A virtual volume method for heterogeneous and anisotropic diffusion-reaction problems on general meshes
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 797-824.

Starting from the recently introduced virtual element method, we construct new diffusion fluxes in two and three dimensions that give birth to symmetric, unconditionally coercive finite volume like schemes for the discretization of heterogeneous and anisotropic diffusion-reaction problems on general, possibly nonconforming meshes. Convergence of the approximate solutions is proved for general tensors and meshes. Error estimates are derived under classical regularity assumptions. Numerical results illustrate the performance of the scheme. The link with the original vertex approximate gradient scheme is emphasized.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016036
Classification : 65N08, 65N12, 65N15
Mots clés : Heterogeneous diffusion-reaction problems, finite volumes, general meshes, virtual element method
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     title = {A virtual volume method for heterogeneous and anisotropic diffusion-reaction problems on general meshes},
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     pages = {797--824},
     publisher = {EDP-Sciences},
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Coatléven, Julien. A virtual volume method for heterogeneous and anisotropic diffusion-reaction problems on general meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 797-824. doi : 10.1051/m2an/2016036. http://www.numdam.org/articles/10.1051/m2an/2016036/

I. Aavatsmark, T. Barkve, O. Boe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media part i: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. | DOI | MR | Zbl

I. Aavatsmark, T. Barkve, O. Boe and T. Mannseth, Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127 (1996) 2–14. | DOI | Zbl

I. Aavatsmark, T. Barkve, O. Boe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. part ii: Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717–1736. | DOI | MR | Zbl

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. Eq. 24 (2008) 1329–1360. | DOI | MR | Zbl

L. Agélas and R. Masson, Convergence of finite volume mpfa o type schemes for heterogeenous anisotropic diffusion problems on general meshes. C.R. Acad. Paris Ser. I 346 (2008). | MR | Zbl

L. Agélas, D.A. Di Pietro and J. Droniou, The g method for heterogeneous anisotropic diffusion on general meshes. ESAIM: M2AN 11 (2010) 597–625. | DOI | Numdam | MR | Zbl

B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini and A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391. | DOI | MR | Zbl

K. Brenner, M. Groza, C. Guichard and R. Masson, Vertex approximate gradient scheme for hybrid dimensional two-phase darcy flows in fractured porous media. Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems. Vol. 78 of Springer Proceedings in Mathematics & Statistics (2014) 507–515. | MR

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Springer (2008). | MR | Zbl

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR | Zbl

F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. | DOI | MR | Zbl

F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005) 1533–1551. | DOI | MR | Zbl

A. Cangiani, G. Manzini, A. Russo and N. Sukumar, Hourglass stabilization and the virtual element method. Int. J. Numer. Methods Eng. 102 (2015) 404–436. | DOI | MR | Zbl

L. Beirao Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl

L. Beirao da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. Springer (2014). | MR | Zbl

J. Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Special Edition “P.D.E. Discretizations on Polygonal Meshes”. M3AS 24 (2014) 1575–1619. | MR | Zbl

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

J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A unified approach to mimetic finite differences, hybrid finite volume and mixed finite volume methods. IMA J. Num. Anal. 31 (2011) 1357–1401. | Zbl

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. In Techniques of scientific computiing, Part III. Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.-L. Lions. North-Holland, Amsterdam (2000) 713–1020. | MR | Zbl

R. Eymard, T. Gallouët and R. Herbin, A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis. C. R. Math., Acad. Sci. Paris 344 (2007) 403–406. | DOI | MR | Zbl

R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilisation and hybrid interfaces. IMA J. Num. Anal. 30 (2010) 1009–1043. | DOI | MR | Zbl

R. Eymard, C. Guichard and R. Herbin, Small-stencil 3d schemes for diffusive flows in porous media. ESAIM: M2AN 46 (2011) 265–290. | DOI | Numdam | MR | Zbl

R. Eymard, C. Guichard, R. Herbin and R. Masson, Vertex-centred discretization of multiphase compositional darcy flows on general meshes. Comput. Geosci. 16 (2012) 987–1005. | DOI | MR

R. Eymard, C. Guichard, R. Herbin and R. Masson, Vertex centred discretization of two-phase darcy flows on general meshes. ESAIM: Proc. 35 (2012) 59–78. | DOI | MR | Zbl

D.A. Di Pietro, Cell centered galerkin methods for diffusive problems. ESAIM: M2AN 46 (2011) 111–144. | DOI | Numdam | MR | Zbl

D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods. Springer (2012). | MR | Zbl

D.A. Di Pietro and A. Ern, Hybrid high-order methods for variable-diffusion problems on general meshes. C. R. Acad. Sci. Paris, Ser. I 353 (2015) 31–34. | DOI | MR | Zbl

R. Herbin R. Eymard and C. Guichard, Benchmark 3d: the vag scheme. Springer proceedings in Mathematics, FVCA6, Prague 2 (2011) 213–222. | Zbl

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