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Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations

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Abstract

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and \(L^2\) norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special cases, the convergence rate can reach \(h^{r+2}\) and even \(h^{2r}\), comparing with \(h^{r+1}\) rate of the counterpart finite element method. Here \(r\) is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.

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Acknowledgments

Zhimin Zhang is partially supported by the US National Science Foundation through grant DMS-111530, the Ministry of Education of China through the Changjiang Scholars program, and Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program. Qingsong Zou is supported in part by the National Natural Science Foundation of China under the grant 11171359 and in part by the Fundamental Research Funds for the Central Universities of China

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Authors and Affiliations

  1. College of Mathematics and Scientific Computing, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Waixiang Cao

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

  3. College of Mathematics and Scientific Computing and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Qingsong Zou

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  1. Waixiang Cao
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  2. Zhimin Zhang
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  3. Qingsong Zou
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Correspondence to Qingsong Zou.

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Cao, W., Zhang, Z. & Zou, Q. Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations. J Sci Comput 56, 566–590 (2013). https://doi.org/10.1007/s10915-013-9691-2

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  • DOI: https://doi.org/10.1007/s10915-013-9691-2

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