Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem
| Autor: | Eike Hermann Müller, Euan A. Spence, Jeffrey Galkowski |
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| Rok vydání: | 2019 |
| Předmět: |
Dirichlet problem
Helmholtz equation Applied Mathematics Boundary (topology) Numerical Analysis (math.NA) 010103 numerical & computational mathematics 01 natural sciences Integral equation Generalized minimal residual method 010101 applied mathematics Computational Mathematics symbols.namesake Mathematics - Analysis of PDEs Helmholtz free energy FOS: Mathematics symbols Applied mathematics Mathematics - Numerical Analysis 35J05 35J25 65N22 65N38 65R20 0101 mathematics Galerkin method Boundary element method Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Numerische Mathematik. 142:329-357 |
| ISSN: | 0945-3245 0029-599X |
| DOI: | 10.1007/s00211-019-01032-y |
| Popis: | We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber $k$ to have the error in the iterative solution bounded independently of $k$ as $k\rightarrow \infty$ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions as $k \rightarrow \infty$ when the obstacle is nontrapping. Version 3 of this submission has been split into Version 4 and arXiv:1807.09719 |
| Databáze: | OpenAIRE |
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