This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.

翻译：本文提出了一个用于设计、实现和分析多尺度谱广义有限元法（MS-GFEM）的抽象框架，该方法最初由 [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406] 提出。MS-GFEM 是一种单位分解法，采用通过局部谱问题构建的最优局部逼近空间。我们建立了一个通用局部逼近理论，证明了在特定假设下，局部自由度呈指数收敛，且收敛速度显式依赖于关键问题参数。该框架适用于一类具有 $L^{\infty}$ 系数的多尺度偏微分方程，涵盖连续和离散有限元设置，包括高度不定问题（对流主导扩散、高频亥姆霍兹方程、阻抗边界条件下的麦克斯韦和弹性波方程）以及高阶问题。值得注意的是，针对所有这些问题的 MS-GFEM，我们证明了局部收敛率为 $O(e^{-cn^{1/d}})$，优于 Babuska 和 Lipton 证明的 $O(e^{-cn^{1/(d+1)}})$ 收敛率。此外，基于 MS-GFEM 的抽象局部逼近理论，我们建立了多尺度偏微分方程低秩近似的统一框架。该框架适用于上述问题，证明相关格林函数在分离域上允许 $O(|\log\epsilon|^{d})$ 项可分离逼近，误差为 $\epsilon>0$。我们的分析改进并推广了 [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] 中的结果，该文献对泊松型问题证明了 $O(|\log\epsilon|^{d+1})$ 项可分离逼近。