On the relationship between overlapping and nonoverlapping domain decomposition methods

It is proven that the two apparently different approaches in domain decomposition, namely the Schwarz-type overlapping domain algorithms and the Schur complement-type nonoverlapping algorithms, are essentially the same: for any given Schwarz algorithm there corresponds a Schur complement algorithm, with a particular preconditioner, which produces the same iterates on the interfaces. This observation was first made by Bjorstad and Widlund [SIAM J. Sci. Statist. Comput., 10 (1989), pp. 1053–1061], who showed that a result of Chan for Schur complement-type preconditioners [T. F. Chan and D. Resasco, Analysis of domain decomposition preconditioners on irregular regions, in Advances in Computer Methods for Partial Differential Equations,VI, R. Vichnevetsky and R. Stepleman, eds., IMACS,1987, pp. 317–322] can be applied to a related Schwarz-type iteration. This paper gives a different proof using a new characterization of the two algorithms as two different methods for solving the reduced interface problem, whi...