Barycentric Interpolation and Exact Integration Formulas for the Finite Volume Element Method

Abstract

This contribution concerns with the construction of a simple and effective technology for the problem of exact integration of interpolation polynomials arising while discretizing partial differential equations by the finite volume element method on simplicial meshes. It is based on the element-wise representation of the local shape functions through barycentric coordinates (barycentric interpolation) and the introducing of classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over the geometrical shapes defined by a barycentric dual mesh. We discuss especially a related problem of the approximation of the diffusion operators with spatially varying diffusion tensors, resulting in asymmetric stiffness matrices. Numerical examples are presented that illustrate the validity of the technology.