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Stabilized dimensional factorization preconditioner for solving incompressible Navier-Stokes equations. (English) Zbl 1433.65331
Summary: In this paper, we propose a stabilized dimensional factorization (SDF) preconditioner for saddle point problems arising from the discretization of Navier-Stokes equations. The idea is based on regularization, block factorization and selective approximation. The spectral properties of the preconditioned matrix are analyzed in details. Based on the analysis, we prescribe a reasonable choice of the regularization matrix \(W\) in the preconditioner. By using the connection with the RDF preconditioner, we determine the relaxation parameter \({\alpha}\) for the problems discretized by uniform grids and stretched grids, respectively. Finally, numerical experiments on the finite element discretizations of both steady and unsteady incompressible flow problems show that the SDF preconditioner is more efficient and robust than the RDF preconditioner, which has been illustrated very competitive with some existing preconditioners.
MSC:
65F08 Preconditioners for iterative methods
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
Software:
IFISS; KKTDirect
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[1] Amestoy, P. R.; Davis, T. A.; Duff, I. S., An approximate minimum degree ordering algorithm, SIAM J. Matrix Anal. Appl., 17, 886-905, (1996) · Zbl 0861.65021
[2] Axelsson, O.; Neytcheva, M., Eigenvalue estimates for preconditioned saddle point matrices, Numer. Linear Algebra Appl., 13, 339-360, (2006) · Zbl 1224.65080
[3] Axelsson, O.; Neytcheva, M., A general approach to analyse preconditioners for two-by-two block matrices, Numer. Linear Algebra Appl., 20, 723-742, (2013) · Zbl 1313.65048
[4] Axelsson, O.; Blaheta, R.; Neytcheva, M., Preconditioning for boundary value problems using elementwise Schur complements, SIAM J. Matrix Anal. Appl., 31, 767-789, (2009) · Zbl 1194.65047
[5] Bai, Z.-Z.; Golub, G. H.; Ng, M. K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24, 603-626, (2003) · Zbl 1036.65032
[6] Benzi, M., Preconditioning techniques for large linear systems: a survey, J. Comput. Phys., 182, 418-477, (2002) · Zbl 1015.65018
[7] Benzi, M.; Golub, G. H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26, 20-41, (2004) · Zbl 1082.65034
[8] Benzi, M.; Guo, X-. P., A dimensional splitting preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math., 61, 66-76, (2011) · Zbl 1302.65074
[9] Benzi, M.; Olshanskii, M. A., An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput., 28, 2095-2113, (2006) · Zbl 1126.76028
[10] Benzi, M.; Olshanskii, M. A., Field-of-values convergence analysis of augmented Lagrangian preconditioners for the linearized Navier-Stokes problem, SIAM J. Numer. Anal., 49, 770-788, (2011) · Zbl 1245.76044
[11] Benzi, M.; Wang, Z., Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 33, 2761-2784, (2011) · Zbl 1298.76114
[12] Benzi, M.; Wang, Z., A parallel implementation of the modified augmented Lagrangian preconditioner for the incompressible Navier-Stokes equations, Numer. Algorithms, 64, 73-84, (2013) · Zbl 1426.76222
[13] Benzi, M.; Wathen, A. J., Some preconditioning techniques for saddle point problems, (Schilders, W.; Van der Vorst, H. A.; Rommes, J., Model Order Reduction: Theory, Research Aspects and Applications. Model Order Reduction: Theory, Research Aspects and Applications, Mathematics in Industry, (2008), Springer-Verlag), 195-211 · Zbl 1152.65425
[14] Benzi, M.; Deparis, S.; Grandperrin, G.; Quarteroni, A., Parameter estimates for the relaxed dimensional factorization preconditioner and application to hemodynamics, Comput. Methods Appl. Mech. Eng., 300, 129-145, (2016) · Zbl 1423.76212
[15] Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137, (2005) · Zbl 1115.65034
[16] Benzi, M.; Ng, M. K.; Niu, Q.; Wang, Z., A relaxed dimensional factorization preconditioner for the incompressible Navier Stokes equations, J. Comput. Phys., 230, 6185-6202, (2011) · Zbl 1419.76433
[17] Benzi, M.; Olshanskii, M. A.; Wang, Z., Modified augmented Lagrangian preconditioners for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 66, 486-508, (2011) · Zbl 1421.76152
[18] R. Bridson, KKTDirect: a direct solver package for saddle-point (KKT) matrices, 2009, Priprint.
[19] Cao, Z.-H., Comparison of performance of iterative methods for singular and nonsingular saddle point linear systems arising from Navier-Stokes equations, Appl. Math. Comput., 174, 630-642, (2006) · Zbl 1089.65024
[20] Cao, Y.; Yao, L.-Q.; Jiang, M.-Q.; Niu, Q., A relaxed HSS preconditioner for saddle point problems from meshfree discretization, J. Comput. Math., 31, 398-421, (2013) · Zbl 1299.65043
[21] Dolean, V.; Jolivet, P.; Nataf, F., An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation, (2015), SIAM · Zbl 1364.65277
[22] Elman, H. C.; Ramage, A.; Silvester, D. J., IFISS: a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw., 33, (2007), Artical 14 · Zbl 1365.65326
[23] Elman, H. C.; Wathen, A. J.; Silvester, D. J., Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford Series in Numerical Mathematics and Scientific Computation, (2005), Oxford University Press: Oxford University Press Oxford · Zbl 1083.76001
[24] Fortin, M.; Glowinsky, R., Augmented Lagrangian Methods Application to the Solution of Boundary Value Problems, Stud. Math. Appl., vol. 15, (1983), North-Holland: North-Holland Amsterdam-New York
[25] Gander, M.; Niu, Q.; Xu, Y. X., Analysis of dimension-wise splitting iteration with selective relaxation for saddle point problem, BIT, 56, 441-465, (2016) · Zbl 1342.76074
[26] 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
[27] Gould, N. I.M.; Hribar, M. E.; Nocedal, J., On the solution of equality constrained quadratic programming problems arising in optimization, SIAM J. Sci. Comput., 23, 1375-1394, (2001)
[28] He, X.; Neytcheva, M.; Serra Capizzano, S., On an augmented Lagrangian-based preconditioning of Oseen type problems, BIT, 51, 865-888, (2011) · Zbl 1269.65030
[29] He, X.; Vuik, C.; Klaij, C., Block-preconditioners for the incompressible Navier-Stokes equations discretized by a finite volume method, J. Numer. Math., 25, 89-105, (2017) · Zbl 1367.65050
[30] He, X.; Vuik, C.; Klaij, C., Combining the augmented Lagrangian preconditioner with the simple Schur complement approximation, SIAM J. Sci. Comput., 40, A1362-A1385, (2018) · Zbl 1392.65053
[31] Ipsen, I. C.F., A note on preconditioning nonsymmetric matrices, SIAM J. Sci. Comput., 23, 1050-1051, (2001) · Zbl 0998.65049
[32] Keller, C.; Gould, N. I.M.; Wathen, A. J., Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21, 1300-1317, (2000) · Zbl 0960.65052
[33] Lacoursière, C.; Linde, M.; Sabelstrom, O., Direct sparse factorization of blocked saddle point matrices, Lect. Notes Comput. Sci., 7134, 324-335, (2012)
[34] Lukšan, L.; Vlček, J., Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems, Numer. Linear Algebra Appl., 5, 219-247, (1998) · Zbl 0937.65066
[35] 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
[36] 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
[37] Niet, A. C.De.; Wubs, F. W., Numerically stable \(LDL^{ T }\)-factorization of F-type saddle point matrices, IMA J. Numer. Appl., 29, 208-234, (2009) · Zbl 1161.65022
[38] 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
[39] Rusten, T.; Winther, R., A preconditioned method for saddle point problems, SIAM J. Matrix Anal. Appl., 13, 887-904, (1992) · Zbl 0760.65033
[40] Saad, Y., Iterative Methods for Sparse Linear Systems, (1996), PWS Publishing Company: PWS Publishing Company Boston · Zbl 1002.65042
[41] Schmid, J., A remark on characteristic polynomials, Am. Math. Mon., 77, 998-999, (1970) · Zbl 0206.03902
[42] Vuik, C., Solution of the discretized incompressible Navier-Stokes equations with the GMRES method, Int. J. Numer. Methods Fluids, 16, 507-523, (1993) · Zbl 0825.76552
[43] Wathen, A. J., Preconditioning, Acta Numer., 24, 329-376, (2015) · Zbl 1316.65039
[44] Wilkinson, J. H., The Algebraic Eigenvalue Problem, (1965), Clarendon Press: Clarendon Press Oxford · Zbl 0258.65037
[45] Zhang, F. Z., Matrix Theory: Basic Results and Techniques, (2011), Springer
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