Robust convergence of multi point flux approximation on rough grids. (English) Zbl 1102.76036

Summary: 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.


76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI Link


[1] Aavatsmark I. (2002). An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 404–432 · Zbl 1094.76550
[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 · Zbl 0859.76048
[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) · Zbl 1110.65106
[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) · Zbl 1128.65093
[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 · Zbl 0839.65116
[6] Arnold D.N., Boffi D., Falk R.S.(2002). Approximation by quadrilateral finite elements. Math. Comp. 71(239): 909–922 · Zbl 0993.65125
[7] Arnold D.N., Boffi D., Falk R.S.(2005). Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 · Zbl 1086.65105
[8] Arnold D.N., Boffi D., Falk R.S., Gastaldi L.(2001). Approximation by quadrilateral finite elements. Commun. Numer. Methods Eng. 17, 805–812 · Zbl 0999.76073
[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 · Zbl 1014.65114
[10] Brezzi F., Fortin M.(1991). Mixed and hybrid finite element methods. Springer, Berlin Heidelberg New York · Zbl 0788.73002
[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 · Zbl 1007.65091
[12] Ciarlet P.G.(1997). The Finite Element Method for Elliptic Problems. Noth-Holland, Amsterdam · Zbl 0887.65121
[13] Edwards M.G.(2002). Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6, 433–452 · Zbl 1036.76034
[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 · Zbl 1089.76037
[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 · Zbl 1124.76030
[17] Klausen R.A., Winther R.: Convergence of multi point flux approximations on quadrilateral grids. Published electronically in Numer. Methods Partial Diff. Eqns. (2006) · Zbl 1106.76043
[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 · Zbl 0813.65120
[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) · Zbl 1110.65099
[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 · Zbl 1121.76040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.