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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 L²-stability is achieved for the discrete solutions and the associated difference quotients, the scheme only guarantees the L¹-stability for the discrete magnetic curl field in the obstacle region. The lack of the global L²-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 L¹-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. [ABSTRACT FROM AUTHOR]

Over the last 10 years, results from [J. M. Melenk and S. Sauter, Math. Comp., 79 (2010), pp. 1871--1914], [J. M. Melenk and S. Sauter, SIAM J. Numer. Anal., 49 (2011), pp. 1210--1243], [S. Esterhazy and J. M. Melenk, Numerical Analysis of Multiscale Problems, Springer, New York, 2012, pp. 285--324] and [J. M. Melenk, A. Parsania, and S. Sauter, J. Sci. Comput., 57 (2013), pp. 536--581] decomposing high-frequency Helmholtz solutions into "low-" and "high-" frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer--Sjöstrand functional calculus [B. Helffer and J. Sjöstrand, Schrödinger Operators, Springer, Berlin, 1989, pp. 118--197] this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjöstrand and Zworski [J. Amer. Math. Soc., 4 (1991), pp. 729--769] thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the hp-finite-element method (hp-FEM) applied to the variable-coefficient Helmholtz equation in the exterior of an analytic Dirichlet obstacle, where the coefficients are analytic in a neighborhood of the obstacle, and (ii) the h-FEM applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the hp-FEM applied to this problem does not suffer from the pollution effect. [ABSTRACT FROM AUTHOR]