%0 Journal Article
%T Evaluation of the Capability of the Multigrid Method in Speeding Up the Convergence of Iterative Methods
%A Iman Harimi
%A Mohsen Saghafian
%J ISRN Computational Mathematics
%D 2012
%R 10.5402/2012/172687
%X 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. 1. Introduction The standard iterative methods like Jacobi and Gauss-Seidel (GS) rapidly damp out the local errors (high-frequency errors) of the solution, but they are extremely slow to remove the global errors (low-frequency errors) [1, 2]. In fact, these methods have a local stencil and may require a large number of iterations to converge. The multigrid method (MG) is one of the most efficient methods for solving linear and nonlinear systems, which can speed up the rate of damping out low-frequency errors. In this method, the high-frequency components of the solution error are damped by an iterative solver, or smoother, on a fine grid, whereas the low-frequency components are transferred to the coarser grid. On the coarser grid, these low-frequency error components appear as high-frequency ones, which are iteratively solved by a smoother. The typical application of the multigrid method is the numerical solution of elliptic partial differential equations [3]. The multigrid methods have also been used successfully for problems in image processing and vision [4]. In the past decades, many researchers including Fedorenko, Bakhvalov, and Brandt have studied and developed the multigrid methods [5¨C10]. The multigrid idea was first introduced by Fedorenko in 1962 and 1964 [5, 6] and then generalized by Bakhvalov in 1966 [7]. The multigrid algorithms were developed to practical applicability by Brandt in 1973 [8]. In 1977, Brandt [9] introduced a multilevel adaptive technique (MLAT) for fast
%U http://www.hindawi.com/journals/isrn.computational.mathematics/2012/172687/