The G method for heterogeneous anisotropic diffusion on general meshes

Agélas, Léo; Di Pietro, Daniele A.; Droniou, Jérôme

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 4, pp. 597-625.

In the present work we introduce a new family of cell-centered Finite Volume schemes for anisotropic and heterogeneous diffusion operators inspired by the MPFA L method. A very general framework for the convergence study of finite volume methods is provided and then used to establish the convergence of the new method. Fairly general meshes are covered and a computable sufficient criterion for coercivity is provided. In order to guarantee consistency in the presence of heterogeneous diffusivity, we introduce a non-standard test space in $H_0^1$(Ω) and prove its density. Thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with classical multi-point Finite Volume methods is provided.

[1] I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405-432. | Zbl

[2] I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on non-orthogonal, curvilinear grids for multi-phase flow, in Proc. of the 4th European Conf. on the Mathematics of Oil Recovery (Røros, Norway), Vol. D (1994).

[3] I. Aavatsmark, G.T. Eigestad, B.T. Mallison, J.M. Nordbotten and E. Øian, A new finite volume approach to efficient discretization on challeging grids, in Proc. SPE 106435, Houston, USA (2005).

[4] I. Aavatsmark, G.T. Eigestad, R.A. Klausen, M.F. Wheeler and I. Yotov, Convergence of a symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11 (2007) 333-345. | Zbl

[5] I. Aavatsmark, G.T. Eigestad, B.T. Mallison and J.M. Nordbotten, A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ. 24 (2008) 1329-1360. | Zbl

[6] L. Agélas and D.A. Di Pietro, A symmetric finite volume scheme for anisotropic heterogeneous second-order elliptic problems, in Finite Volumes for Complex Applications, V.R. Eymard and J.-M. Hérard Eds., John Wiley & Sons (2008) 705-716.

[7] L. Agélas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes. C. R. Acad. Sci. Paris, Sér. I 346 (2008) 1007-1012. | Zbl

[8] L. Agélas and R. Masson, Convergence of finite volume MPFA O type schemes for heterogeneous anisotropic diffusion problems on general meshes. Preprint available at http://hal.archives-ouvertes.fr/hal-00340159/fr (2008). | Zbl

[9] L. Agélas, D.A. Di Pietro and R. Masson, A symmetric and coercive finite volume scheme for multiphase porous media flow with applications in the oil industry, in Finite Volumes for Complex Applications, V.R. Eymard and J.-M. Hérard Eds., John Wiley & Sons (2008) 35-52.

[10] L. Agélas, D.A. Di Pietro, R. Eymard and R. Masson, An abstract analysis framework for nonconforming approximations of diffusion problems on general meshes. IJFV 7 (2010) 1-29.

[11] S. Balay, W.D. Gropp, L.C. Mcinnes and B.F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A.M. Bruaset and H.P. Langtangen Eds., Birkhäuser Press (1997) 163-202. | Zbl

[12] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. Mcinnes, B.F. Smith and H. Zhang, PETSc Web page (2001) www.mcs.anl.gov/petsc.

[13] S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. Mcinnes, B.F. Smith and H. Zhang, PETSc users manual. Tech. Report ANL-95/11 - Revision 2.1.5, Argonne National Laboratory (2004).

[14] F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 45 (2005) 1872-1896. | Zbl

[15] F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Mod. Meths. Appli. Sci. 15 (2005) 1533-1553. | Zbl

[16] F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes with curved faces. Math. Mod. Meths. Appli. Sci. 26 (2006) 275-298. | Zbl

[17] D.A. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math. Comp. (2010), preprint available at http://hal.archives-ouvertes.fr/hal-00278925/fr/. | Zbl

[18] J. Droniou, A density result in Sobolev spaces. J. Math. Pures Appl. 81 (2002) 697-714. | Zbl

[19] J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006) 35-71. | Zbl

[20] J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods. Maths. Models Methods Appl. Sci. 20 (2010) 1-31. | Zbl

[21] M.G. Edwards and C.F. Rogers, A flux continuous scheme for the full tensor pressure equation, in Proc. of the 4th European Conf. on the Mathematics of Oil Recovery (Røros, Norway), Vol. D (1994).

[22] R. Eymard, T. Gallouët and R. Herbin, The finite volume method, Ph.G. Charlet and J.-L. Lions Eds., North Holland (2000). | Zbl

[23] R. Eymard, R. Herbin and J.C. Latché, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes. SIAM J. Numer. Anal. 45 (2007) 1-36. | Zbl

[24] R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. (2009) doi: 10.1093/imanum/drn084. | Zbl

[25] M. Vohralík, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM: M2AN 40 (2006) 367-391. | EuDML | Numdam | Zbl