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Super-localized orthogonal decomposition for high-frequency Helmholtz problems

Freese, Philip; Hauck, Moritz; Peterseim, Daniel

Abstract:

We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber $\kappa$. On a coarse mesh of width $H$, the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width $\ell H$ and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multi-scale methods, the localization error decays super-exponentially as the oversampling parameter $\ell$ is increased. This implies that optimal convergence is observed under the substantially relaxed over-sampling condition $\ell\gtrsim(\log \frac{\kappa}{H})^{(d-1)/d}$ with $d$ denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.


Volltext §
DOI: 10.5445/IR/1000142318
Veröffentlicht am 25.01.2022
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 01.2022
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000142318
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 21 S.
Serie CRC 1173 Preprint ; 2022/7
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter Helmholtz equation, high-frequency, heterogeneous media, numerical homogenization, multi-scale method, super-localization
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft
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