Finite Element Methods

Abstract

This chapter is concerned with numerical solutions of the grating problems which are discussed in Chaps. 2 and 3. There are two challenges for the grating problems: the solutions may have singularity due to possible nonsmooth surfaces and discontinuous media; the problems are formulated in unbounded domains. There are already many numerical methods for modeling the light diffraction by surface relief gratings, such as the S-matrix algorithm [, ], the rigorous coupled-wave approach [, ], the Fourier modal method [, , , ], the method of coordinate transformation [, , ], the boundary perturbation method [, , , , ], the method of transformed field expansion [, , , ], the boundary integral equation method [, , , , , , ], and the finite element method [, , , , , , ]. We present the adaptive finite element methods to overcome the difficulties when solving the diffraction grating problems.