A new high order finite volume element solution on arbitrary triangular and quadrilateral meshes

Abstract

In this work, for the two dimensional anisotropic diffusion problem, the classical kth order finite element solution is postprocessed to obtain a new finite volume element solution, such that the new solution satisfies the local conservation property on a certain dual mesh, and converges to the analytic solution with optimal rates. The postprocessing algorithm has a local nature and can be conducted element by element. The novelty of this paper is the introduction of a new bubble function, which enables us to prove the existence and uniqueness of the postprocessed solution on arbitrary triangular or convex quadrilateral meshes with full anisotropic diffusion tensor. The theoretical findings are also verified by some numerical results.