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Semi-convergence analysis of the GPIU method for singular nonsymmetric saddle-point problems

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Abstract

In this paper, the generalized parameterized inexact Uzawa (GPIU) method is further investigated for solving singular nonsymmetric saddle-point problems. The semi-convergence conditions of this method are derived, which further develop the results obtained in the paper Zhang and Wang, Appl. Math. Comput. 219(9) 4225–4231 (2013). Furthermore, the theoretical results are confirmed by a steady-state Navier-Stokes problem. Numerical experiments demonstrate that the GPIU method is feasible and effective for the ‘leaky’ lid-driven cavity problems with higher viscosity constants, i.e., singular nonsymmetric saddle-point problems with positive real and symmetric dominant (1,1) part.

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References

  1. Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75(254), 791–815 (2006)

    Article  MATH  Google Scholar 

  2. Bai, Z.-Z.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. Numer. Linear Algebra Appl. 16(6), 447–479 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.-Z.: On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems. Computing 89, 171–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27(1), 1–23 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.-Z., Golub, G.H., Li, C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76(257), 287–298 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26(3), 844–863 (2005)

    Article  MathSciNet  Google Scholar 

  8. 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(3), 603–626 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numer. Math. 102(1), 1–38 (2005)

    Article  MathSciNet  Google Scholar 

  10. Bai, Z.-Z., Wang, Z.-Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428, 2900–2932 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benzi, M., Golub, G.H.: A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl. 26(1), 20–41 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadephia (1994)

    Book  Google Scholar 

  14. Björck, A.: Numerical Methods for Least Squares Problems. SIAM, Philadephia (1996)

    Book  MATH  Google Scholar 

  15. Bramble, J., Pasciak, J., Vassilev, A.T.: Uzawa type algorithms for nonsymmetric saddle point problems. Math. Comput. 69(230), 667–689 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bramble, J., Pasciak, J., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34(3), 1072–1092 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chao, Z., Chen, G.-L.: A note on semi-convergence of generalized parameterized inexact Uzawa method for singular saddle point problems

  18. Chao, Z., Zhang, N.-M.: A generalized preconditioned HSS method for singular saddle point problems. Numer. Algor. 66(2), 203–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, F., Jiang, Y.-L.: A generalization of the inexact parameterized Uzawa methods for saddle point problems. Appl. Math. Comput 206(2), 765–771 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, X.-J.: On preconditioned Uzawa methods and SOR methods for saddle-point problems. J. Comput. Appl. Math. 100(2), 207–224 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Elman, H.C.: Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM J. Sci. Comput. 20(4), 1299–1316 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645–1661 (1994)

    Article  MathSciNet  Google Scholar 

  23. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  24. Fortin, M., Brezzi, F.: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991)

    Google Scholar 

  25. Freund, R.W., Nachtigal, N.M.: QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60(1), 315–339 (1991)

    Article  MathSciNet  Google Scholar 

  26. Golub, G.H., Wu, X., Yuan, J.-Y.: SOR-like methods for augmented systems. BIT Numer. Math. 41(1), 71–85 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Haber, E., Modersitzki, J.: Numerical methods for volume preserving image registration. BIT Numer. Math. 20(5), 1621 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Jiang, M.-Q., Cao, Y.: On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems. J. Comput. Appl. Math. 231(2), 973–982 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Krukier, L.A., Krukier, B.L., Ren, Z.-R.: Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems. Numer. Linear Algebra Appl. 21, 152–170 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liang, Z.-Z., Zhang, G.-F.: On block-diagonally preconditioned accelerated parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput. 221, 89–101 (2013)

    Article  MathSciNet  Google Scholar 

  31. Liang, Z.-Z., Zhang, G.-F.: Modified unsymmetric SOR method for saddle-point problems. Appl. Math. Comput. 234, 584–598 (2014)

    Article  MathSciNet  Google Scholar 

  32. Miller, J.H.: On the location of zeros of certain classes of polynomials with applications to numerical analysis. IMA J. Appl. Math. 8(3), 397–406 (1971)

    Article  Google Scholar 

  33. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  34. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, L., Bai, Z.-Z.: Convergence conditions for splitting iteration methods for non-Hermitian linear systems. Linear Algebra Appl. 428, 453–468 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, S.-S., Zhang, G.-F.: Preconditioned AHSS iteration method for singular saddle point problems. Numer. Algor. 63(3), 521–535 (2013)

    Article  Google Scholar 

  37. Zhang, G.-F., Wang, S.-S.: A generalization of parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput. 219(9), 4225–4231 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, N.-M., Lu, T.-T., Wei, Y.-M.: Semi-convergence analysis of Uzawa methods for singular saddle point problems. J. Comput. Appl. Math. 255, 334–345 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, N.-M., Wei, Y.-M.: On the convergence of general stationary iterative methods for range-Hermitian singular linear systems. Numer. Linear Algebra Appl. 17(1), 139–154 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhou, Y.-Y., Zhang, G.-F.: A generalization of parameterized inexact Uzawa method for generalized saddle point problems. Appl. Math. Comput. 215(2), 599–607 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhu, M.-Z.: A generalization of the local Hermitian and skew-Hermitian splitting iteration methods for the non-hermitian saddle point problems. Appl. Math. Comput. 218(17), 8816–8824 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Zhao-Zheng Liang & Guo-Feng Zhang

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  1. Zhao-Zheng Liang
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  2. Guo-Feng Zhang
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Correspondence to Guo-Feng Zhang.

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This work was supported by the National Natural Science Foundation of China (11271174).

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Liang, ZZ., Zhang, GF. Semi-convergence analysis of the GPIU method for singular nonsymmetric saddle-point problems. Numer Algor 70, 151–169 (2015). https://doi.org/10.1007/s11075-014-9939-4

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  • DOI: https://doi.org/10.1007/s11075-014-9939-4

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