A high-order numerical method for solving non-periodic scattering problems in three-dimensional bi-periodic structures

In this paper, we focus on scattering of non-periodic incident fields in three-dimensional bi-periodic structures, as they can not be solved by the classical methods used for the quasi-periodic scattering problems. To solve such non-periodic scattering problems, the Floquet-Bloch transform, which decomposes the unbounded problem into a family of periodic problems in a bounded unit cell, has been applied together with a numerical method in [25]. However, its theoretical result indicates that the computational order is too low. Hence, our aim is to propose a high-order numerical approach by using the Floquet-Bloch transform. To this end, the first crucial part is to analyze the regularity of the Floquet-Bloch of the total field with respect to the Floquet parameter.
The second challenging part is to propose a high-order tailor-made quadrature method adapted to singularities of the Floquet-Bloch of the total field formed by a finite number of circular arcs. Afterward, we obtain the error estimation of the proposed numerical approach. Eventually, the accuracy and efficiency of the mentioned approach are revealed by several numerical examples.