Near-optimal perfectly matched layers for indefinite Helmholtz problems (Q2808270)

In this article, it is illustrated how to construct an absorbing boundary condition for the indefinite Helmholtz problem for an unbounded domain. The authors use a near-best uniform rational interpolant of the inverse square root function on the union of a negative and a positive real interval which has been designed with the help of a classical result by Zolotarev. The authors are able to convert the interpolant into a three-term finite difference discretization of a perfectly matched layer using Krein's interpretation of a Stieltjes continued fraction. Additionally, it is shown that the interpolant converges exponentially in the number of grid points. Further, is is shown that the convergence rate is asymptotically optimal for propagative and evanescent wave modes. Moreover, some numerical results illustrate the theoretical findings such as the accuracy. Finally, a detailed summary, generalizations, and open problems are given.