Asymptotic performance of the Scheduled Relaxation Jacobi method

The performance of the Scheduled Relaxation Jacobi method for levels as high as 25 and mesh size as large as $$4096\times 4096$$ is studied in the present work. The optimal values for the relaxation parameters have been obtained using a search procedure proposed in an earlier study by the same author after suitable modifications and improvements which allow it to be used for the present purpose. It is shown that for a given number of levels and mesh size, the theoretical spectral radius and the speed-up of the method improve as the upper limit for the value of the relaxation factor is increased, albeit at the cost of numerical stability. Since the values for the over-relaxation factors for the cases considered here are high and there are more than one under-relaxation factor, a proper sequencing of the over- and under-relaxed iterations is essential and a strategy for the same is proposed. The five level method with suitably determined optimal values is shown to perform better than a twenty five level method, when used for solving the Neumann–Laplace problem, despite the theoretical spectral radius of the former being less than that of the latter. The numerical calculations demonstrate that the SRJ method with optimal parameters can achieve a reduction in the initial error by four orders of magnitude within 3000 iterations even for a 4096 $$\times $$ 4096 mesh.