On the Recommendations for Reducing CPU Time of Multigrid Preconditioned Gauss–Seidel Method

Abstract

Gauss–Seidel method is one of the simplest available iterative methods for solving systems of linearized equations. It can effectively reduce high-frequency errors but performs poorly with errors of low frequency. Multigrid (MG) utilizes this quality of the point-wise methods by successively coarsening the grid, so that the lowest frequency errors appear as high frequency and can be easily reduced. In this work, optimization study was performed to lower the CPU time of the Multigrid method. We have considered several parameters, such as the number of grid levels used, the number of inner iterations (iterations at each intermediate grid), the overall coarsening and interpolation cycle (V and W), and the number of these cycles in each iteration. A surrogate model is used to predict optimum value for these parameters. In this chapter, MG is used with a Gauss–Seidel solver for a 2D conduction problem with Dirichlet boundary condition on a 256 × 256 structured grid. The results suggest that a W cycle is more efficient than a V cycle and should be executed to the penultimate grid level during both restriction (coarsening) and prolongation.