A Family of Finite Volume Schemes of Arbitrary Order on Rectangular Meshes

Authors:
Zhimin Zhang;Qingsong Zou
Affiliations:
Beijing Computational Science Research Center, Beijing, People's Republic of China 100084 and Department of Mathematics, Wayne State University, Detroit, USA 48202;College of Mathematics and Scientific Computing, Sun Yat-sen University, Guangzhou, People's Republic of China 510275 and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen Un ...
Venue:
Journal of Scientific Computing
Year:
2014

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Abstract

In this paper, we analyze vertex-centered finite volume method (FVM) of any order for elliptic equations on rectangular meshes. The novelty is a unified proof of the inf-sup condition, based on which, we show that the FVM approximation converges to the exact solution with the optimal rate in the energy norm. Furthermore, we discuss superconvergence property of the FVM solution. With the help of this superconvergence result, we find that the FVM solution also converges to the exact solution with the optimal rate in the $$L^2$$L2-norm. Finally, we validate our theory with numerical experiments.