An Adaptive Finite Element Method for the Diffraction Grating Problem with Transparent Boundary Condition

The diffraction grating problem is modeled by a boundary value problem governed by a Helmholtz equation with transparent boundary conditions. An a posteriori error estimate is derived when the truncation of the nonlocal boundary operators takes place. To overcome the difficulty caused by the fact that the truncated Dirichlet-to-Neumann (DtN) mapping does not converge to the original DtN mapping in its operator norm, a duality argument without assuming more regularity than the weak solution is applied. The a posteriori error estimate consists of two parts, the finite element discretization error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter. Based on the a posteriori error control, a finite element adaptive strategy is established for the diffraction grating problem, such that the truncation parameter is determined through the 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.