Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

TL;DR

This paper establishes quantitative bounds for impedance-to-impedance operators on convex polygonal domains, enabling robust and efficient numerical solutions for PDEs, including variable media and obstacle scattering problems.

Contribution

It provides the first rigorous bounds on impedance-to-impedance operators with applications to fast direct PDE solvers, including domain decomposition methods.

Findings

Quantitative norm bounds for impedance operators on convex domains.

Invertibility and frequency bounds for merge operators in PDE reconstruction.

Extension of bounds to obstacle scattering problems.

Abstract

We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.