On the Scalability of Classical One-Level Domain-Decomposition Methods

One-level domain-decomposition methods are in general not scalable, and coarse corrections are needed to obtain scalability. It has however recently been observed in applications in computational chemistry that the classical one-level parallel Schwarz method is surprizingly scalable for the solution of one- and two-dimensional chains of fixed-sized subdomains. We first review some of these recent scalability results of the classical one-level parallel Schwarz method, and then prove similar results for other classical one-level domain-decomposition methods, namely the optimized Schwarz method, the Dirichlet-Neumann method, and the Neumann-Neumann method. We show that the scalability of one-level domain decomposition methods depends critically on the geometry of the domain-decomposition and the boundary conditions imposed on the original problem. We illustrate all our results also with numerical experiments.