Methods of Reflections: relations with Schwarz methods and classical stationary iterations, scalability and preconditioning.

The basic idea of the method of reflections appeared almost two hundred years ago; it is a method of successive approximations for the interaction of particles within a fluid, and it seems intuitively related to the Schwarz domain decomposition methods, the subdomains being the complements of the particle domains. We show in this paper that indeed there is a direct correspondence between the methods of reflections and Schwarz methods in the two particle/subdomain case. This allows us to give a new convergence analysis based on maximum principle techniques with precise convergence estimates that one could not obtain otherwise. We then show however also that in the case of more than two particles/subdomains, the methods of reflections and the Schwarz methods are really different methods, with different convergence properties. Using substructuring techniques from domain decomposition, we then show that the methods of reflections are classical block Jacobi and block Gauss-Seidel methods for the interface traces, and we derive new, relaxed versions of the methods of reflections with better convergence properties. We finally also introduce for the first time coarse corrections for the methods of reflections to make them scalable in the case when the number of particles becomes large. The substructured formulations permit the easy use of the methods of reflections as preconditioners for Krylov methods, and we illustrate scalability and preconditioning properties with numerical experiments.