This paper is concerned with the convergence theory of a perfectly matched layer(PML) method for wave scattering problems in a half plane bounded by a step-like surface. Whena plane wave impinges upon the surface, the scattered waves are composed of an outgoing radiativefield and two known parts. The first part consists of two parallel reflected plane waves of differentphases, which propagate in two different subregions separated by a half-line parallel to the wavedirection. The second part stands for an outgoing corner-scattering field which is discontinuousand represented by a double-layer potential. A piecewise circular PML is defined by introducing twotypes of complex coordinates transformations in the two subregions, respectively. A PML variationalproblem is proposed to approximate the scattered waves. The exponential convergence of the PMLsolution is established by two results based on the technique of Cagniard--de Hoop transform. First,we show that the discontinuous corner-scattering field decays exponentially in the PML. Second, weshow that the transparent boundary condition (TBC) defined by the PML is an exponentially smallperturbation of the original TBC defined by the radiation condition. Numerical examples validatethe theory and demonstrate the effectiveness of the proposed PML.

Publication:

SIAM Journal on Numerical AnalysisVolume 63, Issue 2

https://doi.org/10.1137/24M1654221

Author:

WANGTAO LU

School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, 310027 China; Institute of Fundamental and Transdisciplinary Research, Zhejiang University, Hangzhou, 310027 China

wangtaolu@zju.edu.cn\

WEIYING ZHENG

Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100190 China; School of Mathematical Science, University of Chinese Academy of Sciences, Beijing

zwy@lsec.cc.ac.cn

XIAOPENG ZHU

Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing,100190 China; School of Mathematical Science, University of Chinese Academy of Sciences, Beijing; School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, 310027 China

0623559@zju.edu.cn