Decomposition and diagonalization in solving large systems

Consider the nonlinear equation (*) x = Tx + ƒ with a strictly contractive operator T in some real separable Hilbert space. A well-known procedure to approximate the unique solution x* (ƒ) of (*) is the projection-iteration method which can be characterized as a method of diagonalization. In case that (*) is a large system which can be represented as a system of weakly coupled subsystems, an efficient method to approximate x* (ƒ) is the decomposition method which is a block iteration scheme. One realization of this method is the waveform relaxation method. In this note we combine the diagonalization technique with the decomposition method and derive conditions for the convergence of the resulting iteration scheme