Comparison of Two Types of Preconditioners for Stokes and Linearized Navier-Stokes Equations
To solve saddle point systems efficiently, several preconditioners have been published. There are many methods for constructing preconditioners for linear systems from saddle point problems, for instance, the relaxed dimensional factorization (RDF) preconditioner and the augmented Lagrangian (AL) preconditioner are used for both steady and unsteady Navier-Stokes equations. In this paper we compare the RDF preconditioner with the modified AL (MAL) preconditioner to show which is more effective to solve Navier-Stokes equations. Numerical experiments indicate that the MAL preconditioner is more efficient and robust, especially, for moderate viscosities and stretched grids in steady problems. For unsteady cases, the convergence rate of the RDF preconditioner is slightly faster than the MAL perconditioner in some circumstances, but the parameter of the RDF preconditioner is more sensitive than the MAL preconditioner. Moreover the convergence rate of the MAL preconditioner is still quite acceptable. Therefore we conclude that the MAL preconditioner is more competitive than the RDF preconditioner. These experiments are implemented with IFISS package.
<p>[1] Y. Saad, Iterative Methods for Sparse Linear Systems (2nd edn), SIAM,
Philadelphia, PA, 2003.
[2] H. C. Elman, D. J. Silvester, A. J. Wathen, Finite Elements and Fast
Iterative Solvers: With Applications in Incompressible Fluid Dynamics,
Numer. Math. Sci. Comput., Oxford University Press, Oxford, UK, 2005.
[3] M. Fortin, R. Glowinski, Augmented Lagrangian Methods: Applications
to the Numerical Solution of Boundary Value Problems, Stud. Math.
Appl. 15, North-Holland, Amsterdam, New York, Oxford, 1983.
[4] A. Segal, M. ur Rehman, C. Vuik, Preconditioners for incompressible
Navier-Stokes solvers, Numer. Math. Theor. Meth. Appl., 2010, 3(3):
245-275.
[5] M. Benzi, G. H. Golub, J. Liesen, Numerical solution of saddle point
problems, Acta Numer., 2005, 14: 1-137.
[6] M. Benzi, M. Ng, Q. Niu, Z. Wang, A relaxed dimensional factorization
preconditioner for the incompressible Navier-Stokes equations, J.
Comput. Phys., 2011, 230(6): 6185-6202.
[7] M. Benzi, D. B. Szyld, Existence and uniqueness of splittings for
stationary iterative methods with applications to alternating methods,
Numer. Math., 1997, 76: 309-321.
[8] M. Benzi, X.-P. Guo, A dimensional split preconditioner for Stokes and
linearized Navier-Stokes equations, Appl. Numer. Math., 2011, 61: 66-
76.
[9] M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting
iteration, SIAM J. Matrix Anal. Appl., 2009, 31(2): 360-374.
[10] M. Benzi, M. A. Olshanskii, An augmented Lagrangian-based approach
to the Oseen problem, SIAM J. Sci. Comput., 2006, 28: 2095-2113.
[11] M. Benzi, M. A. Olshanskii, Z. Wang, Modified augmented Lagrangian
preconditioners for the incompressible Navier-Stokes equations, Int. J.
Numer. Meth. Fluids, 2011, 66(4): 486-508.
[12] M. Benzi, Z. Wang, Analysis of augmented Lagrangian-based preconditioners
for the steady incompressible Navier-Stokes equations, SIAM
J. Sci. Comput., 2011, 33(5): 2761-2784.
[13] Y. Cao, M.-Q. Jiang, Y.-L. Zheng, A splitting preconditioner for saddle
point problems, Numer. Linear Algebra Appl., 2011, 18: 875-895.
[14] P. R. Amestoy, T. A. Davis, I. S. Duff, Algorithm 837: AMD, an
approximate minimum degree ordering algorithm, ACM Trans. Math.
Software, 2004, 30: 381-388.
[15] H. C. Elman, A. Ramage, D. J. Silvester, Algorithm 886: IFISS, a Matlab
toolbox for modeling incompressible flow, ACM Trans. Math. Software,
2007, 33.
[16] G. H. Golub, C. Greif, On solving block-structured indefinite linear
systems, SIAM J. Sci. Comput., 2003, 24: 2076-2092.
[17] M. F. Murphy, G. H. Golub, A. J. Wathen, A note on preconditioning
for indefinite linear systems, SIAM J. Matrix Anal., 2000, 21:1300-1317
[18] M. Benzi, J. Liu, Block preconditioning for saddle point systems with
indefinite (1, 1) block, Int. J. Comput. Math., 2007, 84(8): 1117-1129.
[19] M. ur Rehman, C. Vuik, G. Segal, A comparison of preconditioners for
incompressible Navier-Stokes solvers, Int. J. Numer. Meth. Fluids, 2008,
57: 1731-1751.</p>
<p>[1] Y. Saad, Iterative Methods for Sparse Linear Systems (2nd edn), SIAM,
Philadelphia, PA, 2003.
[2] H. C. Elman, D. J. Silvester, A. J. Wathen, Finite Elements and Fast
Iterative Solvers: With Applications in Incompressible Fluid Dynamics,
Numer. Math. Sci. Comput., Oxford University Press, Oxford, UK, 2005.
[3] M. Fortin, R. Glowinski, Augmented Lagrangian Methods: Applications
to the Numerical Solution of Boundary Value Problems, Stud. Math.
Appl. 15, North-Holland, Amsterdam, New York, Oxford, 1983.
[4] A. Segal, M. ur Rehman, C. Vuik, Preconditioners for incompressible
Navier-Stokes solvers, Numer. Math. Theor. Meth. Appl., 2010, 3(3):
245-275.
[5] M. Benzi, G. H. Golub, J. Liesen, Numerical solution of saddle point
problems, Acta Numer., 2005, 14: 1-137.
[6] M. Benzi, M. Ng, Q. Niu, Z. Wang, A relaxed dimensional factorization
preconditioner for the incompressible Navier-Stokes equations, J.
Comput. Phys., 2011, 230(6): 6185-6202.
[7] M. Benzi, D. B. Szyld, Existence and uniqueness of splittings for
stationary iterative methods with applications to alternating methods,
Numer. Math., 1997, 76: 309-321.
[8] M. Benzi, X.-P. Guo, A dimensional split preconditioner for Stokes and
linearized Navier-Stokes equations, Appl. Numer. Math., 2011, 61: 66-
76.
[9] M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting
iteration, SIAM J. Matrix Anal. Appl., 2009, 31(2): 360-374.
[10] M. Benzi, M. A. Olshanskii, An augmented Lagrangian-based approach
to the Oseen problem, SIAM J. Sci. Comput., 2006, 28: 2095-2113.
[11] M. Benzi, M. A. Olshanskii, Z. Wang, Modified augmented Lagrangian
preconditioners for the incompressible Navier-Stokes equations, Int. J.
Numer. Meth. Fluids, 2011, 66(4): 486-508.
[12] M. Benzi, Z. Wang, Analysis of augmented Lagrangian-based preconditioners
for the steady incompressible Navier-Stokes equations, SIAM
J. Sci. Comput., 2011, 33(5): 2761-2784.
[13] Y. Cao, M.-Q. Jiang, Y.-L. Zheng, A splitting preconditioner for saddle
point problems, Numer. Linear Algebra Appl., 2011, 18: 875-895.
[14] P. R. Amestoy, T. A. Davis, I. S. Duff, Algorithm 837: AMD, an
approximate minimum degree ordering algorithm, ACM Trans. Math.
Software, 2004, 30: 381-388.
[15] H. C. Elman, A. Ramage, D. J. Silvester, Algorithm 886: IFISS, a Matlab
toolbox for modeling incompressible flow, ACM Trans. Math. Software,
2007, 33.
[16] G. H. Golub, C. Greif, On solving block-structured indefinite linear
systems, SIAM J. Sci. Comput., 2003, 24: 2076-2092.
[17] M. F. Murphy, G. H. Golub, A. J. Wathen, A note on preconditioning
for indefinite linear systems, SIAM J. Matrix Anal., 2000, 21:1300-1317
[18] M. Benzi, J. Liu, Block preconditioning for saddle point systems with
indefinite (1, 1) block, Int. J. Comput. Math., 2007, 84(8): 1117-1129.
[19] M. ur Rehman, C. Vuik, G. Segal, A comparison of preconditioners for
incompressible Navier-Stokes solvers, Int. J. Numer. Meth. Fluids, 2008,
57: 1731-1751.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65397", author = "Ze-Jun Hu and Ting-Zhu Huang and Ning-Bo Tan", title = "Comparison of Two Types of Preconditioners for Stokes and Linearized Navier-Stokes Equations", abstract = "To solve saddle point systems efficiently, several preconditioners have been published. There are many methods for constructing preconditioners for linear systems from saddle point problems, for instance, the relaxed dimensional factorization (RDF) preconditioner and the augmented Lagrangian (AL) preconditioner are used for both steady and unsteady Navier-Stokes equations. In this paper we compare the RDF preconditioner with the modified AL (MAL) preconditioner to show which is more effective to solve Navier-Stokes equations. Numerical experiments indicate that the MAL preconditioner is more efficient and robust, especially, for moderate viscosities and stretched grids in steady problems. For unsteady cases, the convergence rate of the RDF preconditioner is slightly faster than the MAL perconditioner in some circumstances, but the parameter of the RDF preconditioner is more sensitive than the MAL preconditioner. Moreover the convergence rate of the MAL preconditioner is still quite acceptable. Therefore we conclude that the MAL preconditioner is more competitive than the RDF preconditioner. These experiments are implemented with IFISS package.
", keywords = "Navier-Stokes equations, Krylov subspace method,
preconditioner, dimensional splitting, augmented Lagrangian preconditioner.", volume = "7", number = "5", pages = "911-6", }
