Numerical simulation of meshless method and multigrid method in one dimensional boundary value problem

Abstract

The Element-free Galerkin Method has become a very popular tool for the simulation of mechanical problems with moving boundaries. The internally applied Moving Least Squares interpolation uses in general Gaussian or cubic weighting functions and has compact support. Due to the approximative character of this interpolation the obtained shape functions do not fulfill the interpolation conditions, which causes additional numerical effort for the application of the boundary conditions. The implementation of the element free Galerkin method (EFG) for one dimensional boundary value problem based on Poisson’s equation is presented in this paper. The formulation of the discrete system equations is derived from the governing equations of weak function. Moving least-squares approximation is used in this method. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of Poisson’s problem. Essential boundary conditions are efficiently imposed. The results of EFG method will be compared favourably with closed-form solutions and that of multigrid method and finite difference method.