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

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.

Automated summary

A localized virtual boundary element-meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex 2D geometries. The LVBE-MCM simplifies the traditional virtual boundary element method by locally approximating the physical quantities and avoiding the evaluation of singular integrals. Numerical experiments show that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations.