Lower bounds for the condition numbers of the preconditioned systems are obtained for two-level additive Schwarz preconditioners. They show that the known upper bounds for both second order and fourth order problems are sharp in the case of a small overlap.
Publication Source (Journal or Book title)
SIAM Journal of Scientific Computing
Brenner, S. (2000). Lower bounds for two-level additive schwarz preconditioners with small overlap. SIAM Journal of Scientific Computing, 21 (5), 1657-1669. https://doi.org/10.1137/S1064827598336483