Near-optimal perfectly matched layers for indefinite Helmholtz problems

TL;DR

This paper introduces a near-optimal perfectly matched layer (PML) construction for indefinite Helmholtz problems, achieving exponential convergence and optimal performance for various wave modes using rational interpolation and Krein's interpretation.

Contribution

The paper presents a novel PML construction based on rational interpolation and Krein's continued fractions, providing near-optimal convergence for indefinite Helmholtz problems.

Findings

Exponential convergence of the PML with respect to grid points

Optimal convergence rates for propagative and evanescent waves

Numerical experiments demonstrating effectiveness

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.