An adaptive localized orthogonal decomposition method for Helmholtz problems
Abstract
We develop an adaptive localized orthogonal decomposition (ALOD) method for Helmholtz problems with parametrized loads. Under standard resolution, oversampling, and discrete-stability conditions, we establish, to our knowledge, the first wavenumber-robust residual a posteriori error estimate for a Helmholtz LOD discretization. The estimate is two-sided relative to the fine-grid solution and includes a computable bound for the corrector-localization error. In addition, we design a novel family-aware adaptive algorithm using additive Schwarz corrections to construct a shared regional enrichment space for Helmholtz problems with multiple loads. Energy-weighted proper orthogonal decomposition compresses this space. We establish stability and quasi-optimality of the enriched formulation, with bounds independent of the enrichment space. Numerical experiments demonstrate the effectiveness of ALOD for multiple loads and large wavenumbers.
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BibTeX
@misc{li2026adaptive,
title = {{An adaptive localized orthogonal decomposition method for Helmholtz problems}},
author = {Yuwen Li and Chuxin Que},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.24776},
eprint = {2609.24776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24776},
}
RIS
TY - UNPB
TI - An adaptive localized orthogonal decomposition method for Helmholtz problems
AU - Yuwen Li
AU - Chuxin Que
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.24776
UR - https://arxiv.org/abs/2609.24776
AB - We develop an adaptive localized orthogonal decomposition (ALOD) method for Helmholtz problems with parametrized loads. Under standard resolution, oversampling, and discrete-stability conditions, we establish, to our knowledge, the first wavenumber-robust residual a posteriori error estimate for a Helmholtz LOD discretization. The estimate is two-sided relative to the fine-grid solution and includes a computable bound for the corrector-localization error. In addition, we design a novel family-aware adaptive algorithm using additive Schwarz corrections to construct a shared regional enrichment space for Helmholtz problems with multiple loads. Energy-weighted proper orthogonal decomposition compresses this space. We establish stability and quasi-optimality of the enriched formulation, with bounds independent of the enrichment space. Numerical experiments demonstrate the effectiveness of ALOD for multiple loads and large wavenumbers.
ER -