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

Thomas Beck, Yaiza Canzani and Jeremy Louis Marzuola

Vol. 4 (2022), No. 2, 225–256
DOI: 10.2140/paa.2022.4.225

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 subdomains. 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.

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impedance boundary conditions, Poincaré–Steklov method, boundary trace estimates
Mathematical Subject Classification
Primary: 35J05, 65N55
Received: 26 March 2021
Revised: 3 February 2022
Accepted: 22 March 2022
Published: 16 October 2022
Thomas Beck
Department of Mathematics
Fordham University
Bronx, NY
United States
Yaiza Canzani
Department of Mathematics
University of North Carolina
Chapel Hill, NC
United States
Jeremy Louis Marzuola
Department of Mathematics
University of North Carolina
Chapel Hill, NC
United States