Mixed finite volume methods for elliptic partial differential equation.

摘要

We study the diffusion problem -1˙K1p=f in a polygonal domain W for anisotropic porous media flow. In mixed methods, one introduces the flux variable u:=-K1 p and write the above equation as the system of first order partial differential equations 1˙u-f=0 inW u+K1 p=0in W with either Dirichlet or Neumann boundary condition.;This system can be interpreted as modeling an incompressible single phase flow in a reservoir, ignoring gravitational effects. The matrix K is the mobility k/m , the ratio of permeability tensor to viscosity of the fluid, u is the Darcy velocity and p the pressure. The second equation is the Darcy law and the first represents conservation of mass with f standing for a source or sink term.;We consider both finite volume box methods and finite covolume methods for the above problem. Triangular and rectangular grids will be treated. Error estimates will be presented in each case for both primary unknown p and secondary unknown u. Also the equivalence of these methods with certain FEM nonconforming methods will be established after judicious transformations. As a consequence, a simple SPD system results that all multigrid, preconditioned conjugate gradient methods can be applied effectively. Numerical results are included.