Domain decomposition methods for advection-dominated problems

In the first part, we present some results obtained with two similar parallelizable Schwarz type methods for advection dominated advection-diffusion problems. The first is the fictitious overlapping technique (by P.L. Lions) or an equivalent augmented Lagrangian method (by P. Le Tallec) together with a stabilized Galerkin finite element method. The results of a recent paper [12] for the symmetric case are not applicable. The key observation is a ''downwind'' propagation of the error from subdomain to subdomain along the characteristics of tile hyperbolic limit problem. Furthermore we found similar results with the second method of (minimal) overlapping type. A convergence proof of this method on the continuous level is available for (very) simple behaviour of the characteristics. In a second part of the paper, we extend the proposed methods to the incompressible Stokes and Navier-Stokes problems with a basic stable finite element or finite difference method and present first numerical results.