The authors draw on material from a variety of sources and their own work to produce preliminary results concerning the convergence of a pair of iterative methods based on domain decomposition (the multiplicative and additive Schwarz methods) for solving the discrete equations obtained when the Galerkin method is applied to a class of operator equations. These results are then developed further in the context of certain boundary integral equations of the first kind (boundary element methods), the upshot being that, under appropriate conditions, the rate of convergence is independent of the size of the discretized problem. Experimental results are presented for several examples. Appreciating the details of the derivation requires considerable theoretical background and knowledge of the cited literature.