Many integral equation-based methods are available for problems of time-harmonic electromagnetic scattering from perfect electric conductors. Moreover, there are numerous ways in which the geometry can be represented, numerous ways to represent the relevant surface current and/or charge densities, numerous quadrature methods that can be deployed, and numerous fast methods that can be used to accelerate the solution of the large linear systems which arise from discretization. Among the many issues that arise in such scattering calculations are the avoidance of spurious resonances, the applicability of the chosen method to scatterers of non-trivial topology, the robustness of the method when applied to objects with multiscale features, the stability of the method under mesh refinement, the ease of implementation with high-order basis functions, and the behavior of the method as the frequency tends to zero. Since three-dimensional scattering is a challenging, large-scale problem, many of these issues have been historically difficult to investigate. It is only with the advent of fast algorithms and modern iterative methods that a careful study of these issues can be carried out effectively. In this paper, we use GMRES as our iterative solver and the fast multipole method as our acceleration scheme in order to investigate some of these questions. In particular, we compare the behavior of the following integral equation formulations with regard to the issues noted above: the standard electric, magnetic, and combined field integral equations with standard RWG basis functions, the non-resonant charge-current integral equation, the electric charge-current integral equation, the augmented regularized combined source integral equation and the decoupled potential integral equation DPIE. Various numerical results are provided to demonstrate the behavior of each of these schemes.

翻译：许多基于积分方程的方法可用于理想导体时谐电磁散射问题。此外，几何表示有多种方式，相关表面电流和/或电荷密度有多种表示方式，可部署的求积方法有多种，以及可用于加速离散化产生的大型线性系统求解的快速方法也有多种。此类散射计算中出现的众多问题包括：避免虚假共振、所选方法对非平凡拓扑散射体的适用性、方法应用于多尺度特征目标时的稳健性、网格细化下方法的稳定性、高阶基函数实现的简便性，以及方法随频率趋近于零时的行为。由于三维散射是极具挑战性的大规模问题，其中许多问题在历史上一直难以研究。直到快速算法和现代迭代方法的出现，才能对这些问题进行有效而细致的研究。本文采用GMRES作为迭代求解器，并以快速多极子方法作为加速方案，以探讨其中部分问题。特别地，我们针对上述问题比较了以下积分方程公式的行为：采用标准RWG基函数的电场、磁场及组合场积分方程，非共振电荷-电流积分方程，电场电荷-电流积分方程，增广正则化组合源积分方程，以及解耦势积分方程DPIE。文中提供了多种数值结果以展示每种方案的性能。