Article

Computing and Visualization in Science

, Volume 13, Issue 5, pp 221-228

First online:

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

  • Andreas VogelAffiliated withGoethe Center for Scientific Computing (G-CSC), Goethe-University Frankfurt am Main Email author 
  • , Jinchao XuAffiliated withDepartment of Mathematics, Pennsylvania State University
  • , Gabriel WittumAffiliated withGoethe Center for Scientific Computing (G-CSC), Goethe-University Frankfurt am Main

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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.

Keywords

Finite volume method Vertex-centered Higher order