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A mixed finite volume scheme for anisotropic diffusion problems on any grid

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Abstract

We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. The approximate solution is shown to converge to the continuous one as the size of the mesh tends to 0, and an error estimate is given. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids and for various diffusion matrices.

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References

  1. Aavatsmark I. (2002) An introduction to multipoint flux approximations for quadrilateral grids. Locally conservative numerical methods for flow in porous media. Comput. Geosci. 6, 405–432

    Article  MATH  MathSciNet  Google Scholar 

  2. Aavatsmark I., Barkve T., Boe O., Mannseth T. (1998) Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: derivation of the methods. J. Sci. Comput. 19, 1700–1716

    MATH  MathSciNet  Google Scholar 

  3. Aavatsmark I., Barkve T., Boe O., Mannseth T. (1998) Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results. SIAM J. Sci. Comput. 19, 1717–1736

    Article  MATH  MathSciNet  Google Scholar 

  4. Arbogast T., Cowsar L.C., Wheeler M.F., Yotov I. (2000) Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37(4): 1295–1315

    Article  MATH  MathSciNet  Google Scholar 

  5. Arbogast T., Wheeler M.F., Yotov I. (1997) Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34(2): 828–852

    Article  MATH  MathSciNet  Google Scholar 

  6. Chavent G., Cohen G., Jaffré J. (1984) Discontinuous upwinding and mixed finite elements for two-phase flows in reservoir simulation. Comput. Methods Appl. Mech. Eng. 47, 93–118

    Article  MATH  Google Scholar 

  7. Chen Z. (1998) Expanded mixed finite element methods for linear second-order elliptic problems. I. RAIRO, Modélisation. Math. Anal. Numér. 32(4): 479–499

    MATH  Google Scholar 

  8. Chen Z. (1998) Expanded mixed finite element methods for quasilinear second order elliptic problems. II. RAIRO, Modélisation. Math. Anal. Numér. 32(4): 501–520

    MATH  Google Scholar 

  9. Croisille J-P. (2000) Finite volume box-schemes and mixed methods. Math. Model. Numer. Anal. 34(5): 1087–1106

    Article  MATH  MathSciNet  Google Scholar 

  10. Droniou J. (2003) Error estimates for the convergence of a finite volume discretization of convection-diffusion equations. J. Numer. Math. 11, 1–32

    Article  MATH  MathSciNet  Google Scholar 

  11. Droniou J., Eymard R., Hilhorst D., Zhou X.D. (2003) Convergence of a finite volume - mixed finite element method for a system of a hyperbolic and an elliptic equations. IMA J. Numer. Anal. 23, 507–538

    Article  MATH  MathSciNet  Google Scholar 

  12. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. vol. 7, pp. 713–1020 North Holland, (2000)

  13. Eymard R., Gallouët T., Herbin R. (2004) A finite volume for anisotropic diffusion problems. Comptes Rendus l’Acad. Sci. 339, 299–302

    Article  MATH  Google Scholar 

  14. Eymard, R., Gallouët, T., Herbin, R.: A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal. Adv. Access (2005). DOI: 10.1093/imanum/dri036

  15. Eymard R., Gallouët T., Herbin R. (2001) Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math. 37, 31–53

    Article  MATH  MathSciNet  Google Scholar 

  16. Faille I. (1992) A control volume method to solve an elliptic equation on a two- dimensional irregular mesh. Comput. Methods Appl. Mech. Eng. 100, 275–290

    Article  MATH  MathSciNet  Google Scholar 

  17. Gallouët T., Herbin R., Vignal M.H. (2000) Error estimate for the approximate finite volume solutions of convection diffusion equations with Dirichlet, Neumann or Fourier boundary conditions. SIAM J. Numer. Anal. 37(6): 1935–1972

    Article  MATH  MathSciNet  Google Scholar 

  18. Kershaw D.S. (1981) Differencing of the diffusion equation in Lagrangian hydrodynamic codes. J. Comput. Phys. 39, 375–395

    Article  MATH  MathSciNet  Google Scholar 

  19. Klausen R.A., Russell T.F. (2004) Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci. 8, 341–377

    Article  MathSciNet  Google Scholar 

  20. Kuznetsov Y., Repin S. (2005) Convergence analysis and error estimates for mixed finite element method on distorted meshes. J. Numer. Math. 13(1): 33–51

    Article  MATH  MathSciNet  Google Scholar 

  21. Le Potier C. (2005) A finite volume method for the approximation of highly anisotropic diffusion operators on unstructured meshes. Finite Volumes for Complex Applications IV. Marrakesh, Marocco

    Google Scholar 

  22. Lipnikov K., Morel J., Shashkov M. (2004) Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes (english). J. Comput. Phys. 199, 589–597

    Article  MATH  Google Scholar 

  23. Roberts, J.E., Thomas, J.M.: Mixed and hybrid methods. In: Ciarlet, P.G. et al. (ed.) Handbook of Numerical Analysis, vol. 2, pp. 523–639. North-Holland (1991)

  24. Younes A., Ackerer P., Chavent G. (2004) From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions. Int. J. Numer. Methods Eng. 59, 365–388

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques, UMR CNRS 5149, CC 051, Université Montpellier II, Place Eugène Bataillon, 34095, Montpellier cedex 5, France

    Jérôme Droniou

  2. Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050, Université de Marne-la-Vallée, 5 boulevard Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée Cedex 2, France

    Robert Eymard

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  1. Jérôme Droniou
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  2. Robert Eymard
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Correspondence to Robert Eymard.

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Droniou, J., Eymard, R. A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105, 35–71 (2006). https://doi.org/10.1007/s00211-006-0034-1

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  • DOI: https://doi.org/10.1007/s00211-006-0034-1

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