Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh

Abstract

A numerical method for solving diffusion problems on polyhedral meshes is presented. It is based on a finite volume approximation with the degrees of freedom located in the centers of computational cells. A numerical gradient is defined by a least-squares minimization for each cell, where we suggest a restricted form in the case of discontinuous diffusion coefficient. The flux balanced approximation is proposed without numerically computing the gradient itself at the faces of computational cells in order to find a normal diffusive flux. To apply the method for parallel computations with a 1-ring neighborhood, we use an iterative method to solve the obtained system of algebraic equations. Several numerical examples illustrate some advantages of the proposed method.