Perfectly Matched Layer Method for Acoustic Scattering Problem by a Locally Perturbed Line with Impedance Boundary Condition

In this paper, we study the two-dimensional Helmholtz scattering problem by a locally perturbed line with impedance boundary condition. Different from the problem with Dirichlet boundary condition, the Green function of the Helmholtz equation with impedance boundary condition becomes very complicated and comprises surface waves along the locally perturbed line. A uniaxial perfectly matched layer (UPML) method is proposed to truncate the half plane into a bounded computational domain. The main contribution of this paper is to prove the well-posedness of the PML problem and the exponential convergence of the approximate solution to the exact solution as either the thickness or the medium parameter of PML increases.