Abstract
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.
Highlights
The standard iterative methods like Jacobi and Gauss-Seidel (GS) rapidly damp out the local errors of the solution, but they are extremely slow to remove the global errors [1, 2]
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
These low-frequency error components appear as highfrequency ones, which are iteratively solved by a smoother
Summary
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 1977, Brandt [9] introduced a multilevel adaptive technique (MLAT) for fast numerical solution to the boundary value problems He developed a distributive Gauss-Seidel (DGS) method as a smoother for solving the Navier-Stokes equations [10]. Liang et al [15] investigated a p-multigrid method for solving spectral difference formulations of the scalar wave and Euler equations on unstructured grids They used a lower-upper symmetric Gauss-Seidel (LU-SGS) method as an iterative smoother. Liang et al [16] developed a two-dimensional high-order solver with spectral difference scheme for unsteady incompressible Navier-Stokes equations They used a p-multigrid method to accelerate the convergence rate. The convergence histories for different cases are compared and discussed
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