On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations (Au=b) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of P relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of M consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are strictly different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method where the matrix of the system of equations is preconditioned multiplying it by D=diag(A). Our method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix A (or the corresponding iteration matrix of the underlying weighted Jacobi scheme) is replaced by the (much easier) calculation of the maximum and minimum frequencies derived from a von Neumann analysis of the continuous elliptic operator. This set of weights is also the optimal one for the general problem, resulting in the fastest convergence of all possible SRJ schemes for a given grid structure. The amplification factor of the method can be found analytically and allows for the exact estimation of the number of iterations needed to achieve a desired tolerance. We also show that with the set of weights computed for the optimal SRJ scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter ω in the Successive Overrelaxation (SOR) method in some cases. Finally, we demonstrate with practical examples that our method also works very well for Poisson-like problems in which a high-order discretization of the Laplacian operator is employed (e.g., a 9- or 17-points discretization). This is of interest since the former discretizations do not yield consistently ordered A matrices and, hence, the theory of Young cannot be used to predict the optimal value of the SOR parameter. Furthermore, the optimal SRJ schemes deduced here are advantageous over existing SOR implementations for high-order discretizations of the Laplacian operator in as much as they do not need to resort to multi-coloring schemes for their parallel implementation.