The G method for heterogeneous anisotropic diffusion on general meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 4, p. 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.

DOI : https://doi.org/10.1051/m2an/2010021
Classification:  65N08,  65N12
Keywords: finite volume methods, heterogeneous anisotropic diffusion, MPFA, convergence analysis
@article{M2AN_2010__44_4_597_0,
author = {Ag\'elas, L\'eo and Di Pietro, Daniele A. and Droniou, J\'er\^ome},
title = {The G method for heterogeneous anisotropic diffusion on general meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {4},
year = {2010},
pages = {597-625},
doi = {10.1051/m2an/2010021},
zbl = {1202.65143},
mrnumber = {2683575},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_4_597_0}
}

Agélas, Léo; Di Pietro, Daniele A.; Droniou, Jérôme. The G method for heterogeneous anisotropic diffusion on general meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 4, pp. 597-625. doi : 10.1051/m2an/2010021. http://www.numdam.org/item/M2AN_2010__44_4_597_0/

[1] I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405-432. | Zbl 1094.76550

[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 1128.65093

[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 1230.65114 | Zbl pre05320771

[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 1152.65107

[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 1152.65107

[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 0882.65154

[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 1108.65102

[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 1083.65099

[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 1094.65111

[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 05776268 | Zbl pre05776268

[18] J. Droniou, A density result in Sobolev spaces. J. Math. Pures Appl. 81 (2002) 697-714. | Zbl 1033.46029

[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 1109.65099

[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 1191.65142

[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 0981.65095

[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 1173.76028

[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 1202.65144

[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. | Numdam | Zbl 1116.65121