We consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for three-dimensional Maxwell’s equations. Based on the Dirichlet-to-Neumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An a posteriori error estimate-based adaptive edge finite element method is developed for the variational problem with the truncated DtN operator. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the former is used for local mesh refinements and the latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to demonstrate the competitive behaviour of the proposed method.

中文翻译：

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双周期结构中麦克斯韦方程组的自适应边缘有限元DtN方法

我们通过双周期结构考虑电磁平面波的衍射。本文关注的是三维麦克斯韦方程的衍射光栅问题的数值解。基于Dirichlet-to-Neumann (DtN)算子，利用透明边界条件，在有界域中建立了等价边值问题。针对截断DtN算子的变分问题，提出了一种基于后验误差估计的自适应边缘有限元方法。该估计同时考虑了有限元逼近误差和 DtN 算子的截断误差，其中前者用于局部网格细化，而后者显示出相对于截断参数呈指数衰减。