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A class of finite volume schemes of arbitrary order on nonuniform meshes

Authors

  • Qinghui Zhang,

    1. Department of Scientific Computation and Computer Applications, School of Mathematics and Computational Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, People's Republic of China
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  • Qingsong Zou

    Corresponding author
    1. Department of Scientific Computation and Computer Applications, School of Mathematics and Computational Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, People's Republic of China
    • Correspondence to: Qingsong Zou, Department of Scientific Computation and Computer Applications, School of Mathematics and Computational Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, P. R. China, (e-mail:mcszqs@mail.sysu.edu.cn)

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  • Dedicate to Professor Kaitai Li on occasion of his 75th birthday.

Abstract

In this article, we generalize the bi-inline image finite volume schemes developed in Zhang and Zou, J Sci Comput (in press) for elliptic equations with high smooth solutions to elliptic equations with singular solutions. By designing a special nonuniform rectangular meshes, we construct a class of finite volume schemes of arbitrary order k. Our theoretic analysis shows that if the solution has weak singularity of type inline image, where r is the distance from some target point to some fixed singular point, the H1 norm of our finite volume schemes' discretization error converges with optimal order inline image, while the L2 norm error converges with order inline image. Here, N2 is the cardinality of the underlying mesh. Superconvergence property of the scheme is also discussed. Our theoretic findings have been verified by numerical experiments. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1614–1632, 2014

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