%A Martin Vohralík
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Alexandre Ern; Martin Vohralík. Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 441-444. doi : 10.1016/j.crma.2009.01.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.017/

Bibliographie
Cité par

[1] M. Ainsworth A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. Numer. Anal., Volume 45 (2007) no. 4, pp. 1777-1798

[2] M. Ainsworth, R. Rankin, Fully computable error bounds for discontinuous Galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes, Research Report 9, University of Strathclyde, 2008

[3] S. Cochez-Dhondt; S. Nicaise Equilibrated error estimators for discontinuous Galerkin methods, Numer. Methods Partial Differential Equations, Volume 24 (2008) no. 5, pp. 1236-1252

[4] A. Ern; S. Nicaise; M. Vohralík An accurate $H (div)$ flux reconstruction for discontinuous Galerkin approximations of elliptic problems, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 12, pp. 709-712

[5] A. Ern, A.F. Stephansen, M. Vohralík, Improved energy norm a posteriori error estimation based on flux reconstruction for discontinuous Galerkin methods, HAL Preprint 00193540 version 1 (14-11-2007), 2007

[6] A. Ern, A.F. Stephansen, M. Vohralík, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection–diffusion–reaction problems, HAL Preprint 00193540, submitted for publication, 2008

[7] A. Ern, A.F. Stephansen, P. Zunino, A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., (electronic), 2008 | DOI

[8] K.Y. Kim A posteriori error estimators for locally conservative methods of nonlinear elliptic problems, Appl. Numer. Math., Volume 57 (2007) no. 9, pp. 1065-1080

[9] R. Luce; B.I. Wohlmuth A local a posteriori error estimator based on equilibrated fluxes, SIAM J. Numer. Anal., Volume 42 (2004) no. 4, pp. 1394-1414

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