Advertisement

BIT Numerical Mathematics

, Volume 53, Issue 4, pp 987–1012 | Cite as

Convergence analysis of HSS-multigrid methods for second-order nonselfadjoint elliptic problems

  • Shishun Li
  • Zhengda Huang
Article

Abstract

In this paper, the multigrid methods using Hermitian/skew-Hermitian splitting (HSS) iteration as smoothers are investigated. These smoothers also include the modified additive and multiplicative smoothers which result from subspace decomposition. Without full elliptic regularity assumption, it is shown that the multigrid methods with these smoothers converge uniformly for second-order nonselfadjoint elliptic boundary value problems if the mesh size of the coarsest grid is sufficiently small (but independent of the number of the multigrid levels). Numerical results are reported to confirm the theoretical analysis.

Keywords

Multigrid methods HSS iteration method Additive smoother Multiplicative smoother 

Mathematics Subject Classification (2000)

65N55 65N30 

Notes

Acknowledgements

We would like to thank the referees for their insightful and valuable suggestions, which greatly improved the original manuscript of this paper.

References

  1. 1.
    Arnold, R.: Fourier analysis of a robust multigrid method for convection-diffusion equations. Numer. Math. 71, 365–397 (1995) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bai, Z.-Z.: Splitting iteration methods for non-Hermitian positive definite systems of linear equations. Hokkaido Math. J. 36, 801–814 (2007) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bai, Z.-Z.: Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl. 16, 447–479 (2009) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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, 603–626 (2003) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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, 844–863 (2005) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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, 583–603 (2006) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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, 287–298 (2007) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bai, Z.-Z., Golub, G.H., Ng, M.K.: On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl. 428, 413–440 (2008) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bank, R.E.: A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations. SIAM J. Numer. Anal. 18, 724–743 (1981) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput. 36, 35–51 (1981) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Benzi, M., Golub, G.H.: A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl. 26, 20–41 (2004) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bertaccini, D., Golub, G.H., Serra Capizzano, S., Tablino Possio, C.: Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation. Numer. Math. 99, 441–484 (2005) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Braess, D., Hackbusch, W.: A new convergence proof for the multigrid method including the V-cycle. SIAM J. Numer. Anal. 20, 967–975 (1983) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Bramble, J.H., Pasciak, J.E.: The analysis of smoothers for multigrid algorithms. Math. Comput. 58, 467–488 (1992) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Bramble, J.H., Pasciak, J.E.: New estimates for multilevel algorithms including the V-cycle. Math. Comput. 60, 447–471 (1993) MathSciNetMATHGoogle Scholar
  19. 19.
    Bramble, J.H., Zhang, X.-J.: The analysis of multigrid methods. In: Handbook of Numerical Analysis VII, pp. 173–415. North-Holland, Amsterdam (2000) Google Scholar
  20. 20.
    Bramble, J.H., Pasciak, J.E., Xu, J.-C.: The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems. Math. Comput. 51, 389–414 (1988) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Bramble, J.H., Kwak, D.Y., Pasciak, J.E.: Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J. Numer. Anal. 31, 1746–1763 (1994) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977) CrossRefMATHGoogle Scholar
  23. 23.
    Brenner, S.C.: Convergence of the multigrid V-cycle algorithms for second order boundary value problems without full elliptic regularity. Math. Comput. 71, 507–525 (2002) CrossRefMATHGoogle Scholar
  24. 24.
    Byfut, A., Gedicke, J., Günther, D., Reininghaus, J., Wiedemann, S.: FFW Documentation. Humboldt University of Berlin, Germany (2007) Google Scholar
  25. 25.
    Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978) MATHGoogle Scholar
  26. 26.
    De Zeeuw, P.M.: Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput. Appl. Math. 33, 1–27 (1990) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Douglas, C.C.: Multigrid algorithms with applications to elliptic boundary value problems. SIAM J. Numer. Anal. 21, 236–254 (1984) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Hackbusch, W.: Multigrid methods and applications. Springer, Berlin (1985) CrossRefGoogle Scholar
  29. 29.
    Hamilton, S., Benzi, M., Haber, E.: New multigrid smoothers for the Oseen problem. Numer. Linear Algebra Appl. 17, 557–576 (2010) MathSciNetMATHGoogle Scholar
  30. 30.
    Maitre, J.F., Musy, F.: Multigrid methods: convergence theory in a variational framework. SIAM J. Numer. Anal. 21, 657–671 (1984) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mandel, J.: Multigrid convergence for nonsymmetric, indefinite variational problems and one smoothing step. Appl. Math. Comput. 19, 201–216 (1986) MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Mandel, J., McCormick, S.F., Bank, R.E.: Variational multigrid theory. In: Multigrid methods. Frontiers in Applied Mathematics, pp. 131–177. SIAM, Philadelphia (1987) CrossRefGoogle Scholar
  33. 33.
    Russo, A., Tablino Possio, C.: Preconditioned Hermitian and skew-Hermitian splitting method for finite element approximations of convection-diffusion equations. SIAM J. Matrix Anal. Appl. 31, 997–1018 (2009) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Schatz, A.H., Wang, J.-P.: Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions. Math. Comput. 65, 19–27 (1996) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Simoncini, V., Benzi, M.: Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems. SIAM J. Matrix Anal. Appl. 26, 377–389 (2004) MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Wang, J.-P.: Convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems. SIAM J. Numer. Anal. 30, 275–285 (1993) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Wesseling, P.: An introduction to multigrid methods. Pure and Applied Mathematics. Chichester, New York (1992) MATHGoogle Scholar
  38. 38.
    Xu, J.-C.: An introduction to multilevel methods. Wavelets, multilevel methods, and elliptic PDE’s. In: Numerical Mathematics and Scientific Computation, pp. 213–302. Oxford University Press, New York (1997) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceHenan Polytechnic UniversityJiaozuoP.R. China
  2. 2.Department of MathematicsZhejiang UniversityHangzhouP.R. China

Personalised recommendations