A generalization of the vertex-centered finite volume scheme to arbitrary high order

Abstract

A higher order finite volume method for elliptic problems is proposed for arbitrary order $${p \in \mathbb{N}}$$. Piecewise polynomial basis functions are used as trial functions while the control volumes are constructed by a vertex-centered technique. The discretization is tested on numerical examples utilizing triangles and quadrilaterals in 2D. In these tests the optimal error is achieved in the H 1-norm. The error in the L 2-norm is one order below optimal for even polynomial degrees and optimal for odd degrees.