Radial basis function approximation for Helmholtz problems

Radial basis function (RBF) approximation methods are meshfree and can provide spectral accuracy. Therefore, they are attractive to use for solving partial differential equations with smooth solutions. However, one difficulty when using RBF methods is how to select the best method parameters for the problem at hand. The choices to make include the number of node and center points, the placement of the points, the basis function and, if applicable, the shape parameter of the basis functions. In this study, we first look at a simple one-dimensional model problem. We investigate how the optimal method parameters vary with the problem parameters. Experiments show that the behavior is very regular and can, in some cases, be described by simple relations. We show how the optimal fixed shape parameter can be determined. Then we give some theoretical results for the behavior of the solutions in the case of nearly flat basis functions. A comparison with a second order finite difference method is also performed. We find that, even for this simple problem where the finite difference method is very efficient, the RBF method is less computationally expensive if an accurate solution is desired. Finally, we present experiments and results for more complicated Helmholtz problems in two dimensions.