An oversampled collocation approach of the Wave Based Method for Helmholtz problems
Provides a theoretical and practical framework for regularizing ill-conditioned Trefftz methods in vibroacoustics, enabling reliable high-accuracy solutions.
The paper analyzes the Wave Based Method for Helmholtz problems, showing its discretization functions form a frame rather than a basis, leading to ill-conditioning. By adopting an oversampled collocation approach, the method achieves high accuracy with small norm coefficients, even under extreme ill-conditioning.
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.