  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.
