Super-localized Orthogonal Decomposition for high-frequency Helmholtz problems

TL;DR

This paper introduces a super-localized variant of the Localized Orthogonal Decomposition method for high-frequency Helmholtz problems, achieving exponential decay in localization error and improved convergence with relaxed oversampling conditions.

Contribution

The paper presents a novel super-localized LOD method that enhances accuracy and efficiency for high-frequency Helmholtz problems by utilizing problem-adapted basis functions and super-exponential localization error decay.

Findings

Localization error decays super-exponentially with oversampling parameter

Optimal convergence achieved under relaxed oversampling conditions

Significant performance improvements in heterogeneous media and PMLs

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 oversampling condition $\ell \gtrsim (\log \tfrac{\kappa}{H})^{(d-1)/d}$ 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.