Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation

When numerical methods such as the finite element method (FEM) arc used to solve the Helmholtz equation, the solutions suffer from the so-called pollution effect which leads to inaccurate results, especially for high wave numbers. The main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution, which is known as dispersion. In order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences. In this paper a meshfree method, namely the radial point interpolation method (RPIM), is investigated with respect to the pollution effect in the 2D-case. It is shown that this methodology is able to reduce the dispersion significantly. Two modifications of the RPIM, namely one with polynomial reproduction and another one with a problem-dependent sine/cosine basis, are also described and tested. Numerical experiments are carried out to demonstrate the advantages of the method compared with the FEM. For identical discretizations, the RPIM yields considerably better results than the FEM. Copyright (C) 2008 John Wiley & Sons, Ltd.