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：在作者之前的工作（Zhang等，2022）中，针对二维周期表面散射问题的完美匹配层（PML）近似，已证明其具有指数收敛性。然而，由于奇点的重叠，在证明中必须排除一种例外情形，即波数为半整数的情况。尽管如此，此类情形的数值结果仍呈现快速收敛速率，这促使我们对此进行深入研究。本文聚焦于这些例外情形，证明了离散格式的快速收敛性，并通过数值算例验证了理论结果。