Advertisement

Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 313–340 | Cite as

A new alternating positive semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

  • Yifen Ke
  • Changfeng Ma
  • Zhiru Ren
Research Article
  • 22 Downloads

Abstract

Based on the special positive semidefinite splittings of the saddle point matrix, we propose a new alternating positive semidefinite splitting (APSS) iteration method for the saddle point problem arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current problem. We prove that the new APSS iteration method is unconditionally convergent for both cases of the simple topology and the general topology. The new APSS matrix can be used as a preconditioner to accelerate the convergence rate of Krylov subspace methods. Numerical results show that the new APSS preconditioner is superior to the existing preconditioners.

Keywords

Time-harmonic eddy current problem saddle point problem alternating positive semidefinite splitting (APSS) convergence analysis preconditioner iteration method 

MSC

65F08 65F10 65F50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11301521, 11771467, 11071041), the Natural Science Foundation of Fujian Province (Nos. 2016J01005, 2015J01578), and the National Post-doctoral Program for Innovative Talents (No. BX201700234).

References

  1. 1.
    Arioli M, Manzini G. A null space algorithm for mixed finite-element approximations of Darcy’s equation. Comm Numer Methods Engrg, 2002, 18: 645–657CrossRefzbMATHGoogle Scholar
  2. 2.
    Bai Z Z. Optimal parameters in the HSS-like methods for saddle-point problems. Numer Linear Algebra Appl, 2009, 16: 447–479MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bai Z Z. Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models. Numer Linear Algebra Appl, 2012, 19: 914–936MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bai Z Z, Golub G H. Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J Numer Anal, 2007, 27: 1–23MathSciNetCrossRefzbMATHGoogle 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, 2003, 24: 603–626MathSciNetCrossRefzbMATHGoogle 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, 2004, 98: 1–32MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bai Z Z, Parlett B N, Wang Z Q. On generalized successive overrelaxation methods for augmented linear systems. Numer Math, 2005, 102: 1–38MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bai Z Z, Wang Z Q. On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl, 2008, 428: 2900–2932MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Benzi M. Solution of equality-constrained quadratic programming problems by a projection iterative method. Rend Mat Appl, 1993, 13: 275–296MathSciNetzbMATHGoogle Scholar
  10. 10.
    Benzi M, Golub G H. A preconditioner for generalized saddle point problems. SIAM J Matrix Anal Appl, 2004, 26: 20–41MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Benzi M, Golub G H, Liesen J. Numerical solution of saddle point problems. Acta Numer, 2005, 14: 1–137MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Ser Comput Math, Vol 15. New York: Springer-Verlag, 1991CrossRefzbMATHGoogle Scholar
  13. 13.
    Cao Y, Dong J L, Yu Y M. A relaxed deteriorated PSS preconditioner for non-symmetric saddle point problems from the steady Navier-Stokes equation. J Comput Appl Math, 2015, 273: 41–60MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cao Y, Du J, Niu Q. Shift-splitting preconditioners for saddle point problems. J Comput Appl Math, 2014, 272: 239–250MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen C R, Ma C F. A generalized shift-splitting preconditioner for singular saddle point problems. Appl Math Comput, 2015, 269: 947–955MathSciNetGoogle Scholar
  16. 16.
    Gould N, Orban D, Rees T. Projected Krylov methods for saddle-point systems. SIAM J Matrix Anal Appl, 2014, 35: 1329–1343MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ke Y F, Ma C F. Spectrum analysis of a more general augmentation block preconditioner for generalized saddle point matrices. BIT, 2016, 56: 489–500MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kotiuga P. Topological considerations in coupling magnetic scalar potentials to stream functions describing surface currents. IEEE Trans Magn, 1989, 25: 2925–2927CrossRefGoogle Scholar
  19. 19.
    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, 2014, 21: 152–170MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ren Z R, Cao Y. An alternating positive semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models. IMA J Numer Anal, 2016, 36: 922–946MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rodríguez A, Hernández R. Iterative methods for the saddle-point problem arising from the H C/E I formulation of the eddy current problem. SIAM J Sci Comput, 2009, 31: 3155–3178MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rodríguez A, Hiptmair R, Valli A. A hybrid formulation of eddy current problems. Numer Methods Partial Differential Equations, 2005, 21: 742–763MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rodríguez A, Valli A. Eddy Current Approximation of Maxwell Equations: Theory, Algorithms and Applications. Milan: Springer, 2010CrossRefzbMATHGoogle Scholar
  24. 24.
    Van der Vorst H A. Iterative Krylov Methods for Large Linear Systems. Cambridge: Cambridge Univ Press, 2003CrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang G F, Ren Z R, Zhou Y Y. On HSS-based constraint preconditioners for generalized saddle-point problems. Numer Algorithms, 2011, 57: 273–287MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Computational Geodynamics of Chinese Academy of SciencesUniversity of Chinese Academy of SciencesBeijingChina
  2. 2.College of Mathematics and Informatics & FJKLMAAFujian Normal UniversityFuzhouChina
  3. 3.School of Statistics and MathematicsCentral University of Finance and EconomicsBeijingChina

Personalised recommendations