On nilpotent subdomain iterations

Abstract

Subdomain iterations which lead to a nilpotent iteration operator converge in a finite number of steps, and thus are equivalent to direct solvers. We determine in this paper for which type of decomposition and relaxation parameters Dirichlet-Neumann, Neumann-Neumann, and Optimal Schwarz methods can be nilpotent. We start using a simple one dimensional model problem, and show that Neumann-Neumann and Dirichlet-Neumann cannot lead to a nilpotent iteration matrix in general. Optimal Schwarz methods can be nilpotent for an arbitrary number of subdomains.