A [formula omitted]-finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh

Abstract

In this paper, we study a Q1-finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh. It is known that the coercivity is the basement for some other theoretical results (stability, H1 and L2 error estimates, etc.) and the existing results were mainly obtained on h1+γ-parallelogram meshes with scalar diffusion coefficients. For the cases of full diffusion tensors and arbitrary convex quadrilateral meshes, we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme. More interesting is that, this sufficient condition covers the traditional h1+γ-parallelogram mesh assumption and has an explicit expression, by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h>0. Moreover, an H1 error estimate is obtained without the h1+γ-parallelogram assumption, and some numerical results are also provided to validate the theoretical results.