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A Two-Level Preconditioned Conjugate-Gradient Method in Distorted and Structured Grids

Published online by Cambridge University Press:  03 June 2015

Qiaolin He
Qiaolin He*
Affiliation:
Deparment of Mathematics, Sichuan University, Chengdu 610064, Sichuan, China
*
*Corresponding author. Email: qlhejenny@gmail.com
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Abstract

In this paper, we propose a new two-level preconditioned C-G method which uses the quadratic smoothing and the linear correction in distorted but topo-logically structured grid. The CPU time of this method is less than that of the multigrid preconditioned C-G method (MGCG) using the quadratic element, but their accuracy is almost the same. Numerical experiments and eigenvalue analysis are given and the results show that the proposed two-level preconditioned method is efficient.

Keywords

65F0865N2265N3065N50Preconditionconjugate gradientmultigridfinite element
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 2 , April 2012 , pp. 238 - 249
Copyright
Copyright © Global-Science Press 2012

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References

[1] Fedorenko, R. P., A relaxation method for solving elliptic difference equations, USSR Comput. Math. Math. Phys., 1 (1962), pp. 1092–1096.CrossRefGoogle Scholar
[2] Fedorenko, R. P., The speed of convergence of one iterative process, USSR Comput. Math. Math. Phys., 4 (1964), pp. 227–235.Google Scholar
[3] Bakhvalov, N. S., On the convergence of a relaxation method with natrural constraints on the elliptic operator, USSR Comput. Math. Math. Phys., 6 (1966), pp. 101–113.Google Scholar
[4] Brandt, A., Multi-level Adaptive technique (MLAT) for fast numerical solution to boundary-value problems, in: Cabannes, H. and Temam, R. (eds.), Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, Paris 1972, Lecture Notes in Physics 18, Springer-Verlag, Berlin, 1973.Google Scholar
[5] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comput., 31 (1977), pp. 333–390.Google Scholar
[6] Hackbusch, W., A multi-grid method applied to a boundary problem with variable coefficients in a rectangle, Report 77-17, Institut für Angewandte Mathematik, Universität Köln, 1977.Google Scholar
[7] Hackbusch, W., Multi-Grid Methods and Applications, Springer-Verlag, Heidelberg, 1985.Google Scholar
[8] Stuüben, K. and Trottenberg, U., Multigrid Methods: Fundamental Algorithms, Model Problem Analysis and Applications, in: Hackbusch-Trottenberg [9].Google Scholar
[9] Hackbusch, W. and Trottenberg, U.(eds.) Multi-Grid Methods, Proceedings, Koüln-Porz, Nov. 1981, Lecture Notes in Mathematics 960, Springer-Verlag, Berlin, 1982.Google Scholar
[10] Ren, W.Q. AND Wang, X.P., An iterative grid redistribution method for singular problems in multiple dimensions, Comput. Phys., 159 (2000), pp. 246–273.Google Scholar
[11] Tatebe, O., The multigrid preconditioned conjugate gradient method, in Proceedings of Sixth Copper Mountain Conference on Multigrid Methods, NASA Conference Publication 3224, April 1993, pp. 621–624.Google Scholar
[12] Braess, D., On the combination of the multigrid method and conjugate gradients, in Multi-grid Methods II (Hackbusch, W.and Trottenberg, U., eds.), Vol. 1228 of Lecture Notes in Mathematics, (1986), pp. 52–64, Springer-Verlag.Google Scholar
[13] Shu, S., Sun, D. and Xu, J., An algebraic multigrid method for higher order finite element discretizations, Comput., 77 (2006), pp. 347–377.Google Scholar
[14] Shu, S., Xu, J., Yang, Y. and Yu, H., An algebraic multigrid method for finite element systems on criss-cross grids, Adv. Comput. Math., 25 (2006), pp. 287–304.Google Scholar
[15] Tatebe, O. and Oyanagi, Y., Efficient implementation of the multigrid preconditioned conjugate gradient method on distributed memory machines, in Proceedings of Supercomputing’ 94, pp. 194–203, IEEE Computer Society, November 1994.Google Scholar
[16] Bank, R. E. and Douglas, C. C., Sharp estimates for multigrid rates of convergence with general smoothing and acceleration, SIAM J. Numer. Anal., 22 (1985), pp. 617–633.Google Scholar