Comparisons between multiplicative and additive Schwarz iterations in domain decomposition methods

We consider the multiplicative and additive Schwarz methods for solving linear systems of equations and we compare their asymptotic rate of convergence. Moreover, we compare the multiplicative Schwarz method with the weighted restricted additive Schwarz method. We prove that the multiplicative Schwarz method is the fastest method among these three. Our comparisons can be done in the case of exact and inexact subspaces solves. In addition, we analyse two ways of adding a coarse grid correction – multiplicatively or additively.