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Local flux mimetic finite difference methods

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Abstract

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L 2-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.

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References

  1. Aavatsmark I.: An introduction to multipoint flux approximations for quadrilateral grids. Comp. Geosci. 6, 405–432 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Adams, R.A.: Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London. Pure Appl. Math., vol. 65 (1975)

  4. Arbogast T., Dawson C.N., Keenan P.T., Wheeler M.F., Yotov I.: Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19, 404–425 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Arnold D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arnold D.N., Brezzi F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modèl. Math. Anal. Numèr. 19, 7–32 (1985)

    MATH  MathSciNet  Google Scholar 

  8. Baranger J., Maitre J.-F., Oudin F.: Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30, 445–465 (1996)

    MATH  MathSciNet  Google Scholar 

  9. Berndt M., Lipnikov K., Moulton J.D., Shashkov M.: Convergence of mimetic finite difference discretizations of the diffusion equation. East-West J. Numer. Math. 9, 253–284 (2001)

    MathSciNet  Google Scholar 

  10. Berndt M., Lipnikov K., Shashkov M., Wheeler M.F., Yotov I.: A mortar mimetic finite difference method on non-matching grids. Numer. Math. 102, 203–230 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Berndt M., Lipnikov K., Shashkov M., Wheeler M.F., Yotov I.: Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM J. Numer. Anal. 43, 1728–1749 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Breil J., Maire P.-H.: A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys. 224, 785–823 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Brenner S., Scott L.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994)

    MATH  Google Scholar 

  14. Brezzi F., Douglas J. Jr, Marini L.D.: Two families of mixed elements for second order elliptic problems. Numer. Math. 88, 217–235 (1985)

    Article  MathSciNet  Google Scholar 

  15. Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    MATH  Google Scholar 

  16. Brezzi F., Lipnikov K., Shashkov M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43, 1872–1896 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Brezzi F., Lipnikov K., Simoncini V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Mod. Methods. Appl. Sci. 15, 1533–1552 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Campbell J., Shashkov M.: A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172, 739–765 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1978)

    MATH  Google Scholar 

  20. Crumpton P., Shaw G., Ware A.: Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. J. Comput. Phys. 116, 343–358 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Edwards M.G., Rogers C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259–290 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, vol. VII. Handb. Numer. Anal., VII, pp. 713–1020. North-Holland, Amsterdam (2000)

  23. Eymard R., Gallouët T., Herbin R.: A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal. 26, 326–353 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  25. Grisvard P.: Elliptic Problems in Nonsmooth domains. Pitman, London (1985)

    MATH  Google Scholar 

  26. Hyman J., Shashkov M.: The approximation of boundary conditions for mimetic finite difference methods. Comput. Math. Appl. 36, 79–99 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hyman J., Shashkov M.: Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion. Prog. Electromagnetic Res. 32, 89–121 (2001)

    Article  Google Scholar 

  28. Hyman J., Shashkov M., Steinberg S.: The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132, 130–148 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Klausen R.A., Winther R.: Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differ. Equations 22, 1438–1454 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Klausen R.A., Winther R.: Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104, 317–337 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kuznetsov Y., Repin S.: New mixed finite element method on polygonal and polyhedral meshes. Russian J. Numer. Anal. Math. Modelling 18, 261–278 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  34. Le Potier C.: Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés. C. R. Math. Acad. Sci. Paris 340, 921–926 (2005)

    MATH  MathSciNet  Google Scholar 

  35. Lions J.L., Magenes E.: Non-homogeneous boundary value problems and applications, vol. 1. Springer, Berlin (1972)

    Google Scholar 

  36. Lipnikov K., Shashkov M., Svyatskiy D.: The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys. 211, 473–491 (2005)

    Article  MathSciNet  Google Scholar 

  37. Margolin L., Shashkov M., Smolarkiewicz P.: A discrete operator calculus for finite difference approximations. Comput. Methods Appl. Mech. Eng. 187, 365–383 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  38. Mathew, T.: Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems. PhD thesis, Courant Institute of Mathematical Sciences, New York University, Tech. Rep. 463 (1989)

  39. Micheletti S., Sacco R., Saleri F.: On some mixed finite element methods with numerical integration. SIAM J. Sci. Comput. 23, 245–270 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  40. Morel, J., Moulton, J., Shashkov, M.: Mimetic preconditioners for mixed discretizations of the diffusion equation. Tech. Report LA-UR-01-807, Los Alamos National Laboratory (2001). http://www.ima.umn.edu/talks/workshops/5-11-15.2004/moulton/moulton.pdf.

  41. Morel J., Roberts R., Shashkov M.: A local support-operators diffusion discretization scheme for quadrilateral r − z meshes. J. Comput. Phys. 144, 17–51 (1998)

    Article  MathSciNet  Google Scholar 

  42. Raviart P., Thomas J.-M.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds) Mathematical Aspects of the Finite Element Method., pp. 292–315. Springer, Berlin (1977)

    Chapter  Google Scholar 

  43. Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R.E., (ed.) The Mathematics of Reservoir Simulation, vol. 1, pp. 35–106. Frontiers in Applied Mathematics. SIAM, Philadelphia, PA (1983)

  44. Stüben K.: Algebraic multigrid (AMG): experiences and comparisons. Appl. Math. Comput. 13, 419–452 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  45. Temam R.: Navier–Stokes equations. Theory and numerical analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI (2001)

    Google Scholar 

  46. Vassilevski P.S., Petrova S.I., Lazarov R.D.: Finite difference schemes on triangular cell-centered grids with local refinement. SIAM J. Sci. Statist. Comput. 13, 1287–1313 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  47. Wheeler M.F., Yotov I.: A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44, 2082–2106 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Mail Stop B284, Los Alamos, NM, 87545, USA

    Konstantin Lipnikov & Mikhail Shashkov

  2. Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA, 15260, USA

    Ivan Yotov

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  1. Konstantin Lipnikov
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  2. Mikhail Shashkov
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  3. Ivan Yotov
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Correspondence to Konstantin Lipnikov.

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Lipnikov, K., Shashkov, M. & Yotov, I. Local flux mimetic finite difference methods. Numer. Math. 112, 115–152 (2009). https://doi.org/10.1007/s00211-008-0203-5

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  • DOI: https://doi.org/10.1007/s00211-008-0203-5

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