Exponential convergence of the PML method for periodic surface scattering problems

Speaker:Dr. Ruming Zhang (Karlsruhe Institute of Technology)

Date:October 18,2021 Mon 3:00pm-4:00pm (Beijing time)

Venue: ZOOM ID：924 0584 1092 Password: 674849

Abstract:The main task is to prove that the perfectly matched layers (PML)  method converges exponentially 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, which is scattering problems with periodic surfaces. The result can also be easily extended to locally  perturbed periodic surfaces or periodic layers. Due to technical reasons,  we have to exclude all the half integer valued wavenumbers. 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 from the inverse Floquet-Bloch transform.