Adaptive Finite Element Methods For Diffraction Grating Problems With PML And Few-Mode DtN Truncations

Adaptive finite element methods with perfectly matched layer(PML)technique for one and two-dimensional diffraction gratings is considered.The grating problems are modeled by Helmholtz equation and Maxwell s equations.For each grating problem,the PML is trun-cated by some few-mode Dirichlet to Neumann(DtN)boundary conditions so that those Fourier modes that cannot be well absorbed by the PML pass through without reflections.Even with the Rayleigh resonance or the nearly Rayleigh resonance,convergence of the trun-cated PML solution is proved,whose rate is exponential with respect to the PML parameters and uniform with respect to all modes.An a posteriori error estimate is derived for the finite element discretization on periodic meshes.The a posteriori error estimate consists of two parts,the finite element discretization error and the PML truncation error.Based on the a posteriori error control,a finite element adaptive strategy is established for the diffraction grating problem.For 1D grating problem,numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive algorithm.In addition,we consider solving Poisson's equation with periodic boundary condition on nonperiodic meshes.A finite element method with penalty on the periodic boundary is proposed.The a priori error estimates indicate that the method is convergent with respect to the mesh scale.We also present the a posteriori error estimates under the assumption that the mesh is admissible.