  The Wave Based Method (WBM) is a Trefftz method for the simulation of wave
problems in vibroacoustics. Like other Trefftz methods, it employs a
non-standard discretisation basis consisting of solutions of the partial
differential equation (PDE) at hand. We analyse the convergence and numerical
stability of the Wave Based Method for Helmholtz problems using tools from
approximation theory. We show that the set of discretisation functions more
closely resembles a frame, a redundant set of functions, than a basis. The
redundancy of a frame typically leads to ill-conditioning, which indeed is
common in Trefftz methods. Recent theoretical results on frames for function
approximation suggest that the associated ill-conditioned system matrix can be
successfully regularised, with error bounds available, when using a discrete
least squares approach. While the original Wave Based Method is based on a
weighted residual formulation, in this paper we pursue an oversampled
collocation approach instead. We show that, for smooth scattering obstacles in
two dimensions, the results closely follow the theory of frames. We identify
cases where the method achieves very high accuracy whilst providing a solution
with small norm coefficients, in spite of ill-conditioning. Moreover, the
accurate results are reliably maintained even in parameter regimes associated
with extremely high ill-conditioning.
