Exponentially Convergent Multiscale Methods for 2D High Frequency Heterogeneous Helmholtz Equations

Chen, Yifan; Hou, Thomas Y.; Wang, Yixuan

doi:10.1137/22m1507802

Cited by 3 publications

(1 citation statement)

References 43 publications

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“…plain waves, generalized harmonic polynomials). Moreover, also other multi-scale methods like the multiscale finite element method (MsFEM) [HW97,EH09] have been successfully applied to the Helmholtz problem [FG17,FLCG19,CHW21].…”

Section: Introductionmentioning

confidence: 99%

Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems

Freese,

Hauck,

Peterseim

2024

SIAM J. Sci. Comput.

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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 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 is increased. This implies that optimal convergence is observed under the substantially relaxed oversampling condition (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.

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Section: Introductionmentioning

confidence: 99%

Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems

Freese,

Hauck,

Peterseim

2024

SIAM J. Sci. Comput.

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Generalized multiscale finite element method for language competition modeling I: Offline approach

Ammosov,

Malysheva,

Zamorshchikova

2024

Journal of Computational and Applied Mathematics

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An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation

Giammatteo,

Heinlein,

Schlottbom

2024

IMA Journal of Numerical Analysis

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In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.

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Exponentially Convergent Multiscale Methods for 2D High Frequency Heterogeneous Helmholtz Equations

Cited by 3 publications

References 43 publications

Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems

Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems

Generalized multiscale finite element method for language competition modeling I: Offline approach

An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation

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