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

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1 International Scholarly Research Network ISRN Computational Mathematics Volume 212, Article ID , 5 pages doi:1.542/212/ Research Article Evaluation of the Capability of the Multigrid Method in Speeding Up the Convergence of Iterative Methods Iman Harimi and Mohsen Saghafian Department of Mechanical Engineering, Isfahan University of echnology, Isfahan , Iran Correspondence should be addressed to Iman Harimi, i.harimi@me.iut.ac.ir Received 2 December 211; Accepted 11 January 212 Academic Editor: R. Pandey Copyright 212 I. Harimi and M. Saghafian. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. he performance of the multigrid method and the effect of different grid levels on the convergence rate are evaluated. he two-, three-, and four-level V-cycle multigrid methods with the Gauss-Seidel iterative solver are employed for this purpose. he 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 etending 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 he standard iterative methods like Jacobi and Gauss-Seidel (GS) rapidly damp out the local errors (high-frequency errors) of the solution, but they are etremely 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. he 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 lowfrequency 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 highfrequency ones, which are iteratively solved by a smoother. he typical application of the multigrid method is the numerical solution of elliptic partial differential equations [3]. he 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 1]. he multigrid idea was first introduced by Fedorenko in 1962 and 1964 [5, 6] and then generalized by Bakhvalov in 1966 [7]. he multigrid algorithms were developed to practical applicability by Brandt in 1973 [8]. In 1977, Brandt [9] introduced a multilevel adaptive technique (MLA) 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 [1]. Jameson (1983 and 1985) developed the multigrid scheme to efficiently solve hyperbolic equations [11, 12]. here are many recent studies that have employed the multigrid method to increase the convergence rate. For eample, Vazquez et al. [13] presented a methodology, equipped with multigrid techniques, to speed up the convergence of a fully implicit solver for the RANS equations for incompressible flows. Bruneau and Mortazavi [14] investigated the flow around a square cylinder. hey used a multigrid method with Gauss-Seidel smoother for solution of the problem. In their work, the set of grids was varied, based on the needed accuracy, from the coarsest 2 5 uniform grid to the finest or uniform

2 2 ISRN Computational Mathematics grid. Liang et al. [15] investigatedap-multigrid method for solving spectral difference formulations of the scalar wave and Euler equations on unstructured grids. hey used a lower-upper symmetric Gauss-Seidel (LU-SGS) method as an iterative smoother. hey also employed a Runge-Kutta eplicit method for comparison. Liang et al. [16] developed a two-dimensional high-order solver with spectral difference scheme for unsteady incompressible Navier-Stokes equations. hey used a p-multigrid method to accelerate the convergence rate. In the present work, in order to evaluate the capability of the multigrid method and the effectof different grid levels on the convergence rate, the two-, three-, and four-level V-cycle multigrid algorithms with Gauss-Seidel iterative method as a smoother are studied. A two-frequency function, with high and low frequencies, is defined for the Laplace equation. he convergence histories for different cases are compared and discussed. 2. Problem Statement he one-dimensional Laplace equation and its discretized form (three-point discretization) can be epressed as in (1) and (2). he Dirichlet boundary conditions are set at the boundaries (() = and(5.3) = ). 54 nodes with distance Δ =.1 are placed in the solution domain 2 =, 5.3, (1) 2 i n+1 = 1 ( n+1 i 1 + n 2 i+1). (2) Initial distribution of function involving high- and lowfrequency components is defined in the solution domain, which is presented in (3). According to this equation, two different frequencies including f 1.4 ( 1 = 25.15) and f 2 =.318 ( 2 = π) eist. he function is plotted in Figure 1, and these two frequencies are distinguished in this figure. At first, the Laplace equation is solved directly using the GS method. hen, it is solved by using the GS method equipped with the two-, three-, and four-level multigrid algorithms. () = sin(2), < 4π, () =.8π.1.5 sin(2 + π), 4π< 8, () =.8π sin(2), 8π < 12π, () =.16π sin(2), 12π < < 5.3. (3) 3. Four-Level Multigrid Method (MG4) with Gauss-Seidel Smoother (GS) A four-level multigrid V-cycle and data transition between different grid levels are shown schematically in Figure 2.he levels 1 and 4 represent the finest and coarsest grids, respectively. In the multigrid method, the coarse grid mesh size is typically twice larger than the fine grid one. In the current Figure 1: Initial distribution of in the solution domain. study, the distance between grid points (mesh size) is doubled at each level (Δ,2Δ,4Δ,and 8Δ). It should be noted that the coarser grids must be coarse enough to make the solution inepensive compared to their previous grid levels. In the following sections, the procedure of going from finer to coarser level (restriction) and transferring corrections from coarser to finer level (prolongation) is eplained in detail Smoothing Iterations on the Finest Grid and ransfer the Residual to the Coarser Grid. he Laplace equation is solved using the GS method (2). he number of relaation sweeps on the finest grid is about 3 to 4. It should be noted that complete convergence is not absolutely necessary. he operator L is defined such that L i becomes the discretized form of Laplace equation, as presented in (4). he residuals are obtained using (5) and then transferred to the coarser grid (level 2). G1 denotes the level 1 in this equation. In order to transfer data to the coarser grids, the injection operator (restriction operator) is employed, which projects the data of the finer grid points to the corresponding points at the coarser grid (see Figure 2(b)) X L i = n i+1 2i n + i 1 n (Δ) 2, (4) R i G1 = L i. (5) 3.2. Computing the Correction erm and New Residuals. he correction term can be calculated using (6), where G2 refers to the level 2. his equation can be epanded to yield (7), where Δ is two times of Δ. In this equation, an initial guess of zero is used for τ. Similarly, 3 or 4 iterations of the solver are adequate at this stage τ n+1 i Lτ i G2 = R i G2, (6) = G2.5 { R i G2 (Δ ) 2 + τi+1 n } G2 + τi 1 n+1 G2. (7) he updated residuals are obtained as the sum of the old residuals and the correction term as follows: R i G2,New = R i G2 + τ i G2. (8) 2

3 ISRN Computational Mathematics 3 Δ n Level 1 2Δ Level 2 4Δ Level 3 8Δ (b) Level 4 (a) Figure 2: (a) Four-level V-cycle (b) data transition between different grid levels in MG4. Similarly, in levels 3 and 4, the new residuals and the correction terms can be calculated. It should be noted that Δ is 4 and 8 times of Δ in levels 3 and 4, respectively ransferring Corrections to the Finer Grid. By using a linear interpolation, the correction term obtained in level 4 is transferredbacktolevel3(τ i G3,Return ) and added to the correction term calculated in this level in the restriction process (9). he obtained values (τ i G3,New ) are employed as an initial guess for solving Laplace equation in this level (1). In (1), R i G3,New is the updated residuals obtained in the restriction process τ i G3,New = τ i G3 + τ i G3,Return, (9) X Lτ i G3 = R i G3,New. (1) Similarly, the correction term can be transferred back to level 2. o return to level 1 (finest grid), the correction term obtained in level 2 is transferred to level 1 (τ i G1,Return )and added to the function calculated by (2), as follows: i New = i Old + τ i G1,Return. (11) he convergence criterion is checked after updating the function. If the desired convergence is not achieved, the entire procedure (Sections 3.1 to 3.3) must be repeated. It should be noted that the updated in the last step (11) is considered as an initial guess for new operation (2). In the two- and three-level multigrid algorithms, the prolongation process is started after reaching the levels 2 and 3, respectively. 4. Numerical Results Figures 3 and 4 show the smoothing of the function using Gauss-Seidel (GS) and multigrid (MG) methods after n iterations. It required and 5685 iterations for GS and MG methods, respectively, to reach the convergence criterion of AscanbeseeninFigure 3, for Gauss-Seidel method, the high-frequency error components are damped completely after 3 iterations, whereas iterations (about 38 times more) are needed to remove the low-frequency components. On the other hand, as soon as the highfrequency components are smoothed out, the convergence slows down. According to the obtained results, the number n = n = 3 n = 3 n = 3 n = 3 n = Figure 3: Smoothing the function by GS iterations (n), convergence criterion = able 1: Number of iterations for convergence. Method Convergence criterion GS MG MG MG of iterations required to smooth out the global errors can be reduced to 5685 by applying the two-level multigrid method. he numbers of iterations for convergence in GS and MG method are presented in able 1. he convergence history of the GS and MG methods is plotted in Figures 5 and 6. AscanbeobservedinFigure 5, for the GS method, after the initial iterations, the residuals decrease very slowly. In addition, the slope of the residuals decreases gradually and approaches zero, indicating that the GS method is very time consuming especially for obtaining the eact solution. But for the two-level MG method, the residuals drop rapidly with a relatively constant slope compared to the GS method. Furthermore, according to Figure 6,

4 4 ISRN Computational Mathematics Residual n = n = 3 n = 3 n = 3 X n = 1 n = 2 n = 5685 Figure 4: Smoothing the function by MG2 iterations (n), convergence criterion = Residual Iteration Gauss-Seidel MG2 Figure 5: Comparison of convergence history using Gauss-Seidel and two-level multigrid methods, convergence criterion = the trend of residual variations is significantly similar for the two-, three-, and four-level MG methods. It seems that using the two-level MG method is adequate to speed up the convergence in this study. 5. Conclusion In this study, the capability of the multigrid method in increasing the convergence rate has been evaluated by using the two-, three-, and four-level V-cycle multigrid algorithms (MG2, MG3, and MG4, resp.) and the iterative Gauss-Seidel method (GS). he obtained results confirm that the MG method accelerates the convergence of the solution drastically. For eample, the number of iterations required to reach MG2 MG3 MG4 Iteration Figure 6: Comparison of convergence history using two-, three-, and four-level multigrid methods, convergence criterion = the convergence criterion of decreases from to 5685 by applying an MG2 method. It is found that the GS method is very time consuming to achieve high accuracy. Furthermore, the comparison of the convergence history of MG2, MG3, and MG4 methods reveals that applying the MG2 method is adequate to increase the convergence speed in this problem. In fact, the choice of the coarsest grid level is problem dependent, and the frequency of errors plays an important role in it. References [1] J. C. annehill, D. A. Anderson, and R. H. Pletcher, Comutational Fluid Mechanics and Heat ransfer, aylor&francis, Philadelphia, Pa, USA, 2nd edition, [2] D. M. Young, Iterative Solutions of Large Linear Systems, Academic Press, New York, NY, USA, [3] U. rottenberg, C. W. Oosterlee, and A. Schüller, Multigrid, Academic Press, San Diego, Calif, USA, 21. [4] A. Kenigsberg, R. Kimmel, and I. Yavneh, A multigrid approach for fast geodesic active contours, ech. Rep. CIS-24-6, echnion-israel Institute of echnology, Haifa, Israel, 24. [5] R. P. Fedorenko, A relaation method for solving elliptic difference equations, USSR Computational Mathematics and Mathematical Physics, vol. 1, no. 4, pp , [6] R. P. Fedorenko, he speed of convergence of one iterative process, USSR Computational Mathematics and Mathematical Physics, vol. 4, no. 3, pp , [7] N. S. Bakhvalov, On the convergence of a relaation method with natural constraints on the elliptic operator, USSR Computational Mathematics and Mathematical Physics, vol. 6, no. 5, pp , [8] A. Brandt, Multi-level adaptive technique (MLA) for fast numerical solution to boundary value problems, in Proceedings of the 3rd International Conference on Numerical Methods in Fluid Mechanics I, pp , Springer, Berlin, Germany, 1973.

5 ISRN Computational Mathematics 5 [9] A. Brandt, Multi-level adaptive solutions to boundary-value problems, Mathemathics of Computation, vol. 31, no. 138, pp , [1] A. Brandt, Multi-level adaptive solutions to boundary-value problems, AIAA Journal, vol. 18, no. 1, pp , 198. [11] A. Jameson, Solution of the Euler equations for two dimensional transonic flow by a multigrid method, Applied Mathematics and Computation, vol. 13, no. 3-4, pp , [12] A. Jameson, Multigrid algorithms for compressible flow calculations, in Proceedings of the 2nd European Conference on Multigrid Methods, Princeton University Report MAE 1743, Cologne, Germany, [13] M. Vazquez, M. Ravachol, and M. Mallet, Multigrid applied to a fully implicit FEM solver for turbulent incompressible flows, in Proceedings of the ECCOMAS Computational Fluid Dynamics Conference, Wales, UK, September 21. [14] C. H. Bruneau and I. Mortazavi, Passive control of the flow around a square cylinder using porous media, International Journal for Numerical Methods in Fluids, vol. 46, no. 4, pp , 24. [15] C. Liang, R. Kannan, and Z. J. Wang, A p-multigrid spectral difference method with eplicit and implicit smoothers on unstructured triangular grids, Computers & Fluids, vol. 38, no. 2, pp , 29. [16] C. Liang, A. S. Chan, and A. Jameson, A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier-Stokes equations, Computers & Fluids, vol. 51, no. 1, pp , 211.

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