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Robust convergence of multi point flux approximation on rough grids

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Abstract

This paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids. By rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit. In contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space, and not on a reference cell. As a consequence, less regular grids are allowed. However, the extra cost is that the symmetry of the method is lost.

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References

  1. Aavatsmark I. (2002). An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 404–432

    Article  Google Scholar 

  2. Aavatsmark I., Barkve T., Bøe Ø., Mannseth T.(1996). Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media J. Comput. Phys. 127, 2–14

    Article  MATH  Google Scholar 

  3. Aavatsmark, I., Eigestad, G.T., Klausen, R.A.: Numerical convergence of MPFA for general quadrilateral grids in two and three dimensions. In: IMA Volumes in Mathematics and its Applications (Compatible spatial discretizations for Partial Differential quations) 142, 1–22 (2006)

  4. Aavatsmark, I., Eigestad, G.T., Klausen, R.A., Wheeler, M.F., Yotov, I.: Convergence of a symmetric MPFA method on quadrilateral grids. (Preprints 2005)

  5. Agouzal A., Baranger J., Maitre J.-F., Oudin F.(1995). Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstants coefficients. Application to a convection diffusion problem. East West J. Numer. Math. 3, 237–254

    MATH  MathSciNet  Google Scholar 

  6. Arnold D.N., Boffi D., Falk R.S.(2002). Approximation by quadrilateral finite elements. Math. Comp. 71(239): 909–922

    Article  MATH  MathSciNet  Google Scholar 

  7. Arnold D.N., Boffi D., Falk R.S.(2005). Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451

    Article  MATH  MathSciNet  Google Scholar 

  8. Arnold D.N., Boffi D., Falk R.S., Gastaldi L.(2001). Approximation by quadrilateral finite elements. Commun. Numer. Methods Eng. 17, 805–812

    Article  MATH  MathSciNet  Google Scholar 

  9. Berndt M., Lipnikov K., Moulton J.D., Shashkov M.(2001). Convergence of mimetic finite difference discretizations of the diffusion equation. East West J. Numer. Math. 9, 130–148

    MathSciNet  Google Scholar 

  10. Brezzi F., Fortin M.(1991). Mixed and hybrid finite element methods. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  11. Chou S., Kwak D.Y., Kim K.Y.(2001). A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: The overlapping covolume case. SIAM J. Numer. Anal. 39, 1170–1196

    Article  MATH  MathSciNet  Google Scholar 

  12. Ciarlet P.G.(1997). The Finite Element Method for Elliptic Problems. Noth-Holland, Amsterdam

    Google Scholar 

  13. Edwards M.G.(2002). Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6, 433–452

    Article  MATH  MathSciNet  Google Scholar 

  14. Eigestad G.T., Klausen R.A.(2005). On the convergence of the multi-point flux approximation O-method; numerical experiments for discontinous permeability. Numer. Methods Part. Diff. Eqns. 21(6): 1079–1098

    Article  MATH  MathSciNet  Google Scholar 

  15. Klausen, R.A.: PhD thesis: on locally conservative numerical methods for elliptic problems; Application to reservoir simulation. Dissertation, no 297. Unipub AS, University of Oslo (2003)

  16. 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 

  17. Klausen R.A., Winther R.: Convergence of multi point flux approximations on quadrilateral grids. Published electronically in Numer. Methods Partial Diff. Eqns. (2006)

  18. Wang J., Mathew T.(1994). Mixed finite element methods over quadrilaterals. In: Dimov I.T. Sendov Bl., Vassilevskith P. (eds) Proceedings of the 3th International Conference on Advances in Numerical Methods and Applications, pp. 203–214

  19. Wheeler M.F., Yotov I.: A cell-centered finite difference method on quadrilaterals. In: IMA Volumes in Mathematics and its Applications (Compatible spatial discretizations for Partial Differential quations) 142, 189–206 (2006)

  20. Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method. In: Technical Report TR Math 05-06, Department of Mathematics, University of Pittsburgh

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  1. Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, 0316, Oslo 3, Norway

    Runhild A. Klausen & Ragnar Winther

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  1. Runhild A. Klausen
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  2. Ragnar Winther
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Correspondence to Runhild A. Klausen.

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Klausen, R.A., Winther, R. Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104, 317–337 (2006). https://doi.org/10.1007/s00211-006-0023-4

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  • DOI: https://doi.org/10.1007/s00211-006-0023-4

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