A linearity-preserving technique for finite volume schemes of anisotropic diffusion problems on polygonal meshes

Abstract

In this work we derive a cell-centered finite volume scheme by a linearity-preserving technique. It solves matrix equation obtained by linearity-preserving criterion and gets the desired approximations. The scheme introduces both cell-centered unknowns and vertex unknowns. The latter are auxiliary and eliminated by the surrounding cell-centered unknowns with a new vertex interpolation algorithm derived by the presented linearity-preserving technique. The proposed technique is flexible and is expected to design algorithms for 3D diffusion problems. Nearly optimal accuracy is found from the numerical experiments. More interesting is that the new vertex interpolation algorithm outperforms some commonly used linearity-preserving algorithms on Kershaw meshes and distorted meshes which have proved difficult for most algorithms.