This article is the first of a series of papers dealing with domain decomposition algorithms for implicit solvent models. We show that, in the framework of the COSMO model, with van der Waals molecular cavities and classical charge distributions, the electrostatic energy contribution to the solvation energy, usually computed by solving an integral equation on the whole surface of the molecular cavity, can be computed more efficiently by using an integral equation formulation of Schwarz's domain decomposition method for boundary value problems. In addition, the so-obtained potential energy surface is smooth, which is a critical property to perform geometry optimization and molecular dynamics simulations. The purpose of this first article is to detail the methodology, set up the theoretical foundations of the approach, and study the accuracies and convergence rates of the resulting algorithms. The full efficiency of the method and its applicability to large molecular systems of biological interest is demonstrated elsewhere.