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 $\Lambda \sim k^{-1+\frac{\gamma}{2}}$.

翻译：我们提出并分析了一种用于异质亥姆霍兹问题的两层限制性加性Schwarz预条件子，该方法基于[C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]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造的MS-GFEM近似空间的全局粗网格求解。其推导过程是：首先将MS-GFEM表述为Richardson迭代方法，在不使用过采样技术的情况下，可简化为[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迭代方法以及在GMRES中使用的该预条件子均以速率$\Lambda$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"。由过采样导致的MS-GFEM指数收敛性质，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$\Lambda$不仅与细网格尺寸$h$和子域数量无关，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$\Lambda \sim k^{-1+\frac{\gamma}{2}}$成立。