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.
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Vogel, A., Xu, J. & Wittum, G. A generalization of the vertex-centered finite volume scheme to arbitrary high order. Comput. Visual Sci. 13, 221–228 (2010). https://doi.org/10.1007/s00791-010-0139-z
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DOI: https://doi.org/10.1007/s00791-010-0139-z