Communications in Mathematical Sciences

Volume 20 (2022)

Number 7

Analysis of the time-domain PML problem for the electromagnetic scattering by periodic structures

Pages: 1785 – 1813



Yanli Chen (Department of Mathematics, Northeastern University, Shenyang, China)

Yixian Gao (School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin, China)

Peijun Li (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


This paper is concerned with the time-domain scattering of an electromagnetic plane wave by a periodic structure. An initial boundary value problem is formulated in a bounded domain by applying the perfectly matched layer (PML) technique to the scattering problem imposed in an unbounded domain. Based on the abstract inversion theorem of the Laplace transform and the analysis in the frequency domain, the well-posedness and stability are established for the truncated time-domain PML problem. Moreover, the exponential convergence of the solution for the truncated PML problem is proved by a careful study on the error for the Dirichlet-to-Neumann operators between the original scattering problem and the truncated PML problem.


time-domain Maxwell’s equations, diffraction gratings, transparent boundary condition, perfectly matched layer, well-posedness and stability, convergence

2010 Mathematics Subject Classification

35Q61, 78A25, 78A45, 78M30

The research of Y.C. was supported in part by NSFC grant 12001086. The research of Y.G. was partially supported by NSFC grants 11871140, 12071065, JLSTDP20190201154JC and FRFCU 2412019BJ005. The research of P.L. was supported in part by the NSF grant DMS-1912704.

Received 9 November 2020

Accepted 24 January 2022

Published 21 October 2022