Optimal V-cycle Algebraic Multilevel Preconditioning

Numerical Linear Algebra with Applications

Volumn 5, Issue 5, 1998, Pages 441-459

  Notay, Yvan ^a  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

Author keywords

Acceleration of convergence; Iterative methods for linear systems; Preconditioning

Indexed keywords

EID: 0032252492     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-1506(199809/10)5:5<441::AID-NLA147>3.0.CO;2-J     Document Type: Article

References (42)

1. R. E. Alcouffe, A. Brandt, J. E. Dendy and J. W. Painter. The multi-grid methods for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Stat. Comput., 2, 430-454, 1981.
2. O. Axelsson. On multigrid methods of the two-level type. In W. Hackbusch and U. Trottenberg, editors, Multigrid Methods, Lectures Notes in Mathematics, pages 352-367, Springer-Verlag. Berlin, Heidelberg, New York, 1981.
3. O. Axelsson. The method of diagonal compensation of reduced matrix entries and multilevel iteration. J. Comput. Appl. Math., 38, 31-43, 1991.
4. O. Axelsson. Iterative Solution Methods, University Press, Cambridge, 1994.
5. O. Axelsson and V. A. Barker. Finite Element Solution of Boundary Value Problems. Theory and Computation, Academic Press, New York, 1984.
6. O. Axelsson and V. Eijkhout. Analysis of recursive 5-point/9-point factorization method. In O. Axelsson and L. Kolotilina, editors, Preconditioned Conjugate Gradient Methods, Vol. 1457 of Lecture Notes in Mathematics pages 154-173. Springer-Verlag, Berlin, 1990.
7. O. Axelsson and V. Eijkhout. The nested recursive two level factorization for nine-point difference matrices. SIAM J. Sci. Stat. Comput., 12, 1373-1400, 1991.
8. O. Axelsson and I. Gustafsson. Preconditioning and two-level multigrid methods of arbitrary degree of approximation. Math. Comp., 40, 214-242, 1983.
9. O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48, 479-498, 1986.
10. O. Axelsson and M. Neytcheva. Algebraic multilevel iterations for Stieltjes matrices. Num. Lin. Alg. Appl., 1, 213-236, 1994.
11. O. Axelsson and P. S. Vassilevski. Algebraic multilevel preconditioning methods. i. Numer. Math., 56, 157-177, 1989.
12. O. Axelsson and P. S. Vassilevski. Algebraic multilevel preconditioning methods, II. SIAM J. Numer. Anal., 27, 1569-1590, 1990.
13. R. E. Bank. Hierarchical bases and the finite element method. Acta Numerica, 5, 1-43, 1996.
14. R. E. Bank and T. F. Dupont. Analysis of a two-level scheme for solving finite element equations. Technical Report CNA-159, Center for Numerical Analysis, The University of Texas at Austin, Texas, USA, 1980.
15. R. E. Bank, T. F. Dupont and H. Yserentant. The hierarchical basis multigrid method. Numer. Math., 52, 427-458, 1988.
16. R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM Publications, Philadelphia, 1993.
17. E. F. F. Botta and F. W. Wubs. The convergence behaviour of iterative methods on severely stretched grids. Int. j. numer. methods eng., 36, 3333-3350, 1993.
18. D. Braess. Towards algebraic multigrid for elliptic problems of second order. Computing, 55, 379-393, 1995.
19. P. M. de Zeeuw. Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput. Appl. Math., 33, 1-27, 1990.
20. J. E. Dendy. Black box multigrid. J. Comput. Phys., 48, 366-386, 1982.
21. J. Fuhrman. Outlines of a modular algebraic multilevel method. In AMLI'96: Proceedings of the Conference on Algebraic Multilevel Iteration Methods with Applications, volume 1, pages 141-152, University of Nijmegan, Nijmegan, The Netherlands, 1996.
22. G. H. Golub and C. F. van Loan. Matrix Computations, third edition. The John Hopkins University Press, Baltimore, Maryland, 1996.
23. W. Hackbusch. Multi-grid Methods and Applications. Springer, Berlin, 1985.
24. R. Kettler. Linear multigrid methods for numerical reservoir simulation. Ph.D. thesis, Technical Univesity Delft, Delft, 1987.
25. J. Maitre and F. Musy. The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems. In W. Hackbusch and U. Trottenberg, editors, Multigrid Methods, Lectures Notes in Mathematics, pages 535-544, Springer-Verlag. Berlin, Heidelberg, New York, 1981.
26. S. Nakamura. Computational Methods in Engineering and Science, John Wiley & Sons, New York, 1977.
27. Y. Notay. DRIC: a dynamic version of the RIC method. J. Num. Lin. Alg. with Appl., 1, 511-532, 1994.
28. Y. Notay. Optimal order preconditioning of finite difference matrices. Technical Report GANMN 97-02, Université Libre de Bruxelles, Brussels, Belgium, 1997. Available at http://homepages.ulb.ac.be/~ynotay.
29. Y. Notay. Using approximate inverses in algebraic multilevel methods. Numer. Math., 80, 397-417, 1998.
30. A. Reusken. Fourier analysis of a robust multigrid method for convection-diffusion equations. Numer. Math., 71, 365-398, 1995.
31. A. Reusken. A multigrid method based on incomplete Gaussian elimination. J. Num. Lin. Alg. with Appl., 3, 369-390, 1996.
32. J. W. Ruge and K. Stüben. Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In D. J. Paddon and H. Holstein, editors, Multigrid Methods for Integral and Differential Equations, The Institute of Mathematics and its Applications Conference Series, pages 169-212. Clarendon Press, Oxford, 1985.
33. J. W. Ruge and K. Stuben. Algebraic multigrid (AMG). In S. F. McCormick, editor, Multigrid Methods, volume 3 of Frontiers in Applied Mathematics, pages 73-130, SIAM, Philadelphia, PA, 1987.
34. K. Stüben. Algebraic multigrid (AMG): experiences and comparisons. Appl. Math. Comput., 13, 419-452, 1983.
35. A. van der Ploeg, E. Botta and F. Wubs. Nested grids ILU-decomposition (NGILU). J. Comput. Appl. Math., 66, 515-526, 1996.
36. P. Vassilevski. Nearly optimal iterative methods for solving finite element elliptic equations based on the multilevel splitting of the matrix. Technical Report # 1989-09, Institute for Scientific Computation, University of Wyoming, Laramie, USA, 1989.
37. P. Vassilevski. Hybrid V-cycle algebraic multilevel preconditioners. Math. Comp., 58, 489-512, 1992.
38. P. Vassilevski. On two ways of stabilizing the hierarchical basis multilevel methods. SIAM Review, 39, 18-53, 1997.
39. C. Wagner, W. Kinzelbach and G. Wittum. Schur-complement multigrid. Numer. Math., 75, 523-545, 1997.
40. P. Wesseling. An Introduction to Multigrid Methods. J. Wiley and Sons, Chichester, 1992.
41. G. Wittum. On the robustness of ILU-smoothing. SIAM J. Sci. Statist. Comput., 10, 699-717, 1989.
42. H. Yserentant. On the multi-level splitting of finite element spaces. Numer. Math., 49, 379-412, 1986.