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

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 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.Assumption 1 (V j are non-trapping with respect to (1, 0)). There existy), satisfy 2V j (x, y) + (x − 1, y)•∇V j (x, y) ≥ c (3)for all (x, y) ∈ S, and j = 1, 2.

Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

“…These maps also arise in the formulation of fast direct methods (e.g. [33], [54]) and the recent work [3] analyses these maps in this setting (using complementary techniques to those in [43]). Previous work of some of the authors (e.g., [37,35]) also used PDE theory to analyse overlapping DD preconditioners; while this work was able to cover very general geometries, it was limited to the case when k > 0, corresponding to media with some absorptive properties.…”