, 55:29 | Cite as

Block preconditioners for saddle point systems arising from liquid crystal directors modeling

  • Fatemeh Panjeh Ali Beik
  • Michele Benzi


We analyze two types of block preconditioners for a class of saddle point problems arising from the modeling of liquid crystal directors using finite elements. Spectral properties of the preconditioned matrices are investigated, and numerical experiments are performed to assess the behavior of preconditioned iterations using both exact and inexact versions of the preconditioners.


Liquid crystals Saddle point problems Krylov subspace methods Preconditioning 

Mathematics Subject Classification

65F10 65F08 65F50 



We would like to thank Alison Ramage for providing the test problems used in the numerical experiments. The first author is grateful for the hospitality of the Department of Mathematics and Computer Science at Emory University, where this work was completed.


  1. 1.
    Adler, J.H., Atherton, T.J., Benson, T.R., Emerson, D.B., MacLachlan, S.P.: Energy minimization for liquid crystal equilibrium with electric and flexoelectric effects. SIAM J. Sci. Comput. 37, S157–S176 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adler, J.H., Atherton, T.J., Emerson, D.B., MacLachlan, S.P.: An energy-minimization finite-element approach to the Frank-Oseen model of nematic liquid crystals. SIAM J. Numer. Anal. 53, 2226–2254 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Beik, F.P.A., Benzi, M.: Iterative methods for double saddle point systems. SIAM J. Matrix Anal. Appl. 39, 902–921 (2018)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefMATHGoogle Scholar
  6. 6.
    Ramage, A., Gartland Jr., E.C.: A preconditioned nullspace method for liquid crystal director modeling. SIAM J. Sci. Comput. 35, B226–B247 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Simoncini, V., Szyld, D.B.: Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl. 14, 1–59 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Department of MathematicsVali-e-Asr University of RafsanjanRafsanjanIran
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  3. 3.Scuola Normale SuperiorePisaItaly

Personalised recommendations