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Indirect Boundary Integral Equation Method for the Cauchy Problem of the Laplace Equation

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Abstract

In this paper, we examine the Cauchy problem of the Laplace equation. Motivated by the incompleteness of the single-layer potential function method, we investigate the double-layer potential function method. Through the use of a layer approach to the solution, we devise a numerical method for approximating the solution of the Cauchy problem, which are well known to be highly ill-posed in nature. The ill-posedness is dealt with Tikhonov regularization, whilst the optimal regularization parameter is chosen by Morozov discrepancy principle. Convergence and stability estimates of the proposed method are then given. Finally, some examples are given for the efficiency of the proposed method. Especially, when the single-layer potential function method does not give accurate results for some problems, it is shown that the proposed method is effective and stable.

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Acknowledgments

We would like to thank the editor and the referee for their careful reading and valuable comments which improved the quality of the original submitted manuscript. The research was supported by the open Research Funds of Tianjin Key Lab for Advanced Signal Processing (No. 2016ASP-TJ02), the Natural Science Foundation of China (No. 11501566).

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Authors and Affiliations

  1. Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin, 300300, China

    Yao Sun

  2. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Yao Sun

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  1. Yao Sun
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Sun, Y. Indirect Boundary Integral Equation Method for the Cauchy Problem of the Laplace Equation . J Sci Comput 71, 469–498 (2017). https://doi.org/10.1007/s10915-016-0308-4

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