2012/04/17 by Stéphane K. Lintner, Stephane K. Lintner, Lintner, Stephane K. +2 · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Electromagnetic Compatibility and Measurements #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1204.3699
29 pages
openalex publication_date 2012/04/17 · arxiv created 2013/06/06 · arxiv updated 2013/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We deal with the general problem of scattering by open-arcs in two-dimensional space. We show that this problem can be solved by means of certain second-kind integral equations of the form N S[φ] = f, where N and S are first-kind integral operators whose composition gives rise to a generalized Calderón formula of the form N S = J0τ+ K in a \em weighted, periodized Sobolev space. The N S formulation provides, for the first time, a second-kind integral equation for the open-arc scattering problem with Neumann boundary conditions. Numerical experiments show that, for both the Dirichlet and Neumann boundary conditions, our second-kind integral equations have spectra that are bounded away from zero and infinity as k→ ∞; to the authors' knowledge these are the first integral equations for these problems that possess this desirable property. Our proofs rely on three main elements: 1) Algebraic manipulations enabled by the presence of integral weights; 2) Use of the classical result of continuity of the Cesàro operator; and 3) Explicit characterization of the point spectrum of Jτ0, which, interestingly, can be decomposed into the union of a countable set and an open set, both tightly clustered around -1/4. As shown in a separate contribution, the new approach can be used to construct simple spectrally-accurate numerical solvers and, when used in conjunction with Krylov-subspace solvers such as GMRES, gives rise to dramatic reductions of Krylov-subspace iteration numbers vs. those required by other approaches.