Communications in Mathematical Sciences

Volume 16 (2018)

Number 4

Convergence of the PML solution for elastic wave scattering by biperiodic structures

Pages: 987 – 1016



Xue Jiang (School of Science, Beijing University of Posts and Telecommunications, Beijing, China)

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

Junliang Lv (School of Mathematics, Jilin University, Changchun, China)

Weiying Zheng (Institute of Computational Mathematics and Scientific/Enginnering Computing, C.A.S., Beijing, China; and School of Mathematical Science, University of the Chinese Academy of Sciences, Beijing, China)


This paper is concerned with the analysis of elastic wave scattering of a time-harmonic plane wave by a biperiodic rigid surface, where the wave propagation is governed by the three-dimensional Navier equation. An exact transparent boundary condition is developed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. The Perfectly Matched Layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing a PML equivalent transparent boundary condition. The proofs rely on a careful study of the error between the two transparent boundary operators. The work significantly extends the results from one-dimensional periodic structures to two-dimensional biperiodic structures. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.


elastic wave equation, perfectly matched layer, biperiodic structures, transparent boundary condition

2010 Mathematics Subject Classification

35Q60, 65N30, 78A45

The research of X.J. was supported in part by National Natural Science Foundation of China grant 11771057, 11401040 and 11671052. The research of P.L. was supported in part by the NSF grant DMS-1151308. The research of J.L. was partially supported by Science Challenge Project grant TZ2016002 and by the National Natural Science Foundation of China grant 11301214. The research of W.Z. was supported in part by the National Natural Science Fund for Distinguished Young Scholars 11725106, by China NSF grant 91430215, and by the National Magnetic Confinement Fusion Science Program 2015GB110003.

Received 22 March 2017

Accepted 17 March 2018

Published 31 October 2018