Mixed Covolume Methods for Elliptic Problems on Triangular Grids

We consider a covolume or finite volume 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 represent either the Darcy law and the mass conservation law in anisotropic porous media flow, or Fourier law and energy conservation. The velocity and pressure are approximated by the lowest order Raviart--Thomas space on triangles. We prove its first-order optimal rate of convergence for the approximate velocities in the $L^2$-and $H(\mbox{div};\Omega)$-norms as well as for the approximate pressures in the $L^2$-norm. Numerical experiments are included.

[1]  Do Y. Kwak,et al.  Mixed Covolume Methods on Rectangular Grids For Elliptic Problems , 2000, SIAM J. Numer. Anal..

[2]  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..

[3]  Do Y. Kwak,et al.  Mixed Upwinding Covolume Methods on Rectangular Grids for Convection-Diffusion Problems , 1999, SIAM J. Sci. Comput..

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

[5]  Panayot S. Vassilevski,et al.  An Upwinding Cell-Centered Method with Piecewise Constant Velocity over Covolumes , 1999 .

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

[7]  Douglas N. Arnold,et al.  Preconditioning in Hdiv and applications , 1997, Math. Comput..

[8]  Panayot S. Vassilevski,et al.  On Two Ways of Stabilizing the Hierarchical Basis Multilevel Methods , 1997, SIAM Rev..

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

[10]  R. S. Falk,et al.  PRECONDITIONING IN H (div) AND APPLICATIONS , 1997 .

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

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

[13]  X. Wu,et al.  Analysis and convergence of the MAC scheme. II. Navier-Stokes equations , 1996, Math. Comput..

[14]  O. Axelsson.  Iterative solution methods , 1995 .

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

[16]  Charles I. Goldstein,et al.  Multilevel Iteration for Mixed Finite Element Systems with Penalty , 1993, SIAM J. Sci. Comput..

[17]  P. Vassilevski,et al.  Multilevel iterative methods for mixed finite element discretizations of elliptic problems , 1992 .

[18]  R. Nicolaides.  Direct discretization of planar div-curl problems , 1992 .

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

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

[21]  T. F. Russell,et al.  Finite element and finite difference methods for continuous flows in porous media. , 1800 .