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On the application of the method of fundamental solutions to boundary value problems with jump discontinuities

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  • Alves, Carlos J.S.
  • Valtchev, Svilen S.

Abstract

Two meshfree methods are proposed for the numerical solution of boundary value problems (BVPs) for the Laplace equation, coupled with boundary conditions with jump discontinuities. In the first case, the BVP is solved in two steps, using a subtraction of singularity approach. Here, the singular subproblem is solved analytically while the classical method of fundamental solutions (MFS) is applied for the solution of the regular subproblem. In the second case, the total BVP is solved using a variant of the MFS where its approximation basis is enriched with a set of harmonic functions with singular traces on the boundary of the domain. The same singularity-capturing functions, motivated by the boundary element method (BEM), are used for the singular part of the solution in the first method and for augmenting the MFS basis in the second method. Comparative numerical results are presented for 2D problems with discontinuous Dirichlet boundary conditions. In particular, the inappropriate oscillatory behavior of the classical MFS solution, due to the Gibbs phenomenon, is shown to vanish.

Suggested Citation

  • Alves, Carlos J.S. & Valtchev, Svilen S., 2018. "On the application of the method of fundamental solutions to boundary value problems with jump discontinuities," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 61-74.
    • Handle: RePEc:eee:apmaco:v:320:y:2018:i:c:p:61-74
    DOI: 10.1016/j.amc.2017.09.018
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    Cited by:

    1. Xi, Qiang & Fu, Zhuojia & Wu, Wenjie & Wang, Hui & Wang, Yong, 2021. "A novel localized collocation solver based on Trefftz basis for potential-based inverse electromyography," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    2. Yao Sun & Xiaoliang Wei & Zibo Zhuang & Tian Luan, 2019. "A Numerical Method for Filtering the Noise in the Heat Conduction Problem," Mathematics, MDPI, vol. 7(6), pages 1-13, June.
    3. Yao Sun & Shijie Hao, 2021. "A Numerical Study for the Dirichlet Problem of the Helmholtz Equation," Mathematics, MDPI, vol. 9(16), pages 1-12, August.

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