Title:     Lower bounds for two-level additive Schwarz preconditioners
           for nonconforming finite elements
                     
Authors:   Susanne C. Brenner and Li-yeng Sung

Published: Advances in Computational Mathematics, Z. Chen et.al., ed.,
           Marcel Dekker, 1999, pp. 585-604 (MR# 99j:65196)

Abstract:  Lower bounds for the condition numbers of the preconditioned
           systems are obtained for two-level additive Schwarz preconditioners
           for nonconforming finite element methods.  They show that the known 
           upper bounds for both second order and fourth order problems are 
           sharp in the case of a small overlap.