Delayed Over-Relaxation for iterative methods

We propose a variant of the relaxation step used in the most widespread iterative methods (e.g. Jacobi Over-Relaxation, Successive Over-Relaxation) which combines the iteration at the predicted step, namely (n + 1), with the iteration at step (n - 1). We provide a theoretical analysis of the proposed algorithm by applying such a delayed relaxation step to a generic (convergent) iterative scheme. We prove that, under proper assumptions, this significantly improves the convergence rate of the initial iterative method. As a relevant example, we apply the proposed algorithm to the solution of the Poisson equation, highlighting the advantages in comparison with classical iterative models. (C) 2016 Elsevier Inc. All rights reserved.