Comptes Rendus
no. 3-4
Numerical Analysis
On the conservativity of cell-centered Galerkin methods
[Sur la conservativité des méthodes de Galerkine centrées aux mailles]

Présenté par : Philippe G. Ciarlet

Daniele A. Di Pietro1
1 University of Montpellier-2, I3M, place Eugène-Bataillon, 34057 Montpellier cedex 5, France
Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 155-159.

Dans cette note, on étudie la conservativité de la méthode de Galerkine centrée aux mailles de Di Pietro (2012) [5] et on fournit une expression analytique pour le flux numérique. Le lien avec la méthode SUSHI de Eymard et al. (2010) [10] et avec les méthodes de Galerkine discontinues est aussi détaillé. Les résultats théoriques sont validés à la fois sur des maillages standard et polygonaux.

In this work we investigate the conservativity of the cell-centered Galerkin method of Di Pietro (2012) [5] and provide an analytical expression for the conservative flux. The relation with the SUSHI method of Eymard et al. (2010) [10] and with discontinuous Galerkin methods is also explored. The theoretical results are assessed on a numerical example using standard as well as general polygonal grids.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.03.001
Daniele A. Di Pietro 1

1 University of Montpellier-2, I3M, place Eugène-Bataillon, 34057 Montpellier cedex 5, France 
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Daniele A. Di Pietro. On the conservativity of cell-centered Galerkin methods. Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 155-159. doi : 10.1016/j.crma.2013.03.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.03.001/

[1] L. Agélas; D.A. Di Pietro; J. Droniou The G method for heterogeneous anisotropic diffusion on general meshes, M2AN Math. Model. Numer. Anal., Volume 44 (2010) no. 4, pp. 597-625

[2] E. Burman; B. Stamm Local discontinuous Galerkin method with reduced stabilization for diffusion equations, Commun. Comput. Phys., Volume 5 (2009), pp. 498-524

[3] C. Dawson; S. Sun; M.F. Wheeler Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Engrg., Volume 193 (2004), pp. 2565-2580

[4] D.A. Di Pietro Cell centered Galerkin methods, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 31-34

[5] D.A. Di Pietro Cell centered Galerkin methods for diffusive problems, M2AN Math. Model. Numer. Anal., Volume 46 (2012) no. 1, pp. 111-144

[6] D.A. Di Pietro; A. Ern Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications, vol. 69, Springer-Verlag, Berlin, 2011

[7] D.A. Di Pietro, S. Lemaire, An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow, preprint hal-00753660, 2012.

[8] D.A. Di Pietro; S. Nicaise A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media, Appl. Numer. Math., Volume 63 (2013), pp. 105-116

[9] J. Droniou; R. Eymard A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math., Volume 105 (2006) no. 1, pp. 35-71

[10] R. Eymard; T. Gallouët; R. Herbin Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., Volume 30 (2010) no. 4, pp. 1009-1043

[11] R. Herbin; F. Hubert Benchmark on discretization schemes for anisotropic diffusion problems on general grids (R. Eymard; J.-M. Hérard, eds.), Finite Volumes for Complex Applications V, John Wiley & Sons, 2008, pp. 659-692

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