Numerical Analysis
A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media
Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 673-676.

We propose a cell-centered symmetric scheme which combines the advantages of MPFA (multipoint flux approximation) schemes such as the L or the O scheme and of hybrid schemes: it may be used on general non-conforming meshes, it yields a 9-point stencil on two-dimensional quadrangular meshes, it takes into account the heterogeneous diffusion matrix, it is coercive and it can be shown to converge. The scheme relies on the use of special points, called harmonic averaging points, located at the interfaces of heterogeneity.

Nous proposons un schéma ayant ses inconnues aux centres des mailles, combinant les avantages des schémas à flux multi-points et des schémas hybrides : il possède un stencil à 9 points en 2D, respecte les hétérogénéités de la matrice de diffusion, et il est coercif ; de plus, on peut montrer qu'il converge. Le schéma est basé sur l'utilisation de points situé aux interfaces d'hétérogénéité, en lesquels la formule de la moyenne harmonique est utilisable.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.03.013
Agelas, Léo 1; Eymard, Robert 2; Herbin, Raphaèle 3

1 Institut Français du Pétrole, 1 & 4, avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France
2 Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050, Université Paris-Est, 5, boulevard Descartes Champs-sur-Marne, F-77454 Marne La Vallée Cedex 2, France
3 Laboratoire d'analyse, topologie et probabilités, Université de Provence, 39, rue Joliot-Curie, 13453 Marseille, France
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Agelas, Léo; Eymard, Robert; Herbin, Raphaèle. A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 673-676. doi : 10.1016/j.crma.2009.03.013. http://www.numdam.org/articles/10.1016/j.crma.2009.03.013/

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[5] Eymard, R.; Gallouët, T.; Herbin, R. Benchmark on anisotropic problems, SUSHI: a scheme using stabilization and hybrid interfaces for anisotropic heterogeneous diffusion problems (Eymard, R.; Hérard, J.-M., eds.), Finite Volumes for Complex Applications, vol. V, Wiley, 2008, pp. 801-814

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Work supported by Groupement de Recherche MOMAS, PACEN/CNRS.