This paper addresses the convergence characteristics of an iterative solution scheme of the Neumann-type useful for obtaining homogenized mechanical material properties from a representative volume element. The analysis is based on Eshelby's idea of "equivalent inclusions" and, within the context of mechanical stress/strain analysis, allows modeling of elastically highly heterogeneous bodies with the aid of discrete Fourier transforms. Within the iterative scheme the proof of convergence depends critically upon the choice of an appropriate, auxiliary stiffness matrix, which also determines the speed of convergence. Mathematically speaking it is based on Banach's fixpoint theorem and only results in sufficient convergence conditions. However, all cases of elastic heterogeneity that are of practical importance are covered and some evidence is provided that other choices of auxiliary stiffness may result in faster convergence even if this cannot explicitly be shown within the theoretical framework chosen.