A Unified Framework for Multiscale Spectral Generalized FEMs and Low-Rank Approximations to Multiscale PDEs

Multiscale partial differential equations (PDEs), featuring heterogeneous coefficients oscillating across possibly non-separated scales, pose computational challenges for standard numerical techniques. Over the past two decades, a range of specialized methods has emerged that enables the efficient solution of such problems. Two prominent approaches are numerical multiscale methods with problem-adapted coarse approximation spaces, and structured inverse methods that exploit a low-rank property of the associated Green’s functions to obtain approximate matrix factorizations. 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 Babuska and Lipton (Multiscale Model Simul 9:373–406, 2011). 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 the number of 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 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 Bebendorf and Hackbusch (Numerische Mathematik 95:1–28, 2003) where an \(O(|\log \epsilon |^{d+1})\)-term separable approximation was proved for Poisson-type problems. It provides a rigorous theoretical foundation for diverse structured inverse methods, and also clarifies the intimate connection between approximation mechanisms in such methods and MS-GFEM.