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

Fast convergent PML method for scattering with periodic surfaces: the exceptional case