The Hybrid Element-Free Galerkin Method for 3D Helmholtz Equations

Abstract

In this study, the hybrid element-free Galerkin (HEFG) method is studied to solve the 3D Helmholtz equations. The idea of the dimension splitting method (DSM) is introduced into the improved element-free Galerkin (IEFG) method, thus a sequence of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation, each 2D form can be discretized by using the IEFG method, thus the corresponding 2D discretized equations can be obtained, and these equations can be coupled by employing the finite difference method (FDM) in the dimension splitting direction, thus final formulae of numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors and the convergence are analyzed, respectively, and the numerical results show that the calculation resources can be saved greatly without losing the computational accuracy when using the HEFG method.