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A kernel-free boundary integral method for elliptic PDEs on a doubly connected domain

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Abstract

We present a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a doubly connected domain. We focus our study on boundary value problems (BVP) and interface problems. A unique feature of the KFBIM is that the method does not require an analytical form of the Green’s function for designing quadratures but rather computes boundary or volume integrals by solving an equivalent interface problem on Cartesian mesh. We first decompose the problem defined in a doubly connected into two separate interface problems. The system of boundary integral equations is solved using the Krylov method. The method is second-order accurate in space, and its complexity is linearly proportional to the number of mesh points. Numerical examples demonstrate that the method is robust for variable coefficients PDEs, even for cases with large diffusion coefficients ratio and complex geometries where two interfaces are close.

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Acknowledgements

S. L. and M. K acknowledge the support from the National Science Foundation, Division of Electrical, Communications and Cyber Systems grant ECCS-1927432. S. L. also acknowledges the partial support from the National Science Foundation, Division of Mathematical Sciences grant DMS-1720420. This work is also supported by the National Natural Science Foundation of China (Grant No. DMS-12101553, Grant No. DMS-11771290).

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Authors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, USA

    Yue Cao & Shuwang Li

  2. College of Science, Zhejiang University of Technology, Xihu, 310023, Hangzhou, People’s Republic of China

    Yaning Xie

  3. Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, USA

    Mahesh Krishnamurthy

  4. School of Mathematical Sciences, MOE-LSC and Institute of Natural Science, Shanghai Jiao Tong University, Shanghai, China

    Wenjun Ying

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  1. Yue Cao
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  2. Yaning Xie
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  3. Mahesh Krishnamurthy
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  4. Shuwang Li
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  5. Wenjun Ying
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Correspondence to Wenjun Ying.

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Cao, Y., Xie, Y., Krishnamurthy, M. et al. A kernel-free boundary integral method for elliptic PDEs on a doubly connected domain. J Eng Math 136, 2 (2022). https://doi.org/10.1007/s10665-022-10233-8

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