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Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation

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Abstract

Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains. Then the new bound of Cond is derived for bounded simply connected domains with mixed types of boundary conditions. Numerical results are reported for Motz’s problem by adding singular functions. The values of Cond grow exponentially with respect to the number of fundamental solutions used. Note that there seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Moreover, the expansion coefficients obtained by the MFS are oscillatingly large, to cause the other kind of instability: subtraction cancelation errors in the final harmonic solutions.

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Authors and Affiliations

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

    Zi-Cai Li

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

    Zi-Cai Li

  3. Department of Applied Mathematics, Chung Hua University, HsinChu, Taiwan

    Zi-Cai Li

  4. College of Applied Mathematics, University of Electronic and Science Technology of China, ChengDu, China

    Jin Huang

  5. Department of Applied Mathematics, I-Shou University, Kaohsiung, 804, Taiwan

    Hung-Tsai Huang

Authors
  1. Zi-Cai Li
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  2. Jin Huang
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  3. Hung-Tsai Huang
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Corresponding author

Correspondence to Zi-Cai Li.

Additional information

This paper is dedicated to Professor Robert Schaback on the occasion of his 65th birthday.

Partial results were presented at the first International Workshop on the Method of Fundamental Solutions (MFS2007), Ayia Napa, Cyprus, 11–13 June 2007. The work of J. Huang is supported by the National Natural Science Foundation of China (10871034).

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Li, ZC., Huang, J. & Huang, HT. Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation. Computing 88, 1–29 (2010). https://doi.org/10.1007/s00607-010-0081-4

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  • DOI: https://doi.org/10.1007/s00607-010-0081-4

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