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Super-localized orthogonal decomposition for high-frequency Helmholtz problems
Publikationstyp
Journal Article
Date Issued
2024-07-18
Sprache
English
Volume
46
Issue
4
Start Page
A2377
End Page
A2397
Citation
SIAM Journal on Scientific Computing 46 (4): A2377-A2397 (2024)
Publisher DOI
Scopus ID
Publisher
Society for Industrial and Applied Mathematics
We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber κ . 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 ℓH and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multiscale methods, two- and three-dimensional numerical computations show that the localization error decays super-exponentially as the oversampling parameter ℓ is increased. This suggests that optimal convergence is observed under the substantially relaxed oversampling condition ℓ ≳ (log κ 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.
Subjects
Helmholtz equation
heterogeneous media
high-frequency
multiscale method
numerical homogenization
super-localization
DDC Class
510: Mathematics