To overlap or not to overlap: a note on a domain decomposition method for elliptic problems

More than one hundred years ago, H.A. Schwarz introduced a domain decomposition method in which the original elliptic equation is solved on overlapping subregions, one after another, in an iterative process. A few years ago, Chan and Resasco, introduced a method that they classified as a domain decomposition method using nonoverlapping subdomains. In this note, it is shown that their method is an accelerated version of the classical method. It is also shown that the error propagation operator of the method can be expressed in terms of Schur complements of certain stiffness matrices and that techniques previously developed for the study of iterative substructuring algorithms can be used to derive estimates on the rate of convergence.