On the asymptotic optimality of spectral coarse spaces

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.