Mixed Covolume Methods on Rectangular Grids For Elliptic Problems

We consider a covolume method for a system of first order PDEs resulting from the mixed formulation of the variable-coefficient-matrix Poisson equation with the Neumann boundary condition. The system may be used to represent the Darcy law and the mass conservation law in anisotropic porous media flow. The velocity and pressure are approximated by the lowest order Raviart--Thomas space on rectangles. The method was introduced by Russell [Rigorous Block-centered Discretizations on Irregular Grids: Improved Simulation of Complex Reservoir Systems, Reservoir Simulation Research Corporation, Denver, CO, 1995] as a control-volume mixed method and has been extensively tested by Jones [A Mixed Finite Volume Elementary Method for Accurate Computation of Fluid Velocities in Porous Media, University of Colorado at Denver, 1995] and Cai et al. [Computational Geosciences, 1 (1997), pp. 289--345]. We reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures.

[1]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[2]  Jim E. Jones,et al.  A mixed method Poisson solver for three-dimensional self-gravitating astrophysical fluid dynamical systems , 1993 .

[3]  Do Y. Kwak,et al.  Analysis and convergence of a MAC‐like scheme for the generalized Stokes problem , 1997 .

[4]  Jim E. Jones,et al.  Control‐volume mixed finite element methods , 1996 .

[5]  Jim E. Jones.  A mixed finite volume element method for accurate computation of fluid velocities in porous media , 1996 .

[6]  Qian Li,et al.  Error estimates in L2, H1 and Linfinity in covolume methods for elliptic and parabolic problems: A unified approach , 1999, Math. Comput..

[7]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[8]  Jian Shen,et al.  A block finite difference scheme for second-order elliptic problems with discontinuous coefficients , 1996 .

[9]  So-Hsiang Chou,et al.  Analysis and convergence of a covolume method for the generalized Stokes problem , 1997, Math. Comput..

[10]  Panayot S. Vassilevski,et al.  A general mixed covolume framework for constructing conservative schemes for elliptic problems , 1999, Math. Comput..

[11]  Zhiqiang Cai,et al.  On the accuracy of the finite volume element method for diffusion equations on composite grids , 1990 .

[12]  Do Y. Kwak,et al.  A Covolume Method Based on Rotated Bilinears for the Generalized Stokes Problem , 1998 .

[13]  T. A. Porsching,et al.  COVOLUME-DUAL VARIABLE METHOD FOR THERMALLY EXPANDABLE FLOW ON UNSTRUCTURED TRIANGULAR GRIDS , 1994 .

[14]  P. Vassilevski,et al.  Mixed Covolume Methods for Elliptic Problems on Triangular Grids , 1998 .