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Kernel-Based Meshless Collocation Methods for Solving Coupled Bulk–Surface Partial Differential Equations

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Abstract

A meshless kernel-based method is developed to solve coupled second-order elliptic PDEs in bulk domains and surfaces, subject to Robin boundary conditions. It combines a least-squares kernel collocation method with a surface-type intrinsic approach. Therefore, we can use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, for the search of least-squares solutions in bulks and on surfaces respectively. We first give error estimates for domain-type Robin-boundary problems. Based on this and existing results for surface PDEs, we discuss the theoretical requirements for the employed Sobolev kernels. Then, we select the orders of smoothness for the kernels in bulks and on surfaces. Lastly, several numerical experiments are demonstrated to test the robustness of the coupled method for accuracy and convergence rates under different settings.

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Notes

  1. \((x^2+y^2+z^2+1^2-({1}/{3})^2)^2-4(x^2+y^2)=0\)

  2. \( \sqrt{(x-1)^2+y^2+z^2}\sqrt{(x+1)^2+y^2+z^2}\sqrt{x^2+(y-1)^2+z^2}\sqrt{x^2+(y+1)^2+z^2} -1.1=0\).

  3. \([(x^2+y^2-1)^2+z^2][(y^2+z^2-1)^2+x^2][(x^2+z^2-1)^2+y^2]-0.075^2[1+3(x^2+y^2+z^2)]=0\).

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Authors and Affiliations

  1. Hong Kong Applied Science and Technology Research Institute Company Limited, Shatin, Hong Kong

    Meng Chen

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Leevan Ling

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  1. Meng Chen
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  2. Leevan Ling
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Correspondence to Meng Chen.

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This work was supported by a Hong Kong Research Grant Council GRF Grant and a Hong Kong Baptist University FRG Grant.

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Chen, M., Ling, L. Kernel-Based Meshless Collocation Methods for Solving Coupled Bulk–Surface Partial Differential Equations. J Sci Comput 81, 375–391 (2019). https://doi.org/10.1007/s10915-019-01020-2

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  • DOI: https://doi.org/10.1007/s10915-019-01020-2

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