Super-localized Orthogonal Decomposition for high-frequency Helmholtz problems

TL;DR

The paper tackles high-frequency Helmholtz scattering by introducing the super-localized LOD (SLOD), a variant of localized orthogonal decomposition that builds problem-adapted bases from local snapshots. By identifying local source terms on coarse-element patches whose responses decay rapidly under the Helmholtz solution operator, SLOD achieves super-exponential localization error decay in the oversampling parameter $\ell$, enabling a relaxed oversampling regime $\ell \gtrsim (\log \tfrac{\kappa}{H})^{(d-1)/d}$ and near-optimal convergence. The authors provide a κ-explicit stability and error analysis under a mild stability assumption, together with an a-posteriori localization-error control strategy based on patch-wise singular values. Numerical experiments demonstrate significantly improved offline/online performance, robustness to heterogeneous media, and effective coupling with perfectly matched layers (PML). The approach offers substantial gains over the classical LOD, with potential extensions to other wave models in elasticity and electromagnetism.

Abstract

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 $\ell H$ 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 $\ell \gtrsim (\log \tfracκ{H})^{(d-1)/d}$ 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.

Figures (7)

• Figure 4.1: Illustration of a $\ell$-th order patch for $\ell = 1,\dots,4$ with gray scale indicating the order.
• Figure 4.2: Global ideal LOD basis (left) and local SLOD basis for $\ell = 1$ (right) with their corresponding $L^2$-normalized right-hand sides $g$ in one space dimension for an interior element (top) and an element at the boundary (bottom). The real (resp. imaginary) parts are depicted using solid (resp. dashed) lines.
• Figure 4.3: Singular values $\sigma_m$ of the operator $\Pi_{H,\omega}|_{Y}$ in descending order for several oversampling parameters for $\kappa = 2^5$ (left) and $\kappa = 2^6$ (right).
• Figure 6.1: Localization errors for the SLOD and LOD for several values of $H$ for $\kappa = 2^5$ (left) and $\kappa = 2^6$ (right).
• Figure 6.2: Convergence plots for the SLOD and LOD for several oversampling parameters $\ell$ for $\kappa = 2^5$ (left) and $\kappa = 2^6$ (right).

Theorems & Definitions (16)

• Remark 2.1: Heterogeneous media, scatterers, boundary conditions
• Lemma 3.2: Stability and $\kappa$-independent approximation
• proof
• Remark 4.1: Friedrich's constant
• Lemma 4.3: Coercivity of $a_\omega$
• proof
• Conjecture 4.4: Super-exponential decay
• Remark 4.5: The case $d=1$
• Lemma 4.6: Pessimistic exponential decay
• proof