Stéphane Suleau

Dispersion and pollution of meshless solutions for the Helmholtz equation

It is well known today that the standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wavenumbers due to the pollution effect, consisting mainly of the dispersion, i.e. the numerical wavelength is longer than the exact one. Unless highly refined meshes are used, FEM solutions lead to unacceptable solutions in terms of precision, while the use of very refined meshed increases the cost in terms of computational times. The paper presents an application of the element-free Galerkin method (EFGM) and focuses on the dispersion analysis in 2D. It shows that it is possible to choose the parameters of the method in order to minimize the dispersion and to get extremely good results in comparison with the stabilized FEM. Moreover, the present meshless formulation is not restricted to regular distribution of nodes and a simple but real-life problem is investigated in order to show the improvement in the accuracy of the numerical results w.r. FEM results.

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

The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly refined 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 finite 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 finite element one. Copyright

Methode d'approximation nodale optimisee pour l'acoustique

The element-free Galerkin method is formulated for the Helmholtz problem. It is based on a moving least square method. This article shows that it is possible to determine the speed of propagation of the numerical wave and to tune the parameters of the method in order to minimise the dispersion.