  The PML method is well-known for its exponential convergence rate and easy
implementation for scattering problems with unbounded domains. For
rough-surface scattering problems, authors in [5] proved that the PML method
converges at most algebraically in the physical domain. However, the authors
also asked a question whether exponential convergence still holds for compact
subsets. In [25], one of our authors proved the exponential convergence for
periodic surfaces via the Floquet-Bloch transform when the wavenumber is
positive and not a half integer; when the wavenumber is a positive half
integer, a nearly fourth-order convergence rate was shown in [26]. The
extension of this method to locally perturbed cases is not straightforward,
since the domain is no longer periodic thus the Floquet-Bloch transform doesn't
work, especially when the domain topology is changed. Moreover, the exact decay
rate when the wavenumber is a half integer remains unclear. The purpose of this
paper is to address these two significant issues. For the first topic, the main
idea is to reduce the problem by the DtN map on an artificial curve, then the
convergence rate of the PML is obtained from the investigation of the DtN map.
It shows exactly the same convergence rate as in the unperturbed case. Second,
to illustrate the convergence rate when the wavenumber is a half integer, we
design a specific periodic structure for which the PML converges at the
fourth-order, showing that the algebraic convergence rate is sharp. We adopt a
previously developed high-accuracy PML-BIE solver to exhibit this unexpected
phenomenon.
