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A Finite Volume Scheme for Diffusion Equations on Distorted Quadrilateral Meshes

A finite volume scheme is discussed for discretizing diffusion operators. Both cell-center unknowns and cell-vertex unknowns are used originally in the construction of the finite volume scheme. Then cell-vertex unknowns are eliminated by expressing them locally as linear combinations of cell-center unknowns, which is derived by the following process. First we construct a special control-volume for each cell-vertex and then design a finite volume scheme in this control-volume to obtain a linear relation between cell-vertex unknowns and cell-center unknowns. Hence cell-vertex unknown can be expressed by the combination of neighboring cell-center unknowns, and the resulting scheme has only cell-center unknowns like standard finite difference methods. Specially, the scheme can naturally treat problems with discontinuous coefficients. Its another advantage is that highly distorted meshes can be used without the numerical results being altered remarkably. We prove theoretically that the finite volume scheme is stable and has first-order accuracy on distorted meshes. Moreover, we have tested the scheme on a few elliptic and parabolic equations. Numerical results exhibit the good behavior of our scheme.