Abstract
In this paper, we present a parameterized inexact Uzawa type method for solving a class of large sparse generalized saddle point problems, and analyze its convergence using a different approach from those utilized in the existing literature. Some numerical results are given to illustrate the effectiveness of the proposed method.
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Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Nonlinear Programming. Stanford University Press, Stanford (1958)
Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)
Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71–78 (2015)
Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)
Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27, 1–23 (2007)
Bai, Z.-Z., Li, G.-Q.: Restrictively preconditioned conjugate gradient methods of linear equations. IMA J. Numer. Anal. 23, 561–580 (2003)
Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numer. Math. 102, 1–38 (2005)
Bai, Z.-Z., Wang, Z.-Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428, 2900–2932 (2008)
Benzi, M., Golub, G.H.: A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl. 26(1), 20–41 (2004)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)
Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia (2001)
Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34, 1072–1092 (1997)
Cao, Z.-H.: Fast Uzawa algorithm for generalized saddle point problems. Appl. Numer. Math. 46, 157–171 (2003)
Cao, Y., Jiang, M.-Q., Zheng, Y.-L.: A splitting preconditioner for saddle point problems. Numer. Linear Algebra Appl. 18, 875–895 (2011)
Chen, X.-J.: On preconditioned Uzawa methods and SOR methods for saddle-point problems. J. Comput. Appl. Math. 100, 207–224 (1998)
Chen, F., Jiang, Y.-L.: A generalization of the inexact parameterized Uzawa methods for saddle point problems. Appl. Math. Comput. 206, 765–771 (2008)
Dai, L.-F., Liang, M.-L., Fan, H.-T.: A new generalized parameterized inexact Uzawa method for solving saddle point problems. Appl. Math. Comput. 265, 414–430 (2015)
Elman, H.C.: Preconditioners for saddle point problems arising in computational fluid dynamics. Appl. Numer. Math. 43, 75–89 (2002)
Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994)
Elman, H.C., Silvester, D.J., Wathen, A.J.: Iterative methods for problems in computational fluid dynmics. In: Chan, R., Chan, T.F., Golub G.H. (eds.) Iterative Methods in Scientific Computing, pp. 271–327. Springer, Singapore (1997)
Fan, H.-T., Zheng, B.: A preconditioned GLHSS iteration method for non-Hermitian singular saddle point problems. Comput. Math. Appl. 67, 614–626 (2014)
Miao, S.-X., Cao, Y.: A note on GPIU method for generalized saddle point problems. Appl. Math. Comput. 230, 27–34 (2014)
Rogers, S.E., Kwak, D.: Steady and unsteady solutions of the incompressible Navier–Stokes equations. AIAA J. 29, 603–610 (1991)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Yin, J.-F., Bai, Z.-Z.: The restrictively preconditioned conjugate gradient methods on normal residual for two-by-two linear systems. J. Comput. Math. 26, 240–249 (2008)
Yuan, J.-Y.: Iterative methods for generalized least squares problems. Ph.D. Thesis, IMPA, Rio de Janeiro, Brazil (1993)
Zhang, N.-M., Lub, T.-T., Wei, Y.-M.: Semi-convergence analysis of Uzawa methods for singular saddle point problems. J. Comput. Appl. Math. 255, 334–345 (2014)
Zhang, G.-F., Yang, J.-L., Wang, S.-S.: On generalized parameterized inexact Uzawa method for a block two-by-two linear system. J. Comput. Appl. Math. 255, 193–207 (2014)
Zheng, B., Bai, Z.-Z., Yang, X.: On semi-convergence of parameterized Uzawa methods for singular saddle-point problems. Linear Algebra Appl. 431, 808–817 (2009)
Zhou, Y.-Y., Zhang, G.-F.: A generalization of parameterized inexact Uzawa method for generalized saddle point problems. Appl. Math. Comput. 215, 599–607 (2009)
Acknowledgements
The authors are very grateful to the referees and the editor for their invaluable comments and suggestions, which greatly improve the quality of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11961057), the Science and Technology Project of Gansu Province (Grant no. 21JR1RE287), the Fuxi Scientific Research Innovation Team of Tianshui Normal University (Grant no. FXD2020-03), and the Science Foundation (Grant nos. CXT2019-36, CXJ2020-11), as well as the Education and Teaching Reform Project of Tianshui Normal University (Grant nos. JY202004, JY203008).
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Dai, L., Liang, M. & Li, Q. Convergence Analysis of the Splitting-Based Iterative Method for Solving Generalized Saddle Point Problems. Mediterr. J. Math. 18, 247 (2021). https://doi.org/10.1007/s00009-021-01906-2
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DOI: https://doi.org/10.1007/s00009-021-01906-2