Multilevel Schwarz Methods with Partial Refinement

We consider multilevel additive Schwarz methods with partial reenement. These algorithms are generalizations of the multilevel additive Schwarz methods developed by Dryja and Widlund and many others. We will give two diierent proofs by using quasi-interpolants under two diierent assumptions on selected reenement subregions to show that this class of methods has an optimal condition number. The rst proof is based purely on the localization property of quasi-interpolants. However, the second proof use some results on iterative reenement 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. AMS(MOS) subject classiications. 65F10,65N30 1. Introduction. In this paper, we consider some solution methods of the large linear systems of algebraic equations which arise when working with elliptic nite element approximations on composite meshes. We consider the following linear, self-adjoint, elliptic problems discretized by nite element methods on a bounded Lipschitz polyhedral region in R n .