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A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations. (English) Zbl 1114.74056
Summary: The standard SSOR preconditioner is ineffective for iterative solution of symmetric indefinite linear systems arising from finite element discretization of Biot’s consolidation equations. In this paper, we propose a modified block SSOR preconditioner combined with the Eisenstat-trick implementation [S. C. Eisenstat, SIAM J. Sci. Stat. Comput. 2, 1–4 (1981; Zbl 0474.65020)]. For actual implementation, a pointwise variant of this modified block SSOR preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness. Numerical experiments show that the proposed modified SSOR preconditioned symmetric QMR solver can achieve faster convergence in terms of total runtime than several effective preconditioners published in the recent literature. Moreover, the proposed modified SSOR preconditioners can be generalized to non-symmetric Biot’s systems.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74L10 Soil and rock mechanics
65F50 Computational methods for sparse matrices
Software:
QMRPACK
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References:
[1] . Critical State Soil Mechanics via Finite Elements. Ellis Horwood Ltd.: Chichester, West Sussex, 1952.
[2] Phoon, International Journal for Numerical Methods in Engineering 55 pp 377– (2002)
[3] Iterative Methods for Sparse Linear Systems. PWS Publishing Company: Boston, MA, 1996.
[4] Paige, SIAM Journal on Numerical Analysis 12 pp 617– (1975)
[5] . A new Krylov-subspace method for symmetric indefinite linear system. In Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, (ed.). IMACS: New Brunswick, NJ, 1994; 1253-1256.
[6] Chan, International Journal for Numerical and Analytical Methods in Geomechanics 25 pp 1001– (2001)
[7] Toh, International Journal for Numerical Methods in Engineering 60 pp 1361– (2004)
[8] Phoon, Computers and Structures 82 pp 2401– (2004)
[9] Eisenstat, SIAM Journal on Scientific and Statistical Computing 2 pp 1– (1981)
[10] Mroueh, International Journal for Numerical and Analytical Methods in Geomechanics 23 pp 1961– (1999)
[11] . 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, Mathematical Programming 76 pp 183– (1997)
[13] . Approximate and incomplete factorizations. Technical Report 871, Department of Mathematics, University of Utrecht, 1994.
[14] Iterative Solution Methods. Cambridge University Press: New York, NY, 1995.
[15] Chow, ACM Transactions on Mathematical Software 24 pp 159– (1998)
[16] , , , , , , , , . Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Press: Philadelphia, PA, 1994. · doi:10.1137/1.9781611971538
[17] Duff, BIT 29 pp 635– (1989)
[18] On the existence problem of incomplete factorisation methods. Lapack Working Note 144, UT-CS-99-435, 1999.
[19] Chow, Journal of Computational and Applied Mathematics 86 pp 387– (1997)
[20] . 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, SIAM Journal on Scientific Computing 21 pp 1969– (2000)
[22] Axelsson, BIT 13 pp 443– (1972)
[23] Meijerink, Mathematics of Computation 31 pp 148– (1977)
[24] Gustafsson, BIT 18 pp 142– (1978)
[25] Gambolati, International Journal for Numerical and Analytical Methods in Geomechanics 25 pp 1429– (2001)
[26] Iterative Methods for Large Linear Systems. Cambridge University Press: Cambridge, MA, 2003. · doi:10.1017/CBO9780511615115
[27] Toh, SIAM Journal on Optimization 14 pp 670– (2003)
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