Determination of the optimal relaxation parameters for the solution of the Neumann–Poisson problem on uniform and non-uniform meshes using the Scheduled Relaxation Jacobi Method
Tóm tắt
The Scheduled Relaxation Jacobi (SRJ) method is a promising technique for the solution of the Poisson equation since it is possible to improve the spectral radius of the method through a proper choice of the relaxation parameters and it is also embarrassingly parallel. This is accomplished in the present study by first developing an expression for the spectral radius of the method using matrix stability analysis. The analysis is shown to provide interesting insights into the method such as the narrowing of the spectrum of the eigenvalues of the iteration matrix as well as bounds on the relaxation factors. Next, a fast and robust procedure is proposed for optimizing the values for the relaxation factors and the corresponding number of iterations. The predicted optimal values as well as those given in an earlier work are used to solve a model Poisson problem with Neumann boundary conditions and a comparison of the performance is presented. The method has been extended to cases where a non-uniform mesh is used. Optimal values for the iteration parameters are obtained for a sample non-uniform mesh. Rapid convergence is demonstrated when the SRJ method is used with these values to solve the transformed Poisson equation for the model problem. Extension of the method to levels beyond 5 is shown to be straightforward and optimal values for this case are reported. These show that further reductions in the value of the spectral radius are possible.
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