On the asymptotic optimality of spectral coarse spaces

TL;DR

This paper investigates the effectiveness of spectral coarse spaces in two-level iterative methods, revealing they are not always asymptotically optimal and that alternative coarse spaces can offer better convergence and conditioning.

Contribution

The paper demonstrates that spectral coarse spaces are not necessarily asymptotically optimal and identifies alternative coarse spaces with improved efficiency and conditioning.

Findings

Spectral coarse spaces do not always minimize the asymptotic contraction factor.

Alternative coarse spaces can outperform spectral spaces in efficiency.

Numerical experiments confirm the existence of more effective coarse spaces.

Abstract

This paper is concerned with the asymptotic optimality of spectral coarse spaces for two-level iterative methods. Spectral coarse spaces, namely coarse spaces obtained as the span of the slowest modes of the used one-level smoother, are known to be very efficient and, in some cases, optimal. However, the results of this paper show that spectral coarse spaces do not necessarily minimize the asymptotic contraction factor of a two-level iterative method. Moreover, numerical experiments show that there exist coarse spaces that are asymptotically more efficient and lead to preconditioned systems with improved conditioning properties.