NEAR-OPTIMAL PERFECTLY MATCHED LAYERS FOR INDEFINITE HELMHOLTZ PROBLEMS

Abstract. A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein’s interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included. Key words. Helmholtz equation, Neumann-to-Dirichlet map, perfectly matched layer, rational approximation, Zolotarev problem, continued fraction AMS subject classifications. 35J05, 65N06, 65N55, 30E10, 65D25 1. Introduction. An important task in science and engineering is the numerical solution of a partial differential equation (PDE) on an unbounded spatial domain. Unbounded spatial domains need to be truncated for computational purposes and this turns out to be particularly difficult when the PDE models wave-like phenomena