Multilevel Additive Schwarz Methods

We consider the solution of the algebraic system of equations which result from the dis-cretization of elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that, in the general case, the condition number of the iterative operator grows at most linearly with the number of levels. The bound is independent of the mesh sizes and the number of levels under the H 2-regularity assumption of the equation. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm.