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\begin{document}

\title{A nonuniform mesh method in the Floquet parameter domain for wave
scattered by periodic surfaces}



\author[1]{Ruming  Zhang}%
\author[1]{Tilo Arens}%
\affil[1]{Karlsruhe Institute of Technology}%


\vspace{-1em}



  
  \date{May 10, 2024}


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\begin{abstract}
In this paper, we propose a new nonuniform mesh method to simulate
acoustic scattering problems in two dimensional periodic structures with
non-periodic incident fields numerically. As existing methods are
difficult to extend to higher dimensions, we have designed the new
method with such extensions in mind. With the help of the Floquet-Bloch
transform, the solution to the original scattering problem is written as
an integral of a family of quasi-periodic problems. These are defined in
bounded domains for each value of the Floquet parameter which varies in
a bounded interval. The key step in our method is the numerical
approximation of the integral by a quadrature rule adapted to the
regularity of the family of quasi-periodic solutions. We design a
nonuniform mesh with a Gaussian quadrature rule applied on each
subinterval. We prove that the numerical method converges exponentially
with respect to both the number of subintervals and the number of
Gaussian quadrature points. Some numerical experiments are provided to
illustrate the results.%
\end{abstract}%



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