SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1083-1110, June 2022.
This paper proposes and examines a finite element method (FEM) for a Maxwell obstacle problem in electric shielding. The model is given by a coupled system comprising the Faraday equation and an evolutionary variational inequality (VI) of Ampère--Maxwell-type. Based on the leapfrog (Yee) time-stepping and the Nédélec edge elements, we set up a fully discrete FEM where the obstacle is discretized in such a way that no additional nonlinear solver is required for the computation of the discrete VI. While the $L^2$-stability is achieved for the discrete solutions and the associated difference quotients, the scheme only guarantees the $L^1$-stability for the discrete magnetic curl field in the obstacle region. The lack of the global $L^2$-stability for the magnetic curl field is justified by the low regularity issue in Maxwell obstacle problems and turns to be the main challenge in the convergence analysis. Our convergence proof consists of two main stages. First, exploiting the $L^1$-stability in the obstacle region, we derive a convergence result towards a weaker system involving smooth feasible test functions. In the second step, we recover the original system by enlarging the feasible test function set through a specific constraint preserving mollification process in the spirit of Ern and Guermond [Comput. Methods Appl. Math., 16 (2016), pp. 51--75]. This paper is closed by three-dimensional numerical results of the proposed FEM confirming the theoretical convergence result and, in particular, the Faraday shielding effect.

中文翻译：

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电屏蔽中麦克斯韦障碍问题的数值分析

SIAM 数值分析杂志，第 60 卷，第 3 期，第 1083-1110 页，2022 年 6 月。
本文针对电屏蔽中的麦克斯韦障碍问题提出并检验了有限元法 (FEM)。该模型由包含法拉第方程和安培-麦克斯韦型进化变分不等式 (VI) 的耦合系统给出。基于跳蛙（Yee）时间步长和 Nédélec 边缘元素，我们建立了一个完全离散的 FEM，其中障碍物以这样一种方式离散化，即不需要额外的非线性求解器来计算离散 VI。虽然对于离散解和相关的差商实现了 $L^2$-稳定性，但该方案仅保证障碍区域中离散磁旋流场的 $L^1$-稳定性。麦克斯韦障碍问题中的低规律性问题证明了磁旋场缺乏全局$L^2$-稳定性，并成为收敛性分析中的主要挑战。我们的收敛证明包括两个主要阶段。首先，利用障碍区域中的$L^1$-稳定性，我们得出了一个收敛结果，该结果朝向涉及平滑可行测试函数的较弱系统。在第二步中，我们本着 Ern 和 Guermond [Comput. 方法应用程序。数学，16（2016 年），第 51--75 页]。本文以所提出的有限元法的三维数值结果结束，证实了理论收敛结果，特别是法拉第屏蔽效应。