A Q1-finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh

In this paper, we study a Q(1)-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, H-1 and L-2 error estimates, etc.) and the existing results were mainly obtained on h(1+gamma)-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 h(1+gamma)-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 H-1 error estimate is obtained without the h(1+gamma)-parallelogram assumption, and some numerical results are also provided to validate the theoretical results. (C) 2020 Elsevier B.V. All rights reserved.