Comptes Rendus
no. 12
Numerical Analysis
An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems
[Une reconstruction précise du flux dans H(div) pour des approximations par la méthode de Galerkine discontinue de problèmes elliptiques]

Présenté par : Olivier Pironneau

Alexandre Ern1 ; Serge Nicaise2 ; Martin Vohralík3
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.

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 L2-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.

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 L2-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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.10.036
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},
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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/

[1] M. Ainsworth; J.T. Oden A Posteriori Error Estimation in Finite Element Analysis, John Wiley and Sons, New York, NY, 2000

[2] D.N. Arnold; F. Brezzi; B. Cockburn; D. Marini Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2002) no. 5, pp. 1749-1779

[3] P. Bastian; B. Rivière Superconvergence and H(div) projection for discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids, Volume 42 (2003) no. 10, pp. 1043-1057

[4] E. Burman; P. Zunino A domain decomposition method based on weighted interior penalties for advection–diffusion–reaction problems, SIAM J. Numer. Anal., Volume 44 (2006) no. 4, pp. 1612-1638 (electronic)

[5] S. Cochez-Dhondt, S. Nicaise, Equilibrated error estimators for discontinuous Galerkin methods, Technical report, Université de Valenciennes, 2007, NMPDE, submitted for publication

[6] A. Ern; J.-L. Guermond Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order elliptic PDEs, SIAM J. Numer. Anal., Volume 44 (2006) no. 6, pp. 2363-2388 (electronic)

[7] A. Ern, A.F. Stephansen, A posteriori energy-norm error estimates for advection–diffusion equations approximated by weighted interior penalty methods, Technical Report 364, Ecole nationale des ponts et chaussées, 2007

[8] A. Ern, A.F. Stephansen, M. Vohralík, Improved energy norm a posteriori error estimation based on flux reconstruction for discontinuous Galerkin methods, Technical report, Ecole nationale des ponts et chaussées and Université Pierre et Marie Curie, 2007

[9] A. Ern, A.F. Stephansen, P. Zunino, A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity, Technical Report 332, Ecole nationale des ponts et chaussées, 2007, IMAJNA, submitted for publication

[10] P. Neitaanmaäki; S. Repin Reliable Methods for Computer Simulation: Error Control and a Posteriori Error Estimates, Elsevier, Amsterdam, The Netherlands, 2004

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