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Localized Boundary Knot Method for Solving Two-Dimensional Laplace and Bi-Harmonic Equations

Author

Listed:
  • Jingang Xiong

    (School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China)

  • Jiancong Wen

    (School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China)

  • Yan-Cheng Liu

    (School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China)

Abstract

In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method.

Suggested Citation

  • Jingang Xiong & Jiancong Wen & Yan-Cheng Liu, 2020. "Localized Boundary Knot Method for Solving Two-Dimensional Laplace and Bi-Harmonic Equations," Mathematics, MDPI, vol. 8(8), pages 1-16, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1218-:d:389123
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    Citations

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    Cited by:

    1. Yang Wu & Junli Zhang & Shuang Ding & Yan-Cheng Liu, 2022. "Localized Boundary Knot Method for Solving Two-Dimensional Inverse Cauchy Problems," Mathematics, MDPI, vol. 10(8), pages 1-17, April.
    2. Chih-Yu Liu & Cheng-Yu Ku & Li-Dan Hong & Shih-Meng Hsu, 2021. "Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs," Mathematics, MDPI, vol. 9(13), pages 1-22, June.

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