In the author's previous paper (Zhang et al. 2022), exponential convergence was proved for the perfectly matched layers (PML) approximation of scattering problems with periodic surfaces in 2D. However, due to the overlapping of singularities, an exceptional case, i.e., when the wave number is a half integer, has to be excluded in the proof. However, numerical results for these cases still have fast convergence rate and this motivates us to go deeper into these cases. In this paper, we focus on these cases and prove that the fast convergence result for the discretized form. Numerical examples are also presented to support our theoretical results.

翻译：提交人前一份文件(Zhang等人,2022年)证明,与2D周期表面的散射问题相匹配的层(PML)近似值指数趋同。 然而,由于奇数重叠,一个例外的情况,即当波数半整数时,在证据中必须排除,但是,这些案件的数字结果仍然具有快速汇合率,这促使我们更深入地研究这些案件。在本文中,我们集中关注这些案件,证明离散形式的快速汇合结果。还提供了数字例子来支持我们的理论结果。