An Adaptive Finite Element DtN Method for Maxwell's Equations
An Adaptive Finite Element DtN Method for Maxwell's Equations
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DOI：
10.4208/eajam.2022-289
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发表时间：
2022-02
期刊：
影响因子：
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通讯作者：
G. Bao;Mingming Zhang;Xue Jiang;Peijun Li;Xiaokai Yuan
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文献类型：
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作者：
G. Bao;Mingming Zhang;Xue Jiang;Peijun Li;Xiaokai Yuan
本文研究了时谐电磁波在三维空间中被有界不可穿透障碍物散射的数值解。用麦克斯韦方程组的边值问题来模拟障碍物外域的电磁波传播。基于由无穷级数定义的Dirichlet-to-Neumann （DtN）算子，引入了精确透明边界条件，将散射问题等效化为有界域。针对离散变分问题，提出了一种基于后验误差估计的自适应有限元DtN方法，将DtN算子截断为有限多个项的和。后验误差估计同时考虑了有限元逼近误差和DtN算子的截断误差。后者显示出相对于截断参数呈指数衰减。数值实验验证了该方法的有效性。
This paper is concerned with a numerical solution to the scattering of a time-harmonic electromagnetic wave by a bounded and impenetrable obstacle in three dimensions. The electromagnetic wave propagation is modeled by a boundary value problem of Maxwell's equations in the exterior domain of the obstacle. Based on the Dirichlet-to-Neumann (DtN) operator, which is defined by an infinite series, an exact transparent boundary condition is introduced and the scattering problem is reduced equivalently into a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is developed to solve the discrete variational problem, where the DtN operator is truncated into a sum of finitely many terms. The a posteriori error estimate takes into account both the finite element approximation error and the truncation error of the DtN operator. The latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.