A polygonal finite volume element method for anisotropic diffusion problems.

In this paper, based on the Wachspress generalized barycentric coordinates, we propose and analyze a polygonal finite volume element method (PFVEM) for solving the anisotropic diffusion equation on convex polygonal meshes. In particular, the PFVEM reduces to the classical P 1 -FVEM on triangular meshes and it is not identical to the classical Q 1 -FVEM on quadrilateral meshes. A new proof is given to Proposition 8 in [19] , a result that is crucial to the derivation of the interpolation error estimates in both H 1 and H 2 norms. The original proof is based on a certain geometric assumption, which is shown not always true by a counterexample given in this paper. Moreover, for the error analysis of the PFVEM, we prove the H 2 error estimate of the Wachspress interpolation. Under the coercivity assumption, the optimal H 1 error estimate for the finite volume element solution is obtained. Several numerical examples are presented to show the efficiency and robustness of the proposed method for the heterogeneous and anisotropic diffusion problems on polygonal meshes. [ABSTRACT FROM AUTHOR]