We consider the electric field integral equation (EFIE) modeling the scattering of time-harmonic electromagnetic waves at a perfectly conducting screen. When discretizing the EFIE by means of low-order Galerkin boundary methods (BEM), one obtains linear systems that are ill-conditioned on fine meshes and for low wave numbers [Formula: see text]. This makes iterative solvers perform poorly and entails the use of preconditioning.In order to construct optimal preconditioners for the EFIE on screens, the authors recently derived compact equivalent inverses of the EFIE operator on simple Lipschitz screens in [R. Hiptmair and C. Urzúa-Torres, Compact equivalent inverse of the electric field integral operator on screens, Integral Equations Operator Theory 92 (2020) 9]. This paper elaborates how to use this result to build an optimal operator preconditioner for the EFIE on screens that can be discretized in a stable fashion. Furthermore, the stability of the preconditioner relies only on the stability of the discrete [Formula: see text] duality pairing for scalar functions, instead of the vectorial one. Therefore, this novel approach not only offers [Formula: see text]-independent and [Formula: see text]-robust condition numbers, but it is also easier to implement and accommodates non-uniform meshes without additional computational effort.

中文翻译：

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在屏幕上预处理 EFIE

我们考虑使用电场积分方程 (EFIE) 来模拟时谐电磁波在完美导电屏幕上的散射。当通过低阶 Galerkin 边界方法 (BEM) 对 EFIE 进行离散化时，可以获得在精细网格和低波数上病态的线性系统 [公式：见正文]。这使得迭代求解器的性能很差，并且需要使用预处理。为了在屏幕上为 EFIE 构建最佳预处理器，作者最近在 [R. Hiptmair 和 C. Urzúa-Torres，屏幕上电场积分算子的紧凑等效逆，积分方程算子理论 92 (2020) 9]。本文详细阐述了如何使用该结果为屏幕上的 EFIE 构建一个最佳的算子预处理器，该预处理器可以以稳定的方式离散化。此外，预条件子的稳定性仅依赖于标量函数的离散[公式：见文本]对偶对的稳定性，而不是矢量对偶的稳定性。因此，这种新颖的方法不仅提供了 [公式：参见文本] 独立和 [公式：参见文本] 稳健的条件数，而且还更容易实现并适应非均匀网格，无需额外的计算工作。