发表机构
Zhejiang University（浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出自适应局部正交分解方法求解Helmholtz问题，首次给出波数鲁棒残差后验误差估计，并利用POD压缩共享富集空间，实现高效在线计算。
AI 中文摘要
我们针对具有局部振荡和奇异性的Helmholtz问题，开发了一种自适应局部正交分解（ALOD）方法。主要结果是首次在标准分辨率、过采样和离散稳定性范围内，为Helmholtz LOD离散化建立了波数鲁棒的残差后验误差估计。该估计相对于细网格解是双侧的，并包含可计算的校正子局部化误差界，从而能够分别自适应在线粗空间、细校正子网格和过采样水平。对于给定的载荷族，我们重用指示器已计算的局部残差Riesz代表元作为区域加性Schwarz修正。它们的张成空间形成共享富集，然后通过能量加权本征正交分解（POD）进行压缩以供在线使用。因此，约化空间直接由自适应误差控制机制生成。误差估计对于满足原始LOD测试空间上Galerkin正交性的富集解仍然有效。核提升测试在统一过采样条件下提供稳定、拟最优的耦合，无需额外的全局细网格求解，并且逐成员检查控制压缩引起的扰动。二维实验展示了针对给定载荷族的在线压缩和低秩重用。
英文摘要
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.
Comments24 pages, 5 figures
