Limitations of Nyquist Criteria in the Discretization of 2D Electromagnetic Integral Equations at High Frequency: Spectral Insights into Pollution Effects
Viviana Giunzioni, Adrien Merlini, Francesco P. Andriulli
TL;DR
This work provides a rigorous spectral analysis of common 2D boundary integral formulations for electromagnetic scattering, revealing a pollution mechanism that arises from discretization while maintaining a fixed number of unknowns per wavelength. By decomposing spectral errors into projection and aliasing components and examining hyperbolic, transition, and elliptic regions, the authors show that pollution can affect TE-EFIE and Calderón-stabilized CCFIE, even when well-conditioned, and quantify its impact on currents and scattered fields. A key contribution is a practical filtering strategy that cuts off high-frequency spectral components of the hypersingular operator, yielding bounded spectral, current, and scattering errors and eliminating resonance-induced degradation. The results, supported by detailed numerical experiments on a PEC cylinder, demonstrate the feasibility and effectiveness of operator filtering to mitigate Nyquist-type pollution in high-frequency boundary element simulations, with implications for more robust high-frequency BEM analyses.
Abstract
The use of boundary integral equations in modeling boundary value problems-such as elastic, acoustic, or electromagnetic ones-is well established in the literature and widespread in practical applications. These equations are typically solved numerically using boundary element methods (BEMs), which generally provide accurate and reliable solutions. 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. 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 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 two-dimensional 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. Our analysis, which is capable of tracking the effects of BEM discretization on compositions and sums of different operators, 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.