Title: New results on overlapping Schwarz methods

Speaker: Olof Widlund, Courant Institute, NYU

Date/Time: Wednesday, August 29, 2007, 10:00 - 11:00 am

Location: CSRI Building, Room 148 (Sandia NM)

Brief Abstract: New results have recently been obtained concerning the classical two-level additive Schwarz preconditioners. In the theory for domain decomposition methods, we have often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, we present extensions of the existing theory to accommodate subdomains with much less regular shape.

One important goal is to extend our analytic tools to problems on subdomains that might not even be Lipschitz and to characterize the rates of convergence of our methods in terms of a few, easy to understand, geometrical parameters of the subregions. We believe that this goal now has been reached fully for scalar elliptic and linear elasticity problems in two dimensions.

We have also designed a new family of overlapping Schwarz methods, which in a certain sense is a hybrid algorithm since we borrow and extend coarse spaces from iterative substructuring methods. Methods based on such choices are known to be very robust even in the presence of large local changes of the materials being modeled by the finite element models. An extra attraction is that the overlapping Schwarz methods can be applied directly to problems where the stiffness matrix is available only in its fully assembled form. Important progress has also been made on almost incompressible elasticity and mixed finite element models. One of our results is the first of its kind for saddle point problems and overlapping Schwarz methods.

Joint work with Clark R. Dohrmann, Axel Klawonn, and Oliver Rheinbach