标题： 奈奎斯特准则在高频下二维电磁积分方程离散化的局限性：谱视角下的污染效应洞察 显示英文标题

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

摘要： 边界积分方程在建模边值问题（如弹性、声学或电磁问题）中的应用在文献中已经得到广泛认可，并且在实际应用中也十分普遍。这些方程通常使用边界元方法（BEMs）进行数值求解，该方法通常能提供准确可靠的结果。 当研究的波动现象频率增加时，问题的离散化通常选择以每波长固定数量的未知数来保持精度。在这种情况下，有限维子空间上的BEM，使用分段多项式基函数，通常被认为能够提供有界解精度。 如果这一结论被证明成立，这将构成BEM相对于有限元和有限差分时域方法的一个显著优势，因为后者会受到数值污染的影响。在这项工作中，我们对一些最常用的边界积分算子进行了严格的谱分析，并考察了BEM离散化对广泛使用的积分方程求解二维电磁散射问题的影响，这些问题描述了一个完美电导圆柱的散射。 我们考虑了病态条件方程和良态条件方程，后者的特点是其解算子与频率无关地保持有界。我们的分析能够追踪BEM离散化对不同算子组合和求和的影响，揭示了一种影响各种类型方程的污染形式。通过阐明BEM离散化如何影响精度，我们提出了一种解决方案策略，可以治愈这种所揭示的污染问题。所提出的理论模型的主要优势在于它能够深入洞察该现象的根本原因。 显示英文摘要

摘要： 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.