无网格Galerkin法求解二维Helmholtz方程

An Element-Free Galerkin Method for Solving Two-Dimensional Helmholtz Equation

摘要: 无网格Galerkin法本来是对固体力学问题进行数值模拟的一个重要方法，在本文我们用此方法来对Helmholtz方程进行求解，并对求解所得的数值模拟结果进行了验证。在求解Helmholtz方程的过程中，首先用移动最小二乘近似来构造形函数，然后边值条件由拉格朗日乘子法引入，最后对平面规则域上的一般Helmholtz方程进行了数值模拟，所得到的结果趋向于精确解，且随着节点的增加，其精确度越来越高，验证了该方法具有良好的收敛性。

Abstract: The element-free Galerkin method is originally an important method for numerical simulation of solid mechanics problems. In this paper, we use this method to solve the Helmholtz equation and verify the numerical simulation results. In the process of solving Helmholtz equation, the shape function is constructed by moving least squares approximation, then the boundary condition is introduced by Lagrange multiplier method. Finally, the general Helmholtz equation on the plane rule domain is numerically simulated. The obtained results tend to be exact solutions. Furthermore, high precision and good convergence could be contained as the nodes increased.