Evaluation of the Capability of the Multigrid Method in Speeding Up the Convergence of Iterative Methods

Abstract EN

The performance of the multigrid method and the effect of different grid levels on the convergence rate are evaluated.

The two-, three-, and four-level V-cycle multigrid methods with the Gauss-Seidel iterative solver are employed for this purpose.

The numerical solution of the one-dimensional Laplace equation with the Dirichlet boundary conditions is obtained using these methods.

For the Laplace equation, a two-frequency function involving high- and low-frequency components is defined.

It is observed that, however, the GS method can smooth out the high-frequency error components properly, but because the difference scheme for Laplace equation is remarkably concise, in the fine grids, a very large number of iterations are needed for extending the boundary conditions into the domain.

Furthermore, the obtained results reveal that the number of necessary iterations for convergence is reduced considerably by employing the two-level multigrid algorithm.

But increasing the number of levels of algorithm does not have any significant effect on the convergence rate in this study.