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A mortar mimetic finite difference method on non-matching grids

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Abstract

We consider mimetic finite difference approximations to second order elliptic problems on non-matching multiblock grids. Mortar finite elements are employed on the non-matching interfaces to impose weak flux continuity. Optimal convergence and, in certain cases, superconvergence is established for both the scalar variable and its flux. The theory is confirmed by computational results.

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References

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

    MathSciNet  Google Scholar 

  2. 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. Comp. 19(2), 404–425 (1998)

    MathSciNet  Google Scholar 

  3. Ben Belgacem, F.: The mortar finite element method with Lagrange multipliers. Numer. Math. 84(2), 173–197 (1999)

    Google Scholar 

  4. Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. Nonlinear partial differential equations and their applications In: H. Brezis and J. L. Lions (eds.) Longman Scientific & Technical UK 1994

  5. 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 

  6. 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. to appear

  7. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer-Verlag, New York 1991

  8. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. to appear

  9. 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 

  10. Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland, New York, 1978

  11. Ewing, R.E., Lazarov, R.D., Russell, T.F., Vassilevski, P.S.: Analysis of the mixed finite element method for rectangular Raviart-Thomas elements with local refinement. In: Third International Symposium on Domain Decomposition Methods for Partial Differential Equations T.F. Chan and R. Glowinski and J. Periaux and O. B. Widlund (Eds.), SIAM Philadelphia 1990 pp 98–114

  12. Ewing, R.E., Liu, M., Wang, J.: Superconvergence of mixed finite element approximations over quadrilaterals. SIAM J. Numer. Anal. 36(3) 772–787 (1999)

    Google Scholar 

  13. Ewing, R.E., Wang, J.: Analysis of mixed finite element methods on locally refined grids. Numer. Math. 63, 183–194 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grisvard, P.: Elliptic problems in nonsmooth domains. Pitman, Boston, 1985

  15. Hyman, J., Morel, J., Shashkov, M., Steinberg, S.: Mimetic finite difference methods for diffusion equations. Comp. Geosci. 6(3–4), 333–352 (2002)

    Google Scholar 

  16. Hyman, J.M., 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 

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

    Google Scholar 

  18. J. L. Lions and E. Magenes Non-homogeneous boundary value problems and applications Springer-Verlag, 1, 1972

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Nedelec, J.C.: Mixed finite elements in R3. Numer. Math. 35, 315–341 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  23. Peszyńska, M, Wheeler, M.F., Yotov, I.: Mortar upscaling for multiphase flow in porous media. Comput. Geosci. Comput. Geosci. 6(1), 73–100 (2002)

    Google Scholar 

  24. Raviart, R.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of the Finite Element Method Lecture Notes in Mathematics Springer-Verlag, New York 606, 1977 pp 292–315

  25. Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In Handbook of Numerical Analysis Vol. II Elsevier Science Publishers B.V. 1991 P. G. Ciarlet and J.L. Lions pp 523–639

  26. Shashkov, M., Steinberg, S.: Solving diffusion equations with rough coefficients in rough grids. J. Comput. Phys. 129, 383–405 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Thomas, J.M.: Sur l'analyse numérique des méthods d'éléments finis hybrides et mixtes. PhD thesis, Université Pierre et Marie Curie Paris, 1977

  28. Wang, J., Mathew, T.P.: Mixed finite element method over quadrilaterals Conference on Advances In: Numerical Methods and Applications Dimov I.T. and Sendov B. and Vassilevski P. (Eds.), World Scientific, River Edge NJ 1994 pp 203–214

  29. Wheeler, M.F., Yotov, I.: Multigrid on the interface for mortar mixed finite element methods for elliptic problems. Comput. Meth. Appl. Mech. Eng. 184, 287–302 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wohlmuth, B.I.: A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer Anal 38(3), 989–1012 (2000)

    MathSciNet  Google Scholar 

  31. Yotov, I.: Mixed finite element methods for flow in porous media. Rice University 1996 Houston Texas TR96-09 Dept. Comp. Appl. Math. Rice University and TICAM report 96-23 University of Texas at Austin

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Authors and Affiliations

  1. Los Alamos National Laboratory, Mail Stop B284, Los Alamos, NM, 87545, USA

    Markus Berndt, Konstantin Lipnikov & Mikhail Shashkov

  2. Institute for Computational Engineering and Sciences (ICES), Department of Aerospace Engineering and Engineering Mechanics, and Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX, 78712, USA

    Mary F. Wheeler

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

    Ivan Yotov

Authors
  1. Markus Berndt
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  2. Konstantin Lipnikov
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  3. Mikhail Shashkov
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  4. Mary F. Wheeler
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  5. Ivan Yotov
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Corresponding author

Correspondence to Markus Berndt.

Additional information

Supported by the US Department of Energy, under contractW-7405-ENG-36. LA-UR-04-4740.

Partially supported by NSF grants EIA 0121523 and DMS 0411413, by NPACI grant UCSD 10181410, and by DOE grant DE-FGO2-04ER25617.

Partially supported by NSF grants DMS 0107389, DMS 0112239 and DMS 0411694 and by DOE grant DE-FG02-04ER25618.

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Berndt, M., Lipnikov, K., Shashkov, M. et al. A mortar mimetic finite difference method on non-matching grids. Numer. Math. 102, 203–230 (2005). https://doi.org/10.1007/s00211-005-0631-4

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  • DOI: https://doi.org/10.1007/s00211-005-0631-4

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