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A Simple, Accurate and Semi-Analytical Meshless Method for Solving Laplace and Helmholtz Equations in Complex Two-Dimensional Geometries

Author

Listed:
  • Xingxing Yue

    (College of Materials Science and Engineering, Qingdao University, Qingdao 266071, China)

  • Buwen Jiang

    (College of Materials Science and Engineering, Qingdao University, Qingdao 266071, China)

  • Xiaoxuan Xue

    (College of Materials Science and Engineering, Qingdao University, Qingdao 266071, China)

  • Chao Yang

    (College of Materials Science and Engineering, Qingdao University, Qingdao 266071, China)

Abstract

A localized virtual boundary element–meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries. “Localized” refers to employing the moving least square method to locally approximate the physical quantities of the computational domain after introducing the traditional virtual boundary element method. The LVBE-MCM is a semi-analytical and domain-type meshless collocation method that is based on the fundamental solution of the governing equation, which is different from the traditional virtual boundary element method. When it comes to 2D problems, the LVBE-MCM only needs to calculate the numerical integration on the circular virtual boundary. It avoids the evaluation of singular/strong singular/hypersingular integrals seen in the boundary element method. Compared to the difficulty of selecting the virtual boundary and evaluating singular integrals, the LVBE-MCM is simple and straightforward. Numerical experiments, including irregular and doubly connected domains, demonstrate that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations.

Suggested Citation

  • Xingxing Yue & Buwen Jiang & Xiaoxuan Xue & Chao Yang, 2022. "A Simple, Accurate and Semi-Analytical Meshless Method for Solving Laplace and Helmholtz Equations in Complex Two-Dimensional Geometries," Mathematics, MDPI, vol. 10(5), pages 1-9, March.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:833-:d:764975
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    References listed on IDEAS

    as
    1. Chai, Yingbin & Li, Wei & Liu, Zuyuan, 2022. "Analysis of transient wave propagation dynamics using the enriched finite element method with interpolation cover functions," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    2. Wang, Fajie & Zhao, Qinghai & Chen, Zengtao & Fan, Chia-Ming, 2021. "Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains," Applied Mathematics and Computation, Elsevier, vol. 397(C).
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    Cited by:

    1. Chih-Yu Liu & Cheng-Yu Ku, 2022. "A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems," Mathematics, MDPI, vol. 10(10), pages 1-23, May.

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