Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
three-by-three block saddle-point problems; matrix splitting; convergence; preconditioning, GMRES method
three-by-three block saddle-point problems; matrix splitting; convergence; preconditioning, GMRES method
Summary:
In this paper, to solve the three-by-three block saddle-point problem, a new block triangular (NBT) preconditioner is established, which can effectively avoid the solving difficulty that the coefficient matrices of linear subsystems are Schur complement matrices when the block preconditioner is applied to the Krylov subspace method. Theoretical analysis shows that the iteration method produced by the NBT preconditioner is unconditionally convergent. Besides, some spectral properties are also discussed. Finally, numerical experiments are provided to show the effectiveness of the NBT preconditioner.
References:
[1] Abdolmaleki, M., Karimi, S., Salkuyeh, D. K.: A new block-diagonal preconditioner for a class of $3 \times 3$ block saddle point problems. Mediterr. J. Math. 19 (2022), Article ID 43, 15 pages. DOI 10.1007/s00009-021-01973-5 | MR 4371180 | Zbl 1481.65048
[2] Aslani, H., Salkuyeh, D. K.: Semi-convergence of the APSS method for a class of nonsymmetric three-by-three singular saddle point problems. Available at https://arxiv.org/abs/2208.00814 (2022), 17 pages. MR 4591480
[3] Aslani, H., Salkuyeh, D. K., Beik, F. P. A.: On the preconditioning of three-by-three block saddle point problems. Filomat 15 (2021), 5181-5194. DOI 10.2298/FIL2115181A | MR 4394237
[4] Cao, Y.: Shift-splitting preconditioners for a class of block three-by-three saddle point problems. Appl. Math. Lett. 96 (2019), 40-46. DOI 10.1016/j.aml.2019.04.006 | MR 3946364 | Zbl 07111438
[5] Cao, Y.: A block positive-semidefinite splitting preconditioner for generalized saddle point linear systems. J. Comput. Appl. Math. 374 (2020), Article ID 112787, 15 pages. DOI 10.1016/j.cam.2020.112787 | MR 4067982 | Zbl 1434.65087
[6] Degond, P., Raviart, P.-A.: An analysis of the Darwin model of approximation to Maxwell's equations. Forum Math. 4 (1992), 13-44. DOI 10.1515/form.1992.4.13 | MR 1142472 | Zbl 0755.35137
[7] Elman, H. C., Ramage, A., Silvester, D. J.: Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33 (2007), Article ID 14, 18 pages. DOI 10.1145/1236463.1236469 | MR 2326956 | Zbl 1365.65326
[8] Han, D., Yuan, X.: Local linear convergence of the alternating direction method of multipliers for quadratic programs. SIAM J. Numer. Anal. 51 (2013), 3446-3457. DOI 10.1137/120886753 | MR 3143838 | Zbl 1285.90033
[9] Hu, K., Xu, J.: Structure-preserving finite element methods for stationary MHD models. Math. Comput. 88 (2019), 553-581. DOI 10.1090/mcom/3341 | MR 3882276 | Zbl 1405.65151
[10] Huang, N.: Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems. Numer. Algorithms 85 (2020), 1233-1254. DOI 10.1007/s11075-019-00863-y | MR 4190815 | Zbl 1455.65049
[11] Huang, N., Dai, Y.-H., Hu, Q.: Uzawa methods for a class of block three-by-three saddle-point problems. Numer. Linear Algebra Appl. 26 (2019), Article ID e2265, 26 pages. DOI 10.1002/nla.2265 | MR 4033762 | Zbl 1463.65046
[12] Huang, N., Ma, C.-F.: Spectral analysis of the preconditioned system for the $3 \times 3$ block saddle point problem. Numer. Algorithms 81 (2019), 421-444. DOI 10.1007/s11075-018-0555-6 | MR 3953154 | Zbl 1454.65019
[13] Huang, Y.-M.: A practical formula for computing optimal parameters in the HSS iteration methods. J. Comput. Appl. Math. 255 (2014), 142-149. DOI 10.1016/j.cam.2013.01.023 | MR 3093411 | Zbl 1291.65100
[14] Huang, Z.-G., Wang, L.-G., Xu, Z., Cui, J.-J.: An efficient preconditioned variant of the PSS preconditioner for generalized saddle point problems. Appl. Math. Comput. 376 (2020), Article ID 125110, 26 pages. DOI 10.1016/j.amc.2020.125110 | MR 4068949 | Zbl 1474.65063
[15] Meng, L., Li, J., Miao, S.-X.: A variant of relaxed alternating positive semi-definite splitting preconditioner for double saddle point problems. Japan J. Ind. Appl. Math. 38 (2021), 979-998. DOI 10.1007/s13160-021-00467-x | MR 4304920 | Zbl 1483.65048
[16] Monk, P.: Analysis of a finite element method for Maxwell's equations. SIAM J. Numer. Anal. 29 (1992), 714-729. DOI 10.1137/0729045 | MR 1163353 | Zbl 0761.65097
[17] Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadephia (2003). DOI 10.1137/1.9780898718003 | MR 1990645 | Zbl 1031.65046
[18] Salkuyeh, D. K., Aslani, H., Liang, Z.-Z.: An alternating positive semidefinite splitting preconditioner for the three-by-three block saddle point problems. Math. Commun. 26 (2021), 177-195. MR 4297389 | Zbl 07424441
[19] Wang, L., Zhang, K.: Generalized shift-splitting preconditioner for saddle point problems with block three-by-three structure. Open Access Library J. 6 (2019), Article ID e5968, 13 pages. DOI 10.4236/oalib.1105968 | MR 3615979
[20] Wang, N.-N., Li, J.-C.: On parameterized block symmetric positive definite preconditioners for a class of block three-by-three saddle point problems. J. Comput. Appl. Math. 405 (2022), Article ID 113959, 15 pages. DOI 10.1016/j.cam.2021.113959 | MR 4355119 | Zbl 1480.65067
[21] Xie, X., Li, H.-B.: A note on preconditioning for the $3 \times 3$ block saddle point problem. Comput. Math. Appl. 79 (2020), 3289-3296. DOI 10.1016/j.camwa.2020.01.022 | MR 4094767 | Zbl 1452.65054
[22] Young, D. M.: Iterative Solution of Large Linear Systems. Computer Science and Applied Mathematics. Academic Press, New York (1971). DOI 10.1016/c2013-0-11733-3 | MR 0305568 | Zbl 0231.65034
[23] Zhang, N., Li, R.-X., Li, J.: Lopsided shift-splitting preconditioner for saddle point problems with three-by-three structure. Comput. Appl. Math. 41 (2022), Articles ID 261, 16 pages. DOI 10.1007/s40314-022-01944-w | MR 4458078 | Zbl 1513.65061
[24] Zhu, J.-L., Wu, Y.-J., Yang, A.-L.: A two-parameter block triangular preconditioner for double saddle point problem arising from liquid crystal directors modeling. Numer. Algorithms 89 (2022), 987-1006. DOI 10.1007/s11075-021-01142-5 | MR 4376676 | Zbl 1484.65058
Search
Partner of
