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

2023/11/15 by Chupeng Ma, Ma, Chupeng · 4 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2311.08761

openalex publication_date 2023/11/15 · openalex created_date 2023/11/17 · openalex updated_date 2026/07/28

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^∞-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.

Cited by

• Multiscale Spectral Generalized Finite Element Methods for Discontinuous Galerkin Schemes
• Robust spectral preconditioning for high-Péclet number convection-diffusion
• Optimal Spectral Approximation in the Overlaps for Generalized Finite Element Methods
• Multiscale model reduction and two-level Schwarz preconditioner for H(curl) elliptic problems

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