An investigation of global radial basis function collocation methods applied to Helmholtz problems

TL;DR

This paper analyzes global RBF collocation methods for Helmholtz problems, focusing on convergence, stability, and parameter selection, with theoretical insights and numerical experiments in curved geometries.

Contribution

It provides a comprehensive analysis of RBF collocation methods for Helmholtz equations, including convergence behavior, flat limit analysis, and practical parameter strategies.

Findings

Exponential convergence for smooth solutions in ideal conditions

Identification of singularities in collocation matrices at certain parameters

Guidelines for choosing shape parameters to balance accuracy and stability

Abstract

Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severely ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyse and characterise the convergence properties as a function of the number of unknowns and for different shape parameter ranges. We provide theoretical results for the flat limit of the PDE solutions and investigate when the non-symmetric collocation matrices become singular. We also provide practical strategies for choosing the method parameters and evaluate the results on Helmholtz problems in a curved waveguide geometry.