Optimal Spectral Approximation in the Overlaps for Generalized Finite Element Methods

TL;DR

This paper introduces a localized spectral approximation method for elliptic PDEs with rough coefficients, achieving nearly exponential decay in error and improving computational efficiency over traditional approaches.

Contribution

It develops a new local eigenvalue problem approach that reduces computational complexity and proves nearly exponential error decay, enhancing spectral approximation in generalized finite element methods.

Findings

Nearly exponential decay of local approximation errors

Reduced problem size and faster spectral computations

Effective as a preconditioner with proven optimal convergence

Abstract

In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on rings around the boundary of local subdomains. Compared to the corresponding method that solves eigenvalue problems on the whole subdomains, the problem size and the bandwidth of the resulting system matrices are substantially reduced, resulting in faster spectral computations. We prove a nearly exponential a priori decay result for the local approximation errors of the proposed method, which implies the nearly exponential decay of the overall approximation error of the method. The proposed method can also be used as a preconditioner, and only a slight adaptation of our theory is necessary to prove the optimal convergence of the preconditioned iteration. Numerical experiments are presented to support the effectiveness of the proposed method and to investigate its coefficient robustness.