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AIMS Mathematics

2023, Volume 8, Issue 12: 30501-30510. doi: 10.3934/math.20231558
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Structured backward errors analysis for generalized saddle point problems arising from the incompressible Navier-Stokes equations

  • School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China
  • Received: 08 April 2023 Revised: 10 May 2023 Accepted: 23 May 2023 Published: 10 November 2023

  • MSC : 15A06, 65F10, 65F50, 65G50

  • Recently, a number of fast iteration methods for the solution of the structured linear system arising from the incompressible Navier-Stokes equations have been proposed by some authors. In order to evaluate the strong stability of these numerical algorithms, in this paper we deal with the structured backward error analysis for this type of structured linear system and present the explicit formula of the structured backward error. Based on the structured backward error, we perform some numerical experiments to compare the availability of some existing numerical algorithms.

    Citation: Peng Lv. Structured backward errors analysis for generalized saddle point problems arising from the incompressible Navier-Stokes equations[J]. AIMS Mathematics, 2023, 8(12): 30501-30510. doi: 10.3934/math.20231558

    Related Papers:

  • Recently, a number of fast iteration methods for the solution of the structured linear system arising from the incompressible Navier-Stokes equations have been proposed by some authors. In order to evaluate the strong stability of these numerical algorithms, in this paper we deal with the structured backward error analysis for this type of structured linear system and present the explicit formula of the structured backward error. Based on the structured backward error, we perform some numerical experiments to compare the availability of some existing numerical algorithms.


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    [1] M. Benzi, X. P. Guo, A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math., 61 (2011), 66–76.
    [2] M. Benzi, M. K. Ng, Q. Niu, Z. Wang, A relaxed dimensional factorization preconditioner for the incompressible Navier-Stokes equations, J. Comput. Phys., 230 (2011), 6185–6202.
    [3] J. R. Bunch, W. James Demmel, C. F. Van Loan, The strong stability of algorithms for solving symmetric linear systems, SIAM J. Matrix Anal. Appl., 10 (1989), 494–499. https://doi.org/10.1137/0610035 doi: 10.1137/0610035
    [4] Y. Cao, S. X. Miao, Y. S. Cui, A relaxed splitting preconditioner for genenralized saddle point problems, Comput. Appl. Math., 34 (2015), 865–879. https://doi.org/10.1007/s40314-014-0150-y doi: 10.1007/s40314-014-0150-y
    [5] Y. Cao, L. Q. Yao, M. Q. Jiang, A modified dimensional split preconditioner for generalized saddle point problems, J. Comput. Appl. Math., 250 (2013), 70–82. https://doi.org/10.1016/j.cam.2013.02.017 doi: 10.1016/j.cam.2013.02.017
    [6] X. S. Chen, W. Li, X. Chen, J. Liu, Structured backward errors for generalized saddle point systems, Linear Algebra Appl., 436 (2012), 3109–3119. https://doi.org/10.1016/j.laa.2011.10.012 doi: 10.1016/j.laa.2011.10.012
    [7] G. Cheng, J. C. Li, A relaxed upper and lower triangular splitting preconditioner for the linearized Navier–Stokes equation, Comput. Math. Appl., 80 (2020), 43–60. https://doi.org/10.1016/j.camwa.2020.02.025 doi: 10.1016/j.camwa.2020.02.025
    [8] H. C. Elman, A. Ramage, D. J. Silvester, Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Software, 33 (2007), 14. https://doi.org/10.1145/1236463.1236469 doi: 10.1145/1236463.1236469
    [9] H. C. Elman, D. J. Silvester, A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford: Oxford University Press, 2014.
    [10] H. T. Fan, X. Y. Zhu, A modified relaxed splitting preconditioner for generalized saddle point problems from the incompressible Navier-Stokes equations, Appl. Math. Lett., 55 (2016), 18–26. https://doi.org/10.1016/j.aml.2015.11.011 doi: 10.1016/j.aml.2015.11.011
    [11] G. H. Golub, C. F. Van Loan, Matrix Computations, Baltimore: The Johns Hopkins University Press, 2013.
    [12] A. Graham, Kronecker Products and Matrix Calculus with Application, New York: Wiley, 1981.
    [13] N. J. Higham, Accuracy and Stability of Numerical Algorithms, Philadelphia: SIAM, 2002. https://doi.org/10.1137/1.9780898718027
    [14] Y. M. Huang, A practical formula for computing optimal parameters in the HSS iteration methods, J. Comput. Appl. Math., 255 (2014), 142–149. https://doi.org/10.1016/j.cam.2013.01.023 doi: 10.1016/j.cam.2013.01.023
    [15] Y. F. Ke, C. F. Ma, An inexact modified relaxed splitting preconditioner for the generalized saddle point problems from the incompressible Navier-Stokes equation, Numer. Algor., 75 (2017), 1103–1121. https://doi.org/10.1007/s11075-016-0233-5 doi: 10.1007/s11075-016-0233-5
    [16] Y. J. Li, X. Y. Zhu, H. T. Fan, Relaxed block upper–lower triangular preconditioner for generalized saddle point problems from the incompressible Navier-Stokes equations, J. Comput. Appl. Math., 364 (2020), 112329. https://doi.org/10.1016/j.cam.2019.06.045 doi: 10.1016/j.cam.2019.06.045
    [17] P. Lv, B. Zheng, Structured backward error analysis for a class of block three-by-three saddle point problems, Numer. Algor., 90 (2022), 59–78. https://doi.org/10.1007/s11075-021-01179-6 doi: 10.1007/s11075-021-01179-6
    [18] L. S. Meng, Y. W. He, S. X. Miao, Structured backward errors for two kinds of generalized saddle point systems, Linear Multilinear Algebra, 70 (2022), 1345–1355. https://doi.org/10.1080/03081087.2020.1760193 doi: 10.1080/03081087.2020.1760193
    [19] J. L. Rigal, J. Gaches, On the compatibility of a given solution with the data of a linear system, J. Assoc. Comput. Mach., 14 (1967), 543–548. https://doi.org/10.1145/321406.321416 doi: 10.1145/321406.321416
    [20] J. G. Sun, Structured backward errors for KKT systems, Linear Algebra Appl., 288 (1999), 75–88. https://doi.org/10.1016/S0024-3795(98)10184-2 doi: 10.1016/S0024-3795(98)10184-2
    [21] J. G. Sun, Matrix Perturbation Analysis, Beijing: Science Press, 2001.
    [22] N. B. Tan, T. Z. Huang, Z. J. Hu, A relaxed splitting preconditioner for the incompressible Navier–Stokes equations, J. Appl. Math., 2012 (2012), 402490. https://doi.org/10.1155/2012/402490 doi: 10.1155/2012/402490
    [23] H. Xiang, Y. M. Wei, On normwise structured backward errors for saddle point systems, SIAM J. Matrix Anal. Appl., 29 (2007), 838–849. https://doi.org/10.1137/060663684 doi: 10.1137/060663684
    [24] B. Zheng, P. Lv, Structured backward error analysis for generalized saddle point problems, Adv. Comput. Math., 46 (2020), 34. https://doi.org/10.1007/s10444-020-09787-x doi: 10.1007/s10444-020-09787-x
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