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arXiv · 2609.24776

An adaptive localized orthogonal decomposition method for Helmholtz problems

Yuwen Li·Chuxin Que
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Abstract

We develop an adaptive localized orthogonal decomposition (ALOD) method for Helmholtz problems with localized oscillations and singularities. The main result is the first wavenumber-robust residual a posteriori error estimate for a Helmholtz LOD discretization within the standard resolution, oversampling, and discrete-stability regime. The estimate is two-sided relative to the fine-grid solution and includes a computable bound for the corrector-localization error, enabling separate adaptation of the online coarse space, the fine corrector mesh, and the oversampling level. For a prescribed load family, we reuse the local residual Riesz representatives already computed by the indicator as regional additive Schwarz corrections. Their span forms a shared enrichment, which is then compressed by energy-weighted proper orthogonal decomposition (POD) for online use. Thus, the reduced space is generated directly by the adaptive error-control mechanism. The error estimate remains valid for enriched solutions satisfying Galerkin orthogonality on the original LOD test space. Kernel-lifted tests provide stable, quasi-optimal coupling under a unified oversampling condition without additional global fine-grid solves, and memberwise checks control the perturbation caused by compression. Two-dimensional experiments demonstrate online compression and low-rank reuse for the prescribed load families.

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Yuwen Li, Chuxin Que. 2026-09-21. An adaptive localized orthogonal decomposition method for Helmholtz problems. https://arxiv.org/abs/2609.24776

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