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Superconvergence of Immersed Finite Volume Methods for One-Dimensional Interface Problems

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Abstract

In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in \(H^1\)- and \(L^2\)-norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order \(p+2\) at the roots of generalized Lobatto polynomials, and the flux is superconvergent of order \(p+1\) at generalized Gauss points on each element including the interface element. Furthermore, for diffusion interface problems, the convergence rates for IFVM solution at the mesh points and the flux at generalized Gauss points can both be raised to 2p. These superconvergence results are consistent with those for the standard finite volume methods. Numerical examples are provided to confirm our theoretical analysis.

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  • 05 December 2017

    The original version of this article unfortunately contained an error.

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Waixiang Cao & Qingsong Zou

  2. Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS, 39762, USA

    Xu Zhang

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

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

    Zhimin Zhang

Authors
  1. Waixiang Cao
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  2. Xu Zhang
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  3. Zhimin Zhang
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  4. Qingsong Zou
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Corresponding author

Correspondence to Xu Zhang.

Additional information

The work of W. Cao was supported in part by the NSFC Grant 11501026, and the China Postdoctoral Science Foundation Grants 2016T90027, 2015M570026. The work of Z. Zhang was supported in part by the NSFC Grants 11471031, 91430216, and U1530401; and NSF Grant DMS-1419040. The work of Q. Zou was supported in part by the following Grants: the special project High performance computing of National Key Research and Development Program 2016YFB0200604, the NSFC 11571384, the Guangdong Provincial NSF 2014A030313179, and the Fundamental Research Funds for the Central Universities 16lgjc80.

A correction to this article is available online at https://doi.org/10.1007/s10915-017-0609-2.

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Cao, W., Zhang, X., Zhang, Z. et al. Superconvergence of Immersed Finite Volume Methods for One-Dimensional Interface Problems. J Sci Comput 73, 543–565 (2017). https://doi.org/10.1007/s10915-017-0532-6

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  • DOI: https://doi.org/10.1007/s10915-017-0532-6

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