2018/02/04 by Daan Huybrechs, Huybrechs, Daan, Anda-Elena Olteanu +1 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Digital Filter Design and Implementation #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1802.01090
openalex publication_date 2018/02/04 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
The Wave Based Method (WBM) is a Trefftz method for the simulation of wave\nproblems in vibroacoustics. Like other Trefftz methods, it employs a\nnon-standard discretisation basis consisting of solutions of the partial\ndifferential equation (PDE) at hand. We analyse the convergence and numerical\nstability of the Wave Based Method for Helmholtz problems using tools from\napproximation theory. We show that the set of discretisation functions more\nclosely resembles a frame, a redundant set of functions, than a basis. The\nredundancy of a frame typically leads to ill-conditioning, which indeed is\ncommon in Trefftz methods. Recent theoretical results on frames for function\napproximation suggest that the associated ill-conditioned system matrix can be\nsuccessfully regularised, with error bounds available, when using a discrete\nleast squares approach. While the original Wave Based Method is based on a\nweighted residual formulation, in this paper we pursue an oversampled\ncollocation approach instead. We show that, for smooth scattering obstacles in\ntwo dimensions, the results closely follow the theory of frames. We identify\ncases where the method achieves very high accuracy whilst providing a solution\nwith small norm coefficients, in spite of ill-conditioning. Moreover, the\naccurate results are reliably maintained even in parameter regimes associated\nwith extremely high ill-conditioning.\n