Analysis of Perfectly Matched Layer operators for acoustic scattering on manifolds with quasicylindrical ends

TL;DR

This paper demonstrates the stability and exponential convergence of PML methods for acoustic scattering on manifolds with quasicylindrical ends, modeling long-range geometric perturbations of waveguides.

Contribution

It introduces a new PML construction for manifolds with quasicylindrical ends and proves exponential convergence of solutions as PML length increases.

Findings

PMLs are stable and exponentially convergent for the considered manifolds.

Finite PML problems are uniquely solvable.

Solutions approximate scattered waves with exponentially decreasing error.

Abstract

We prove stability and exponential convergence of the Perfectly Matched Layer (PML) method for acoustic scattering on manifolds with axial analytic quasicylindrical ends. These manifolds model long-range geometric perturbations (e.g. bending or stretching) of tubular waveguides filled with homogeneous or inhomogeneous media. We construct non-reflective infinite PMLs replacing the metric on a part of the manifold by a non-degenerate complex symmetric tensor field. We prove that the problem with PMLs of finite length is uniquely solvable and solutions to this problem locally approximate scattered solutions with an error that exponentially tends to zero as the length of PMLs tends to infinity.