On Wellposedness and Convergence of UPML method for analyzing wave scattering in layered media

TL;DR

This paper develops a rigorous theoretical framework for the UPML method in 2D layered acoustic scattering, proving wellposedness, resonance-free conditions, and exponential convergence of the UPML solution to the true solution.

Contribution

It establishes the first comprehensive wellposedness and convergence theory for UPML in layered media without constraints on wavenumbers or absorption strength.

Findings

UPML problem is unconditionally resonance free.

Green function converges exponentially fast as absorption increases.

The UPML solution converges exponentially to the true solution.

Abstract

This paper proposes a novel method to establish the wellposedness and convergence theory of the uniaxial-perfectly-matched-layer (UPML) method in solving a two-dimensional acoustic scattering problem due to a compactly supported source, where the medium consists of two layers separated by the horizontal axis. When perfectly matched layer (PML) is used to truncate the vertical variable only, the medium structure becomes a closed waveguide. The Green function due to a primary source point in this waveguide can be constructed explicitly based on variable separations and Fourier transformations. In the horizontal direction, by properly placing periodical PMLs and locating periodic source points imaged by the primary source point, the exciting waveguide Green functions by those imaging points can be assembled to construct the Green function due to the primary source point for the two-layer medium truncated by a UPML. Incorporated with Green's identities, this UPML Green function directly leads to the wellposedness of the acoustic scattering problem in a UPML truncation with no constraints about wavenumbers or UPML absorbing strength. Consequently, we firstly prove that such a UPML truncating problem is unconditionally resonance free. Moreover, we show, under quite general conditions, that this UPML Green function converges to the exact layered Green function exponentially fast as absorbing strength of the UPML increases, which in turn gives rise to the exponential convergence of the solution of the UPML problem towards the original solution.