Journal of Scientific Computing

, Volume 58, Issue 2, pp 308–330 | Cite as

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

  • Zhimin Zhang
  • Qingsong Zou


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 \(L^2\)-norm. Finally, we validate our theory with numerical experiments.


High order Finite volume method Inf-sup condition Superconvergence 

Mathematics Subject Classification

Primary 65N30 Secondary 45N08 



The authors would like to thank a Ph.D. student, Waixiang Cao for her assistance in the numerical experiments.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Beijing Computational Science Research CenterBeijingPeople’s Republic of China
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA
  3. 3.College of Mathematics and Scientific ComputingSun Yat-sen UniversityGuangzhouPeople’s Republic of China
  4. 4.Guangdong Province Key Laboratory of Computational ScienceSun Yat-sen UniversityGuangzhouPeople’s Republic of China

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