An Adaptive Finite Element DtN Method for Maxwell's Equations in Biperiodic Structures
For computational electromagnetics, this provides an adaptive algorithm for grating problems, though it is an incremental improvement over existing DtN-based methods.
The paper develops an adaptive finite element method with a Dirichlet-to-Neumann boundary operator for solving Maxwell's equations in biperiodic structures, achieving error control through a posteriori estimates that separate finite element and truncation errors. Numerical experiments show competitive performance.
Consider the diffraction of an electromagnetic plane wave by a biperiodic structure where the wave propagation is governed by the three-dimensional Maxwell equations. Based on transparent boundary condition, the grating problem is formulated into a boundary value problem in a bounded domain. Using a duality argument technique, we derive an a posteriori error estimate for the finite element method with the truncation of the nonlocal Dirichlet-to-Neumann (DtN) boundary operator. The a posteriori error consists of both the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is developed with error controlled by the a posterior error estimate, which determines the truncation parameter through the truncation error and adjusts the mesh through the finite element approximation error. Numerical experiments are presented to demonstrate the competitive behavior of the proposed adaptive method.