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This work presents an abstract framework for the design, implementation, andanalysis 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 approximationspaces constructed from local spectral problems. We establish a general localapproximation theory demonstrating exponential convergence with respect tolocal degrees of freedom under certain assumptions, with explicit dependence onkey problem parameters. Our framework applies to a broad class of multiscalePDEs with $L^{\infty}$-coefficients in both continuous and discrete, finiteelement settings, including highly indefinite problems (convection-dominateddiffusion, as well as the high-frequency Helmholtz, Maxwell and elastic waveequations with impedance boundary conditions), and higher-order problems.Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEMfor all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown byBabuska and Lipton.  Moreover, based on the abstract local approximation theory for MS-GFEM, weestablish a unified framework for showing low-rank approximations to multiscalePDEs. This framework applies to the aforementioned problems, proving that theassociated Green's functions admit an $O(|\log\epsilon|^{d})$-term separableapproximation on well-separated domains with error $\epsilon>0$. Our analysisimproves 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 forPoisson-type problems.

A unified framework for multiscale spectral generalized FEMs and low-rank approximations to multiscale PDEs