Limitations of Nyquist Criteria in the Discretization of 2D Electromagnetic Integral Equations at High Frequency: Spectral Insights into Pollution Effects

Electromagnetic integral equations are typically solved numerically using boundary element methods (BEMs). When the frequency of the wave phenomenon under study increases, the discretization of the problem is typically chosen to maintain a fixed number of unknowns per wavelength. Under these conditions, the BEM over finite-dimensional subspaces of piecewise polynomial basis functions is commonly believed to provide a bounded solution accuracy, at least for non-trapping geometries. If proven, this would constitute a significant advantage of the BEM with respect to finite element and finite difference time domain methods, which, in contrast, are affected by numerical dispersion, that causes the number of unknowns per wavelength required to achieve a prescribed solution accuracy to increase with frequency, phenomenon known as pollution. In this work, we conduct a rigorous spectral analysis of some of the most commonly used boundary integral operators and examine the impact of the BEM discretization on the solution accuracy of widely used integral equations modeling the electromagnetic scattering from a perfectly electrically conducting cylinder. We consider both ill-conditioned and well-conditioned equations, the latter being characterized by solution operators bounded independently of frequency. Contrary to the common belief, our analysis reveals a form of pollution that affects, in different measures, equations of both kinds. After elucidating the mechanism by which the BEM discretization impacts accuracy, we propose a solution strategy that can cure the pollution problem thus evidenced. The defining strength of the proposed theoretical model lies in its capacity to deliver deep insight into the root causes of the phenomenon.