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We prove quantitative norm bounds for a family of operators involvingimpedance boundary conditions on convex, polygonal domains. A robust numericalconstruction of Helmholtz scattering solutions in variable media via theDirichlet-to-Neumann operator involves a decomposition of the domain into asequence of rectangles of varying scales and constructingimpedance-to-impedance boundary operators on each subdomain. Our estimates inparticular ensure the invertibility, with quantitative bounds in the frequency,of the merge operators required to reconstruct the originalDirichlet-to-Neumann operator in terms of these impedance-to-impedanceoperators of the sub-domains. A key step in our proof is to obtain Neumann andDirichlet boundary trace estimates on solutions of the impedance problem, whichare of independent interest. In addition to the variable media setting, we alsoconstruct bounds for similar merge operators in the obstacle scatteringproblem.

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