Numerical Convergence of the MPFA O-Method for General Quadrilateral Grids in Two and Three Dimensions

This paper presents the MPFA O-method for quadrilateral grids, and gives convergence rates for the potential and the normal velocities. The convergence rates are estimated from numerical experiments. If the potential is in H 1+α , α>0, the found L 2 convergence order on rough grids in physical space is min{2, 2α} for the potential and min{1, α} for the normal velocities. For smooth grids the convergence order for the normal velocities increases to min{2,α}. The O-method is exact for uniform flow on rough grids. This also holds in three dimensions, where the cells may have nonplanar surfaces.

[1]  Ivar Aavatsmark,et al.  Discretization on Unstructured Grids For Inhomogeneous, Anisotropic Media. Part II: Discussion And Numerical Results , 1998, SIAM J. Sci. Comput..

[2]  Michael G. Edwards,et al.  Finite volume discretization with imposed flux continuity for the general tensor pressure equation , 1998 .

[3]  T. Mannseth,et al.  A Class of Discretization Methods for Structured and Unstructured Grids in Anisotropic, Inhomogeneous Media , 1996 .

[4]  Michael G. Edwards,et al.  Unstructured, Control-Volume Distributed, Full-Tensor Finite-Volume Schemes with Flow Based Grids , 2002 .

[5]  Ivar Aavatsmark,et al.  Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods , 1998, SIAM J. Sci. Comput..

[6]  Mary F. Wheeler,et al.  Enhanced Velocity Mixed Finite Element Methods for Flow in Multiblock Domains , 2002 .

[7]  M. Shashkov,et al.  The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials , 1997 .

[8]  B. Rivière,et al.  Part II. Discontinuous Galerkin method applied to a single phase flow in porous media , 2000 .

[9]  S. Verma,et al.  Two- and Three-Dimensional Flexible Grids for Reservoir Simulation , 1996 .

[10]  W. E. Bardsley.  An algorithm for calculating an upper bound to x2 for testing binomial homogeneity with some observations uncategorized , 1980 .

[11]  I. Aavatsmark,et al.  Control-Volume Discretization Methods for 3D Quadrilateral Grids in Inhomogeneous, Anisotropic Reservoirs , 1998 .

[12]  J. D. Wilson,et al.  Shape Functions for Velocity Interpolation in General Hexahedral Cells , 2002 .

[13]  I. Aavatsmark,et al.  An Introduction to Multipoint Flux Approximations for Quadrilateral Grids , 2002 .

[14]  Nonconforming Finite Volume Methods , 2002 .

[15]  Mikhail Shashkov,et al.  Solving Diffusion Equations with Rough Coefficients in Rough Grids , 1996 .

[16]  Ivar Aavatsmark,et al.  Monotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous media , 2005 .

[17]  G. T. Eigestad,et al.  On the convergence of the multi-point flux approximation O-method: Numerical experiments for discontinuous permeability , 2005 .

[18]  Ivar Aavatsmark,et al.  Discretization on Non-Orthogonal, Quadrilateral Grids for Inhomogeneous, Anisotropic Media , 1996 .

[19]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .