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Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods

Advances in Computational Mathematics volume 36, pages 79–97 (2012)Cite this article

Abstract

The accuracy of numerical solutions near singular points is crucial for numerical methods. In this paper we develop an efficient mechanical quadrature method (MQM) with high accuracy. The following advantages of MQM show that it is very promising and beneficial for practical applications: (1) the \( O(h_{\rm {max}}^{3})\) convergence rate; (2) the \(O(h_{\rm {max}}^{5})\) convergence rate after splitting extrapolation; (3) Cond = \(O(h_{\rm {min}}^{-1})\); (4) the explicit discrete matrix entries. In this paper, the above theoretical results are briefly addressed and then verified by numerical experiments. The solutions of MQM are more accurate than those of other methods. Note that for the discontinuous model in Li et al. (Eng Anal Bound Elem 29:59–75, 2005), the highly accurate solutions of MQM may even compete with those of the collocation Trefftz method.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Jin Huang & Guang Zeng

  2. Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306, USA

    Xiaoming He

  3. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Xiaoming He

  4. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Zi-Cai Li

  5. Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Zi-Cai Li

Authors
  1. Jin Huang
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  2. Guang Zeng
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  3. Xiaoming He
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  4. Zi-Cai Li
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Corresponding author

Correspondence to Xiaoming He.

Additional information

Communicated by Yuesheng Xu.

The work is supported by the National Natural Science Foundation of China (10871034).

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Huang, J., Zeng, G., He, X. et al. Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods. Adv Comput Math 36, 79–97 (2012). https://doi.org/10.1007/s10444-011-9181-8

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  • DOI: https://doi.org/10.1007/s10444-011-9181-8

Keywords

  • First-kind boundary integral equation
  • Mechanical quadrature method
  • Splitting extrapolation
  • A posteriori estimate
  • Singularity
  • Stability analysis

Mathematics Subject Classifications (2010)

  • 45B05
  • 45E99
  • 65B05