Advertisement

BIT Numerical Mathematics

, Volume 56, Issue 2, pp 441–465 | Cite as

Analysis of a new dimension-wise splitting iteration with selective relaxation for saddle point problems

  • Martin J. Gander
  • Qiang Niu
  • Yingxiang Xu
Article

Abstract

We propose a new dimension-wise splitting with selective relaxation (DSSR) method for saddle point systems arising from the discretization of the incompressible Navier–Stokes equations. Using Fourier analysis, we determine the optimal choice of the relaxation parameter that leads to the best performance of the iterative method for the Stokes and the steady Oseen equations. We also explore numerically the influence of boundary conditions on the optimal choice of the parameter, the use of inner and outer iterations, and the performance for a lid driven cavity flow.

Keywords

Splitting iterations Optimized relaxation parameter Stokes Oseen 

Mathematics Subject Classification

65F10 65N22 

Notes

Acknowledgments

The authors would like to thank the organizing committee for the wonderful conference NASC2014, where the authors met each other and started their collaboration on this interesting topic. They are also very thankful for the constructive comments of the anonymous referees, which substantially enhanced the content and structure of this manuscript.

References

  1. 1.
    Bai, Z.-Z.: Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl. 16(6), 447–479 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bai, Z.-Z., Golub, G.H., Li, C.-K.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput. 28(2), 583–603 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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. Comp. 76(257), 287–298 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefMATHGoogle Scholar
  6. 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), 1–32 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bennequin, D., Gander, M.J., Halpern, L.: A homographic best approximation problem with application to optimized Schwarz waveform relaxation. Math. Comp. 78(265), 185–223 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bennequin, D., Gander, M.J., Gouarin, L., Halpern, L.: Optimized Schwarz waveform relaxation for advection reaction diffusion equations in two dimensions. Numer. Math. (2016) (to appear)Google Scholar
  9. 9.
    Benzi, M., Gander, M.J., Golub, G.H.: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT 43(5), 881–900 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14(1), 1–137 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Benzi, M., Guo, X.-P.: A dimensional split preconditioner for Stokes and linearized Navier–Stokes equations. Appl. Numer. Math. 61(1), 66–76 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Benzi, M., Ng, M.K., Niu, Q., Wang, Z.: A relaxed dimensional factorization preconditioner for the incompressible Navier–Stokes equations. J. Comput. Phys. 230(16), 6185–6202 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Brown, P.N., Walker, H.F.: GMRES on (nearly) singular systems. SIAM J. Matrix Anal. Appl. 18(1), 37–51 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cao, Y., Du, J., Niu, Q.: Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 272, 239–250 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Elman, H.C., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.S.: Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27(5), 1651–1668 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Elman, H.C., Howle, V.E., Shadid, J., Silvester, D.J., Tuminaro, R.S.: Least squares preconditioners for stabilized discretizations of the Navier–Stokes equations. SIAM J. Sci. Comput. 30(1), 290–311 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Elman, H.C., Tuminaro, R.S.: Boundary conditions in approximate commutator preconditioners for the Navier–Stokes equations. Electron. Trans. Numer. Anal. 35, 257–280 (2009)MathSciNetMATHGoogle Scholar
  18. 18.
    Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gander, M.J.: Schwarz methods over the course of time. ETNA 31, 228–255 (2008)MathSciNetMATHGoogle Scholar
  20. 20.
    Gander, M.J., Halpern, L.: Optimized Schwarz waveform relaxation for advection reaction diffusion problems. SIAM J. Numer. Anal. 45(2), 666–697 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gander, M.J., Xu, Y.: Optimized Schwarz methods for circular domain decompositions with overlap. SIAM J. Numer. Anal. 52(4), 1981–2004 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Gander, M.J., Xu, Y.: Optimized Schwarz methods with nonoverlapping circular domain decomposition. Math. Comp. (2016) (to appear 2016)Google Scholar
  23. 23.
    Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8(12), 2182–2189 (1965)CrossRefMATHGoogle Scholar
  24. 24.
    Lebedev, V.I.: Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I. USSR Comput. Math. Math. Phys. 4(3), 69–92 (1964)CrossRefMATHGoogle Scholar
  25. 25.
    Reichel, L., Ye, Q.: Breakdown-free GMRES for singular systems. SIAM J. Matrix Anal. Appl. 26(4), 1001–1021 (2005)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, SIAM (2003)Google Scholar
  27. 27.
    Welch, J.E., Harlow, F.H., Shannon, J.P., Daly, B.J.: The MAC method-a computing technique for solving viscous, incompressible, transient fluid-flow problems involving free surfaces. Tech. Rep., Los Alamos Scientific Laboratory, University of California, New Mexico (1965)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Section de MathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.Department of Mathematical SciencesXi’an Jiaotong-Liverpool UniversitySuzhouChina
  3. 3.School of Mathematics and StatisticsNortheast Normal UniversityChangchunChina

Personalised recommendations