The Diffraction Gratings Truncated By The PML And Few-mode DtN Boundary Conditions And The A Post- Eriori Error Estimates For The Finite Element Discretiz- Ations

The finite element method with PML for one-dimensional gratings is considered. The PML medium used to be truncated by Dirichlet boundary conditions with the assumption that no Rayleigh resonance occurs, as those nearly Rayleigh resonant modes cannot be absorbed effectively by PML and may generate reflected waves at the Dirichlet truncated boundaries.In this thesis, we truncate the PML with few-mode DtN boundary con-ditions to let those nearly Rayleigh resonant modes pass through without reflection. Even with the Rayleigh resonance, it is shown that the truncated PML solution converges to the original scattering solution in the computa-tional domain at an exponential rate with respect to either the PML model medium property or the thickness of the PML layer. An a posteriori upper bound for the error between the finite element solution and the original scat-tering solution in the computational domain is established. The a posteriori error estimate may be used to choose the PML parameters and the thickness of the PML layer and to mark elements for refinement in an adaptive finite element algorithm.