Finite Element Approximation of Time Harmonic Waves in Periodic Structures

Consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material. It is shown that the scattering (diffraction) problem may be modeled by a Helmholtz equation with transparent boundary conditions. The diffraction problem may be solved by a finite element method. In this paper, optimal error estimates for the finite element method are established. The error estimates are also established when the truncation of the nonlocal transparent boundary operators takes place.