The Corrected Finite Volume Element Methods for Diffusion Equations Satisfying Discrete Extremum Principle

Abstract

In this paper, we correct the finite volume element methods for diffusion equations on general triangular and quadrilateral meshes. First, we decompose the numerical fluxes of original schemes into two parts, i.e., the principal part with a two-point flux structure and the defective part. And then with the help of local extremums, we transform the original numerical fluxes into nonlinear numerical fluxes, which can be expressed as a nonlinear combination of two-point fluxes. It is proved that the corrected schemes satisfy the discrete strong extremum principle without restrictions on the diffusion coefficient and meshes. Numerical results indicate that the corrected schemes not only satisfy the discrete strong extremum principle but also preserve the convergence order of the original finite volume element methods.