A Dimension Coupling Method for Three-Dimensional Helmholtz and Poisson’s Equations

Abstract

In this study, by introducing the finite element method (FEM) into the improved element-free Galerkin (IEFG) method, the dimension coupling method (DCM) is proposed for solving three-dimensional (3D) Helmholtz and Poisson’s equations efficiently. The dimensional splitting method is introduced into the corresponding governing equations, thus 3D equations can be split into a series of 2D ones. The IEFG method is selected to discretize those 2D forms, thus the discretized equations are derived easily by using the weak forms. In the third direction, the FEM is selected to couple those discretized equations, thus the final linear equation of 3D equation is derived. In numerical examples, the good convergence of the DCM for Helmholtz and Poisson’s equations is shown. The solutions of numerical examples show that the computational efficiency of the IEFG method is improved significantly without losing the computational accuracy when the DCM is used. In addition, the DCM can enhance the computational efficiency of the hybrid element-free Galerkin (HEFG) method significantly without too many layers when the natural boundary conditions exist in the splitting direction.