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

TL;DR

This paper provides a spectral analysis of boundary integral operators in 2D electromagnetic scattering, revealing pollution effects in BEM discretization at high frequencies and proposing strategies to mitigate these issues.

Contribution

It offers a rigorous spectral framework to understand pollution effects in BEM for electromagnetic problems and introduces a solution strategy to address discretization-induced inaccuracies.

Findings

Pollution affects both well-conditioned and ill-conditioned equations.

Discretization impacts the accuracy of boundary integral operators.

A proposed method can mitigate pollution effects.

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.