Solving electromagnetic scattering problems at resonance frequencies

Abstract

The ‘‘resonance problem’’ is that at certain values of the wave number k (the resonant k’s), the second-kind integral equation for solving scattering problems can become extremely ill-conditioned. This adversely affects both the accuracy and speed of numerical solutions. We consider transverse-magnetic scattering from a conductor (Dirichlet problem). The integral equation (derived using double-layer potentials) is discretized using approximately fourth-order convergent quadrature formulas. At resonant k’s for circular and elliptical scatterers, we find very large condition numbers for the discrete matrices [up to O(107) ], generally leading to poor solutions. We apply two approaches to alleviate the resonance problem. The first is to use a different integral equation, based on both single- and double-layer potentials. This leads to low condition numbers and good solutions at resonant k. The second method is to use the original second-kind integral equation, introduce a small imaginary part in k, and extrapolate back to the real axis. Solutions obtained by the two methods are in excellent agreement. The extrapolation technique will be particularly useful in the case of the exterior Neumann problem, when the application of the first technique will be numerically more difficult. By solving the resonance problem, we ensure that fast and accurate solutions are obtainable at any arbitrary wave number.