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

TL;DR

The paper develops an abstract Hilbert-space framework for multiscale spectral generalized FEM (MS-GFEM) that builds optimal local approximation spaces via local spectral problems and oversampling, with an abstract convergence theory showing exponential decay of local errors and quasi-optimal global convergence. A unified low-rank framework then links these local approximation properties to separability of Green's functions and solution operators, yielding $O(||^{d})$-type separable approximations on well-separated domains. The contributions include two explicit, problem-parameter-dependent assumptions (Caccioppoli-type inequality and weak approximation), proofs of sharper local convergence rates $O(e^{-c n^{1/d}})$, extensions to higher-order problems, and applicability to a broad class of multiscale PDEs (including convection-diffusion, H(curl) elliptic, Helmholtz, Maxwell, elastic waves, and fourth-order problems) in both continuous and discrete FE settings. The framework furnishes a rigorous basis for efficient, parallelizable multiscale solvers and provides a route to sharp complexity estimates via low-rank representations of Green's functions, with practical implications for model-order reduction and scalable PDE computation.

Abstract

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 $O(|\logε|^{d})$-term separable approximation on well-separated domains with error $ε>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\logε|^{d+1})$-term separable approximation was proved for Poisson-type problems.

Figures (2)

• Figure 1: Illustration of a subdomain $\omega_i$ that lies within the interior of $\Omega$ (left) and one that intersects the outer boundary $\partial \Omega$ (right) with associated oversampling domains $\omega_i^{\ast}$.
• Figure 1: Illustration of the decomposition of $D_{c} = D^{\ast}\setminus D$ (the colored region). Here $D_{c} = D_{c,1}\cup D_{c,2}$, where $D_{c,1}$ and $D_{c,2}$ are the unions of the red and green colored regions, respectively.

Theorems & Definitions (115)

• Remark 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Theorem 2.8
• Proof 1
• Definition 2.10: Local particular functions
• Remark 2.11
• Proposition 2.12
• Remark 2.14