654 Multilevel Schwarz Methods with Partial Refinement H. Cheng We consider multilevel additive Schwarz methods with partial refinement. These algorithms are generalizations of the multilevel additive Schwarz methods developed by Dryja and Widlund and many others. We will give two different proofs by using quasi-interpolants under two different assumptions on selected refinement subregions to show that this class of methods has an optimal condition number. The first proof is based purely on the localization property of quasi-interpolants. However, the second proof use some results on iterative refinement methods. As a by-product, the multiplicative versions which corresponds to the FAC algorithms with inexact solvers consisting of one Gauss-Seidel or damped Jacobi iteration have optimal rates of convergence. Finally, some numerical results are presented for these methods.