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 Bjørstad 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, which also allows immediate generalizations to other more complicated domains with more than two interior interfaces.