Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition

We consider the e cient numerical solution of the Helmholtz equation in a rectangular domain with a perfectly matched layer (PML) or an absorbing boundary condition (ABC). Standard bilinear (trilinear) nite-element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction-type fast direct solver. We present numerical studies to estimate the re ection of waves caused by an absorbing boundary and a PML, and we optimize certain parameters of the layer to minimize the re ection.