Block triangular preconditioners for symmetric saddle-point problems.(English)Zbl 1053.65033
Author’s abstract: We study the spectral properties and the computational performance of a block triangular preconditioner for the solution of the general symmetric saddle-point problem. We provide estimates for the region containing both the nonreal and the real eigenvalues. Moreover, we show that an indefinite inner product can be employed to devise an efficient short-term recurrence Krylov subspace solver to be used with the analyzed preconditioner. Numerical experiments on a variety of application problems are also reported.
Reviewer: Xavier Antoine (Pasadena)
MSC:
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 65F10 | Iterative numerical methods for linear systems |
Keywords:
saddle-point systems;iterative methods;preconditioning;Krylov subspace;numerical experimentsCite
Full Text:DOI
References:
| [1] | AEA Technology, Harwell Subroutine Library, Harwell Laboratory, Oxfordshire, December 1995; AEA Technology, Harwell Subroutine Library, Harwell Laboratory, Oxfordshire, December 1995 |
| [2] | Axelsson, O., Preconditioning of indefinite problems by regularization, SIAM J. Numer. Anal., 16, 58-69 (1979) ·Zbl 0416.65071 |
| [3] | Axelsson, O.; Barker, V. A., Finite Element Solution of Boundary Value Problems (1984), Academic Press: Academic Press Orlando, FL ·Zbl 0537.65072 |
| [4] | O. Axelsson, M. Neytcheva, Preconditioning methods for constrained optimization problems with applications for the linear elasticity equations, Tech. Rep., Department of Mathematics, University of Nijmegen, Nijmegen, 2003; O. Axelsson, M. Neytcheva, Preconditioning methods for constrained optimization problems with applications for the linear elasticity equations, Tech. Rep., Department of Mathematics, University of Nijmegen, Nijmegen, 2003 ·Zbl 1071.65527 |
| [5] | Axelsson, O.; Neytcheva, M., Preconditioning methods for linear systems arising in constrained optimization problems, Numer. Linear Algebra Appl., 10, 3-31 (2003) ·Zbl 1071.65527 |
| [6] | D. Boffi, L. Gastaldi, Analysis of finite element approximation of evolution problems in mixed form, Tech. Rep. 1286, IAN-CNR, 2002; D. Boffi, L. Gastaldi, Analysis of finite element approximation of evolution problems in mixed form, Tech. Rep. 1286, IAN-CNR, 2002 ·Zbl 1080.65089 |
| [7] | Bramble, J. H.; Pasciak, J. E., A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp., 50, 1-17 (1988) ·Zbl 0643.65017 |
| [8] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer: Springer New York ·Zbl 0788.73002 |
| [9] | E. Chow, Y. Saad, Private communication, 1998; E. Chow, Y. Saad, Private communication, 1998 |
| [10] | Durazzi, C.; Ruggiero, V., Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems, Numer. Linear Algebra Appl., 10, 8, 673-688 (2003) ·Zbl 1071.65512 |
| [11] | Elman, H., Multigrid and Krylov subspace methods for the discrete Stokes equations, Internat. J. Numer. Methods Fluids, 22, 755-770 (1996) ·Zbl 0865.76078 |
| [12] | Elman, H.; Silvester, D.; Wathen, A., Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90, 665-688 (2002) ·Zbl 1143.76531 |
| [13] | Elman, H. C.; Silvester, D. J.; Wathen, A. J., Finite Elements and Fast Iterative Solvers (2003), Oxford University Press: Oxford University Press Oxford |
| [14] | Fischer, B.; Ramage, A.; Silvester, D. J.; Wathen, A. J., Minimum residual methods for augmented systems, BIT, 38, 527-543 (1998) ·Zbl 0914.65026 |
| [15] | D.R. Fokkema, Subspace Methods for Linear, Nonlinear, and Eigen Problems, Ph.D. Thesis, Utrecht University, Utrecht, 1996; D.R. Fokkema, Subspace Methods for Linear, Nonlinear, and Eigen Problems, Ph.D. Thesis, Utrecht University, Utrecht, 1996 |
| [16] | Freund, R. W., Quasi-kernel polynomials and convergence results for quasi-minimal residual iterations, (Braess, D.; Schumaker, L. L., Numerical Methods in Approximation Theory, vol. 9 (1992), Birkhäuser: Birkhäuser Basel), 77-95 ·Zbl 0814.65035 |
| [17] | Freund, R. W.; Nachtigal, N. M., Software for simplified Lanczos and QMR algorithms, Appl. Numer. Math., 19, 319-341 (1995) ·Zbl 0853.65041 |
| [18] | Golub, G. H.; Wathen, A. J., An iteration for indefinite systems and its application to the Navier-Stokes equations, SIAM J. Sci. Comput., 19, 530-539 (1998) ·Zbl 0912.76053 |
| [19] | Johnson, C.; Thomée, V., Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér., 15, 41-78 (1981) ·Zbl 0476.65074 |
| [20] | Klawonn, A., Block-triangular preconditioners for saddle-point problems with a penalty term, SIAM J. Sci. Comput., 19, 172-184 (1998) ·Zbl 0917.73069 |
| [21] | Klawonn, A., An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput., 19, 540-552 (1998) ·Zbl 0912.65018 |
| [22] | Klawonn, A.; Starke, G., Block triangular preconditioners for nonsymmetric saddle point problems: Field-of-values analysis, Numer. Math., 81, 577-594 (1999) ·Zbl 0922.65021 |
| [23] | Krzyzanowski, P., On block preconditioners for nonsymmetric saddle point problems, SIAM J. Sci. Comput., 23, 157-169 (2001) ·Zbl 0998.65048 |
| [24] | The MathWorks, Inc., MATLAB User’s Guide, Natick, MA 01760, May 2001; The MathWorks, Inc., MATLAB User’s Guide, Natick, MA 01760, May 2001 |
| [25] | Murphy, M. F.; Golub, G. H.; Wathen, A. J., A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21, 1969-1972 (2000) ·Zbl 0959.65063 |
| [26] | Perugia, I.; Simoncini, V., Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations, Numer. Linear Algebra Appl., 7, 585-616 (2000) ·Zbl 1051.65038 |
| [27] | Rusten, T.; Winther, R., A preconditioned iterative method for saddle point problems, SIAM J. Matrix Anal. Appl., 13, 887-904 (1992) ·Zbl 0760.65033 |
| [28] | Saad, Y., Iterative Methods for Sparse Linear Systems (1996), The PWS Publishing Company: The PWS Publishing Company Boston, MA ·Zbl 1002.65042 |
| [29] | Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7, 856-869 (1986) ·Zbl 0599.65018 |
| [30] | Silvester, D.; Wathen, A., Fast iterative solution of stabilized Stokes systems, Part II: Using general block preconditioners, SIAM J. Numer. Anal., 31, 1352-1367 (1994) ·Zbl 0810.76044 |
| [31] | Sleijpen, G. L.G.; Fokkema, D. R., Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum, ETNA Electronic Trans. Numer. Anal., 1, 11-32 (1993) ·Zbl 0820.65016 |
| [32] | Wathen, A.; Fischer, B.; Silvester, D., The convergence rate of the minimal residual method for the Stokes problem, Numer. Math., 71, 121-134 (1995) ·Zbl 0837.65026 |
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