Abstract

The main task in this talk is to prove that the perfectly matched layers (PML) method has high order converge with respect to the PML parameter, for scattering problems with periodic surfaces. A linear convergence has already been proved for the PML method for scattering problems with rough surfaces in a paper by S.N. Chandler-Wilder and P. Monk in 2009. At the end of that paper, three important questions are asked, and the third question is if exponential convergence holds locally. In this talk, we answer this question for a special case, i.e., scattering problems with periodic surfaces. The main idea of the proof is to apply the Floquet-Bloch transform to write the problem into an equivalent family of quasi-periodic problems, and then study the analytic extension of the quasi-periodic problems with respect to the Floquet-Bloch parameters. Then the Cauchy integral formula is applied for piecewise analytic functions to avoid linear convergent points. Finally the exponential convergence is proved for almost all 2D cases and 3D cases with small wavenumbers, and high order convergence is proved for 3D cases with larger wavenumbers.