The eigenvalue spectrum of domain decomposed preconditioners

In the past several years, domain decomposition has been a very popular topic, motivated by the ease of parallelisation. In a large class of domain decomposition methods, the continuity of the solution over the internal subdomain boundaries is obtained through an iterative process. When solving the Schur complement equations for the unknowns on the internal boundaries, an elliptic problem has to be solved on each subdomain in each iteration step. Therefore, doubts have been raised as to the efficiency of these methods. An alternative solution is using preconditioners for the original problem that can be inverted in a "domain decomposed" way. In this paper, we describe and study a class of preconditioners that are combinations of preconditioners for the subdomain and the interface problems. We derive some properties of the eigenvalue spectrum of the preconditioned system, relating it to the eigenspectra of the subdomain problems. We show some numerical examples to illustrate these properties.