NUMERICAL ANALYSIS FOR MAXWELL OBSTACLE PROBLEMS IN ELECTRIC SHIELDING

摘要

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 Ampere-Maxwell-type. Based on the leapfrog (Yee) time-stepping and the Nedelec 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 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.