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：本文提出了多尺度谱广义有限元法（MS-GFEM）设计、实现与分析的一个抽象框架。MS-GFEM是一种基于单位分解的数值多尺度方法，最初由[Babuska和Lipton, 2011]提出，其采用通过局部谱问题构造的最优局部逼近空间。我们建立了一套通用的局部逼近理论，在特定假设下证明了该方法关于局部自由度的指数收敛性，并明确揭示了关键问题参数的依赖关系。该框架适用于具有$L^{\infty}$系数的广泛多尺度偏微分方程，涵盖连续与离散的有限元设定，包括高度不定问题（对流占优扩散方程，以及具有阻抗边界条件的高频亥姆霍兹方程、麦克斯韦方程和弹性波方程）和高阶问题。特别地，我们证明了MS-GFEM对所有这些问题均具有$O(e^{-cn^{1/d}})$的局部收敛速率，改进了Babuska和Lipton给出的$O(e^{-cn^{1/(d+1)}})$速率。此外，基于MS-GFEM的抽象局部逼近理论，我们建立了一个用于证明多尺度偏微分方程低秩逼近的统一框架。该框架适用于上述所有问题，证明了相关格林函数在良好分离区域上存在误差为$\epsilon>0$的$O(|\log\epsilon|^{d})$项可分离逼近。我们的分析改进并推广了[Bebendorf和Hackbusch, 2003]的结果，该研究针对泊松类问题证明了$O(|\log\epsilon|^{d+1})$项的可分离逼近。