One-dimensional dispersion analysis for the element-free Galerkin method for the Helmholtz equation

The standard "nite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly re"ned meshes are used, leading to unacceptable resolution times. The paper presents an application of the element-free Galerkin method (EFG) and focuses on the dispersion analysis in one dimension. It shows that, if the basis contains the solution of the homogenized Helmholtz equation, it is possible to eliminate the dispersion in a very natural way while it is not the case for the "nite element methods. For the general case, it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion. Finally, theoretical developments are validated by numerical experiments showing that, for the same distribution of nodes, the element-free Galerkin method solution is much more accurate than the "nite element one