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\begin{document}

\title{The time-harmonic electromagnetic wave scattering by a bi-periodic
elastic body}



\author[1]{Tielei Zhu}%
\author[2]{Changkun Wei}%
\author[3]{Jiaqing Yang}%
\affil[1]{School of Mathematics and Statistics Xi’an Jiaotong University Xi’an Shaanxi 710049 China}%
\affil[2]{School of Mathematics and Statistics Beijing Jiaotong University Beijing 100044}%
\affil[3]{School of Mathematics and Statistics Xi’an Jiaotong University Xi’an Shaanxi 710049}%


\vspace{-1em}



  
  \date{June 10, 2024}


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\begin{abstract}
This paper concentrates on an interaction scattering problem between the
time-harmonic electromagnetic waves and an unbounded periodic elastic
medium. The uniqueness results of the interaction problem are
established for small frequencies or all frequencies except a discrete
set in both the absorbing and non-absorbing medium, and then the
existence of solutions is derived by the classical Fredholm alternative.
The perfectly matched layer (PML) method is proposed to truncate the
unbounded scattering domain to a bounded computational domain. We prove
the well-posedness of the solution for the truncated PML problem, where
a homogeneous boundary condition is imposed on the outer boundary of the
PML. The exponential convergence of the PML method is established in
terms of the thickness and parameters of the PML. The proof is based on
the PML extension and the exponential decay properties of the modified
fundamental solution.%
\end{abstract}%



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