A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations

International Journal for Numerical Methods in Engineering

Volumn 65, Issue 6, 2006, Pages 785-807

  Chen, X ^a   Toh, K C ^a   Phoon, K K ^a  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Biot's consolidation equations; Generalized Jacobi preconditioner; Modified symmetric successive over relaxation preconditioner; Preconditioned symmetric quasi minimal residual method

Indexed keywords

CONSOLIDATION; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; ITERATIVE METHODS; MATRIX ALGEBRA; PROBLEM SOLVING; SOIL MECHANICS;

BIOT CONSOLIDATION EQUATIONS; GENERALIZED JACOBI PRECONDITIONER; PRECONDITIONED SYMMETRIC QUASI-MINIMAL RESIDUAL METHOD; SYMMETRIC SUCCESSIVE OVER-RELAXATION (SSOR) PRECONDITIONER;

LINEAR EQUATIONS;

CONSOLIDATION; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; ITERATIVE METHODS; LINEAR EQUATIONS; MATRIX ALGEBRA; PROBLEM SOLVING; SOIL MECHANICS;

EID: 31644438565     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.1461     Document Type: Article

References (27)

1. Britto AM, Gunn MJ. Critical State Soil Mechanics via Finite Elements. Ellis Horwood Ltd.: Chichester, West Sussex, 1952.
2. Phoon KK, Toh KC, Chan SH, Lee FH. An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations. International Journal for Numerical Methods in Engineering 2002; 55(4): 377-400.
3. Saad Y. Iterative Methods for Sparse Linear Systems. PWS Publishing Company: Boston, MA, 1996.
4. Paige CC, Saunders MA. Solution of sparse indefinite systems of linear equations. SIAM Journal on Numerical Analysis 1975; 12:617-629.
5. Freund RW, Nachtigal NM. A new Krylov-subspace method for symmetric indefinite linear system. In Proceedings of the l4th IMACS World Congress on Computational and Applied Mathematics, Ames WF (ed.). IMACS: New Brunswick, NJ, 1994; 1253-1256.
6. Chan SH, Phoon KK, Lee FH. A modified Jacobi preconditioner for solving ill-conditioned Biot's consolidation equations using symmetric quasi-minimal residual method. International Journal for Numerical and Analytical Methods in Geomechanics 2001; 25:1001-1025.
7. Toh KC, Phoon KK, Chan SH. Block preconditioners for symmetric indefinite linear systems. International Journal for Numerical Methods in Engineering 2004; 60:1361-1381.
8. Phoon KK, Toh KC, Chen X. Block constrained versus generalized Jacobi preconditioners for iterative solution of large-scale Biot's FEM equations. Computers and Structures 2004; 82:2401-2411.
9. Eisenstat SC. Efficient implementation of a class of preconditioned conjugate gradient methods. SIAM Journal on Scientific and Statistical Computing 1981; 2:1-4.
10. Mroueh H, Shahrour I. Use of sparse iterative methods for the resolution of three-dimensional soil/structure interaction problems. International Journal for Numerical and Analytical Methods in Geomechanics 1999; 23:1961-1975.
11. Toh KC, Phoon KK. Further observations on generalized Jacobi preconditioning for iterative solution of Biot's FEM equations. Working Paper, Department of Mathematics, National University of Singapore, 2005.
12. Freund RW, Jarre F. A QMR-based interior-point algorithm for solving linear programs. Mathematical Programming 1997; 76:183-210.
13. Chan TF, Van der Vorst HA. Approximate and incomplete factorizations. Technical Report 871, Department of Mathematics, University of Utrecht, 1994.
14. Axelsson O. Iterative Solution Methods. Cambridge University Press: New York, NY, 1995.
15. Chow E, Heroux MA. An object-oriented framework for block preconditioning. ACM Transactions on Mathematical Software (TOMS) 1998; 24:159-183.
16. Barrett R, Berry M, Chan TF, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, Van der Vorst HA. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Press: Philadelphia, PA, 1994.
17. Duff IS, Meurant GA. The effect of ordering on preconditioned conjugate gradients. BIT 1989; 29:635-657.
18. Eijkhout V. On the existence problem of incomplete factorisation methods. Lapack Working Note 144, UT-CS-99-435, 1999.
19. Chow E, Saad Y. Experimental study of ILU preconditioners for indefinite matrices. Journal of Computational and Applied Mathematics 1997; 86:387-414.
20. Pommerell C, Fichtner W. PILS: an iterative linear solver package for ill-conditioned systems. Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, Albuquerque, New Mexico, U.S.A., 1991; 588-599.
21. Murphy MF, Golub GH, Wathen AJ. A note on preconditioning for indefinite linear systems. SIAM Journal on Scientific Computing 2000; 21:1969-1972.
22. Axelsson O. A generalized SSOR method. BIT 1972; 13:443-467.
23. Meijerink JA, Van der Vorst HA. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation 1977; 31:148-162.
24. Gustafsson I. A class of first order factorization methods. BIT 1978; 18:142-156.
25. Gambolati G, Pini G, Ferronato M. Numerical performance of projection methods in finite element consolidation models. International Journal for Numerical and Analytical Methods in Geomechanics 2001; 25:1429-1447.
26. Van der Vorst HA. Iterative Methods for Large Linear Systems. Cambridge University Press: Cambridge, MA, 2003.
27. Toh KC. Solving large scale semidefinite programs via an iterative solver on the augmented systems. SIAM Journal on Optimization 2003; 14:670-698.