Skip to main content
SpringerLink
Log in

A Splitting in Time Scheme and Augmented Lagrangian Method for a Nematic Liquid Crystal Problem

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We study the numerical approximation of nematic liquid crystal flows governed by a Ericksen–Leslie problem. This problem couples the incompressible Navier–Stokes dynamic with a gradient flow system related to the orientation unitary vector of molecules. First, a two sub-step viscosity-splitting time scheme is proposed. The first sub-step couples diffusion and convection terms whereas the second one is concerned with diffusion terms and constraints (divergence free and unit director field). Then, in the first sub-step we use a Gauss–Seidel decoupling algorithm, and in the second sub-step, we use Uzawa type algorithms on augmented Lagrangian functionals to overcome the divergence free and the unit director field constraints. From the computational point of view, it is a fully decoupled linear scheme (where all systems to solve are for scalar variables). Some numerical experiments in 2D domains are carried out by using only linear finite elements in space, confirming at least numerically the viability and the convergence of our scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Badia, S., Guillén-González, F., Gutiérrez-Santacreu, J.V.: Finite element approximation of nematic liquid crystal flows using a saddle-point structure. J. Comput. Phys. 230, 1686–1706 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Blasco, J., Codina, R.: Error estimates for an operator-splitting method for incompressible flows. Appl. Numer. Math. 51, 1–17 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Becker, R., Feng, X., Prohl, A.: Finite element approximations of the Ericksen–Leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46(4), 1704–1731 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. Studies in Applied Mathematics. SIAM, Philadelphia (1989)

    Book  Google Scholar 

  5. Glowinski, R., Lin, P., Pan, X.-B.: An operator-splitting method for a liquid crystal model. Comput. Phys. Comm. 152, 242–252 (2003)

    Article  Google Scholar 

  6. Glowinski, R., Pan, T.W., Periaux, J.: Distributed Lagrange multiplier methods for incompressible viscous flow around moving rigid bodies. Comput. Methods Appl. Mech. Eng. 151, 181–184 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.: A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies: application to particulate flow. Comput. Methods Appl. Mech. Eng. 184, 241–267 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Girault, V., Raviart, P.A.: Finite element methods for Navier–Stokes equations: theory and algorithms. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  9. Girault, V., Guillén-González, F.: Mixed formulation, approximation and decoupling algorithm for a nematic liquid crystals model. Math. Comput. 80(274), 781–819 (2011)

    Article  MATH  Google Scholar 

  10. Guillén-González, F., Gutiérrez-Santacreu, J.V.: A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model. ESAIM: M2AN 47(5), 1433–1464 (2013)

    Article  MATH  Google Scholar 

  11. Guillén-González, F., Redondo-Neble, M.V.: New error estimates for a viscosity-splitting scheme in time for the three-dimensional Navier–Stokes equations. IMA J. Numer. Anal. 31(2), 556–579 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Guillén-González, F., Redondo-Neble, M.V.: Spatial error estimates for a finite element viscosity-splitting scheme for the Navier–Stokes equations. Int. J. Numer. Anal. Methods 10(4), 826–844 (2013)

    MATH  Google Scholar 

  13. Koko, J.: Calcul Scientifique avec Matlab. Elippses, Paris (2009)

    Google Scholar 

  14. Lin, P., Liu, C.: Simulations of singularity dynamics in liquid crystal flows: a \(C^0\) finite element approach. J. Comput. Phys. 215, 348–362 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, C., Walkington, N.: Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37(3), 726–741 (2000)

    Article  MathSciNet  Google Scholar 

  16. Liu, C., Walkington, N.: Mixed methods for the approximation of liquid crystal flows. M2AN 36(2), 205–222 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Prohl, A.: Computational micro-magnetism. Advances in Numerical Mathematics. Teubner, Stuttgart (2001)

    Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the anonymous referees for their helpful and valuable suggestions and remarks, which greatly improved the earlier version of this paper.

Author information

Authors and Affiliations

  1. Departamento de Ecuaciones Diferenciales y Análisis Numérico and IMUS, Universidad de Sevilla, Aptdo. 1160, 41080, Seville, Spain

    F. Guillén-González

  2. ISIMA/LIMOS, Université Blaise Pascal – CNRS UMR 6158, Campus des Cézeaux, BP 10125, 63173, Aubière Cedex, France

    J. Koko

Authors
  1. F. Guillén-González
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Koko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Guillén-González.

Additional information

F. Guillén-González is partially supported by the Spanish project MTM2012-32325 and also acknowledges support by the University of Clermont-Ferrand II during his stay as a “Professeur invité”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guillén-González, F., Koko, J. A Splitting in Time Scheme and Augmented Lagrangian Method for a Nematic Liquid Crystal Problem. J Sci Comput 65, 1129–1144 (2015). https://doi.org/10.1007/s10915-015-0002-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0002-y

Keywords

Search

Navigation