We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The preconditioner uses local solves with impedance boundary conditions, and a global coarse solve based on the MS-GFEM approximation space constructed from local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson iterative method, and without using an oversampling technique, reduces to the preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv 2402.06905]. We prove that both the Richardson iterative method and the preconditioner used within GMRES converge at a rate of $\Lambda$ under some reasonable conditions, where $\Lambda$ denotes the error of the underlying MS-GFEM \rs{approximation}. Notably, the convergence proof of GMRES does not rely on the `Elman theory'. An exponential convergence property of MS-GFEM, resulting from oversampling, ensures that only a few iterations are needed for convergence with a small coarse space. Moreover, the convergence rate $\Lambda$ is not only independent of the fine-mesh size $h$ and the number of subdomains, but decays with increasing wavenumber $k$. In particular, in the constant-coefficient case, with $h\sim k^{-1-\gamma}$ for some $\gamma\in (0,1]$, it holds that $\Lambda \sim k^{-1+\frac{\gamma}{2}}$. We present extensive numerical experiments to illustrate the performance of the preconditioner, including 2D and 3D benchmark geophysics tests, and a high-contrast coefficient example arising in applications.

翻译：本文提出并分析了一种用于异构亥姆霍兹问题的双层限制加性施瓦茨预条件子，其基于[C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1548]中提出的多尺度谱广义有限元法。该预条件子采用具有阻抗边界条件的局部求解，以及一个基于由局部特征问题构建的MS-GFEM近似空间的全局粗网格求解。其推导首先将MS-GFEM表述为Richardson迭代法，并且在不使用过采样技术的情况下，简化为[Q. Hu and Z. Li, arXiv 2402.06905]中最近提出并分析的预条件子。我们证明，在一些合理的条件下，Richardson迭代法和在GMRES中使用的预条件子均以速率$\Lambda$收敛，其中$\Lambda$表示底层MS-GFEM近似的误差。值得注意的是，GMRES的收敛性证明不依赖于"Elman理论"。MS-GFEM由过采样产生的指数收敛特性，确保了仅需少量迭代即可在较小的粗网格空间下收敛。此外，收敛速率$\Lambda$不仅独立于细网格尺寸$h$和子域数量，而且随着波数$k$的增加而衰减。特别地，在常系数情况下，当$h\sim k^{-1-\gamma}$（其中$\gamma\in (0,1]$）时，有$\Lambda \sim k^{-1+\frac{\gamma}{2}}$。我们提供了大量的数值实验来说明该预条件子的性能，包括二维和三维基准地球物理测试，以及一个应用中产生的高对比度系数示例。