A block triangular preconditioner for a class of three-by-three block saddle point problems. (English) Zbl 1515.65070
Summary: This paper deals with solving a class of three-by-three block saddle point problems. The systems are solved by preconditioning techniques. Based on an iterative method, we construct a block upper triangular preconditioner. The convergence of the presented method is studied in details. Finally, some numerical experiments are given to demonstrate the superiority of the proposed preconditioner over some existing ones.
MSC:
| 65F08 | Preconditioners for iterative methods |
| 65F10 | Iterative numerical methods for linear systems |
| 65F50 | Computational methods for sparse matrices |
References:
| [1] | Abdolmaleki, M.; Karimi, S.; Salkuyeh, DK, A new block-diagonal preconditioner for a class of \(3 \times 3\) block saddle point problems, Mediterr. J. Math., 19, 1-15 (2022) · Zbl 1481.65048 · doi:10.1007/s00009-021-01973-5 |
| [2] | Assous, F.; Degond, P.; Heintze, E.; Raviart, PA; Segre, J., On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys., 109, 222-237 (1993) · Zbl 0795.65087 · doi:10.1006/jcph.1993.1214 |
| [3] | Beik, FPA; Benzi, M., Iterative methods for double saddle point systems, SIAM J. Matrix Anal. Appl., 39, 902-921 (2018) · Zbl 1391.65062 · doi:10.1137/17M1121226 |
| [4] | Beik, FPA; Benzi, M., Block preconditioners for saddle point systems arising from liquid crystal directors modeling, CALCOLO, 55, 29 (2018) · Zbl 1416.65079 · doi:10.1007/s10092-018-0271-6 |
| [5] | Benzi, M., Preconditioning techniques for large linear systems: a survey, J. Comput. Phys., 182, 418-477 (2002) · Zbl 1015.65018 · doi:10.1006/jcph.2002.7176 |
| [6] | Benzi, M.; Golub, GH; Liesen, J., Numerical Solution of Saddle Point Problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034 · doi:10.1017/S0962492904000212 |
| [7] | Bertsekas, DP, Nonlinear Programming (1999), Nashua: Athena Scientic, Nashua · Zbl 1015.90077 |
| [8] | Cao, Z-H, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math., 57, 899-910 (2007) · Zbl 1118.65021 · doi:10.1016/j.apnum.2006.08.001 |
| [9] | Cao, Y., Shift-splitting preconditioners for a class of block three-by-three saddle point problems, Appl. Math. Lett., 96, 40-46 (2019) · Zbl 1524.65120 · doi:10.1016/j.aml.2019.04.006 |
| [10] | Cao, Y., A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models, Comput. Math. Appl., 77, 1124-1143 (2019) · Zbl 1442.65039 · doi:10.1016/j.camwa.2018.10.046 |
| [11] | Cao, Y.; Du, J.; Niu, Q., Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math., 270, 239-250 (2014) · Zbl 1303.65012 · doi:10.1016/j.cam.2014.05.017 |
| [12] | Cao, Y.; Li, S.; Yao, L., A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems, Appl. Math. Lett., 49, 20-27 (2015) · Zbl 1382.65075 · doi:10.1016/j.aml.2015.04.001 |
| [13] | Cao, Y.; Miao, S-X; Ren, Z-R, On preconditioned generalized shift-splitting iteration methods for saddle point problems, Comput. Math. Appl., 74, 859-872 (2017) · Zbl 1384.65025 · doi:10.1016/j.camwa.2017.05.031 |
| [14] | Chen, C-R; Ma, C-F, A generalized shift-splitting preconditioner for singular saddle point problems, Appl. Math. Comput., 269, 947-955 (2015) · Zbl 1410.65096 |
| [15] | Chen, Z-M; Du, Q.; Zou, J., Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer Anal., 37, 1542-1570 (2000) · Zbl 0964.78017 · doi:10.1137/S0036142998349977 |
| [16] | Ciarlet, P.; Zou, J., Finite element convergence for the Darwin model to Maxwell’s equations, RAIRO Math. Modelling Numer. Anal., 31, 213-249 (1997) · Zbl 0887.65121 · doi:10.1051/m2an/1997310202131 |
| [17] | Elman, HC; Silvester, DJ; Wathen, AJ, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90, 665-688 (2002) · Zbl 1143.76531 · doi:10.1007/s002110100300 |
| [18] | Estrin, R.; Greif, C., Towards an optimal condition number of certain augmented Lagrangian-type saddle-point matrices, Numer. Linear Algebra Appl., 23, 693-705 (2016) · Zbl 1413.65163 · doi:10.1002/nla.2050 |
| [19] | Gould, NIM; Orban, D.; Toint, PL, CUTEr and SifDec, a constrained and unconstrained testing environment, revisited, ACM Trans. Math. Softw., 29, 373-394 (2003) · Zbl 1068.90526 · doi:10.1145/962437.962439 |
| [20] | Han, DR; Yuan, XM, Local linear convergence of the alternating direction method of multipliers for quadratic programs, SIAM J. Numer. Anal., 51, 3446-3457 (2013) · Zbl 1285.90033 · doi:10.1137/120886753 |
| [21] | Huang, N., Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems, Numer. Algor., 85, 1233-1254 (2020) · Zbl 1455.65049 · doi:10.1007/s11075-019-00863-y |
| [22] | Huang, N.; Ma, C-F, Spectral analysis of the preconditioned system for the \(3 \times 3\) block saddle point problem, Numer. Algor., 81, 421-444 (2019) · Zbl 1454.65019 · doi:10.1007/s11075-018-0555-6 |
| [23] | Ke, Y-F; Ma, C-F, The parameterized preconditioner for the generalized saddle point problems from the incompressible Navier-Stokes equations, J. Comput. Appl. Math., 37, 3385-3398 (2018) · Zbl 1451.65031 |
| [24] | Paige, CC; Saunders, MA, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12, 617-629 (1975) · Zbl 0319.65025 · doi:10.1137/0712047 |
| [25] | Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469 (1993) · Zbl 0780.65022 · doi:10.1137/0914028 |
| [26] | Saad, Y., Iterative Methods for Sparse Linear Systems (1995), New York: PWS Press, New York · Zbl 1002.65042 |
| [27] | Salkuyeh, DK; Rahimian, M., A modification of the generalized shift-splitting method for singular saddle point problems, Comput. Math. Appl., 74, 2940-2949 (2017) · Zbl 1398.65050 · doi:10.1016/j.camwa.2017.07.029 |
| [28] | Salkuyeh, DK; Masoudi, M.; Hezari, D., On the generalized shift-splitting preconditioner for saddle point problems, Appl. Math. Lett., 48, 55-61 (2015) · Zbl 1325.65048 · doi:10.1016/j.aml.2015.02.026 |
| [29] | Shen, Q-Q; Shi, Q., Generalized shift-splitting preconditioners for nonsingular and singular generalized saddle point problems, Comput. Math. Appl., 72, 632-641 (2016) · Zbl 1359.65044 · doi:10.1016/j.camwa.2016.05.022 |
| [30] | Xie, X.; Li, H-B, A note on preconditioning for the \(3 \times 3\) block saddle point problem, Comput. Math. Appl., 79, 3289-3296 (2020) · Zbl 1452.65054 · doi:10.1016/j.camwa.2020.01.022 |
| [31] | Young, DM, Iterative Solution or Large Linear Systems (1971), New York: Academic Press, New York · Zbl 0231.65034 |
| [32] | Yuan, J-Y, Numerical methods for generalized least squares problems, J. Comput. Appl. Math., 66, 571-584 (1996) · Zbl 0858.65043 · doi:10.1016/0377-0427(95)00167-0 |
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