Meshfree Methods for the Helmholtz Equation with Variable Wave Speed

In this paper, we develop one-level and multilevel meshfree radial basis functions (RBF) collocation methods for solving the Helmholtz equation with variable wave speed on a bounded connected Lipschitz domain. The approximate solution is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. We prove the convergence of one-level and multilevel collocation method for the modelling problem in Sobolev spaces. The convergence rates depend on the regularity of the solution, the smoothness of the computational domain, the bounds of frequency and wave speed, the approximation of scaled kernel-based spaces, the increasing rules of scattered data, and the selection of scaling parameters.