An adaptive finite element PML method for the elastic wave scattering problem in periodic structures

¹ School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, P.R. China.
jxue@lsec.cc.ac.cn
² Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
lipeijun@math.purdue.edu
³ School of Mathematics, Jilin University, Changchun 130012, P.R. China.
lvjl@jlu.edu.cn
⁴ LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Science, University of Chinese Academy of Sciences, 100190, P.R. China.
zwy@lsec.cc.ac.cn

Received: 23 December 2016
Revised: 22 March 2017
Accepted: 5 April 2017

Abstract

An adaptive finite element method is presented for the elastic scattering of a time-harmonic plane wave by a periodic surface. First, the unbounded physical domain is truncated into a bounded computational domain by introducing the perfectly matched layer (PML) technique. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing an equivalent transparent boundary condition. Second, an a posteriori error estimate is deduced for the discrete problem and is used to determine the finite elements for refinements and to determine the PML parameters. Numerical experiments are included to demonstrate the competitive behavior of the proposed adaptive method.