An Adaptive Finite Element Method for the Diffraction Grating Problem with PML and Few-Mode DtN Truncations

The diffraction grating problem is modeled by a Helmholtz equation with PML boundary conditions. The PML is truncated by some few-mode Dirichlet to Neumann boundary conditions so that those Fourier modes that cannot be well absorbed by the PML pass through without reflections. Convergence of the truncated 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. The a posteriori error estimate consists of two parts, the finite element discretization error and the PML truncation error which decays exponentially with respect to the PML parameters and uniformly with respect to all modes. Based on the a posteriori error control, a finite element adaptive strategy is established for the diffraction grating problem, such that the PML parameters are determined through the PML truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive algorithm.