2018/07/25 by Jeffrey Galkowski, Euan A. Spence, Galkowski, Jeffrey +1
Computer Science · Engineering · Mathematics · #31B10 #31B25 #35J05 #35J25 #65R20 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1807.09719
openalex publication_date 2018/07/25 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz\nsingle- and double-layer boundary-integral operators as mappings from\nL2(\∂ \Ω)\→ H1(\∂ \Ω) (where \∂\Ω\nis the boundary of the obstacle). The new bounds are obtained using estimates\non the restriction to the boundary of quasimodes of the Laplacian, building on\nrecent work by the first author and collaborators.\n Our main motivation for considering these operators is that they appear in\nthe standard second-kind boundary-integral formulations, posed in L2(\∂\n\Ω), of the exterior Dirichlet problem for the Helmholtz equation. Our new\nwavenumber-explicit L2(\∂ \Ω)\→ H1(\∂ \Ω)\nbounds can then be used in a wavenumber-explicit version of the classic\ncompact-perturbation analysis of Galerkin discretisations of these second-kind\nequations; this is done in the companion paper [Galkowski, M "uller, Spence,\narXiv 1608.01035].\n