Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

TL;DR

This paper proves a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains and develops a PML method that approximates wave solutions with exponentially decreasing error as layer length increases.

Contribution

It introduces a new limiting absorption principle for quasi-cylindrical domains and constructs a PML method with proven exponential convergence.

Findings

Established a limiting absorption principle for these domains.

Constructed a PML method with finite length layers.

Proved exponential error decay in approximations.

Abstract

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.