The fictitious eigenvalue difficulty in the T-matrix and the BEM methods for exterior wave scattering

Abstract

Exterior time-harmonic wave scattering by finite obstacles in infinite domains has been studied extensively by both boundary element methods (BEM) and transfer-matrix (T-matrix) methods. The former is essentially a local discretization numerical approach, while the latter is basically a global ‘‘eigenfunction’’ expansion approach and thus these two approaches represent in some sense an ‘‘alpha and omega’’ of approximation solution method. Since these are both based on a Helmholtz integral equation formulation, which is known to fail at certain eigenvalues of a related interior problem, one could expect corresponding failures in these approximate methods. Such failures are well known in the BEM literature; they are recognized in principle but are essentially ignored in practice in the T-matrix literature. This paper studies this disparity through a simple example where the T-matrix approach is seen to fail over such a narrow frequency band near the interior eigenvalues that finding such failures in a standard solution procedure using discrete frequency values is highly unlikely. However, a related eigenfunction expansion method yields the same broadband failure as found in BEM indicating that this is not simply a result of eigenfunction versus element accuracy.