Hostname: page-component-7b7567c8c-lpg2g Total loading time: 0 Render date: 2026-11-15T14:37:36.858Z Has data issue: false hasContentIssue false
  • English
  • Français

A linear mixed finite element scheme for a nematicEricksen–Leslie liquid crystal model∗

Published online by Cambridge University Press:  30 July 2013

F.M. Guillén-González and
J.V. Gutiérrez-Santacreu
F.M. Guillén-González
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain.. guillen@us.es .
J.V. Gutiérrez-Santacreu
Affiliation:
Dpto. Matemática Aplicada I, University of Sevilla, Av. Reina Mercedes s/n, 41012 Sevilla, Spain.; juanvi@us.es .

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this work we study a fully discrete mixed scheme, based on continuous finite elementsin space and a linear semi-implicit first-order integration in time, approximating anEricksen–Leslie nematic liquid crystal model by means of aGinzburg–Landau penalized problem. Conditional stability of this schemeis proved via a discrete version of the energy law satisfied by thecontinuous problem, and conditional convergence towards generalized Young measure-valuedsolutions to the Ericksen–Leslie problem is showed when the discreteparameters (in time and space) and the penalty parameter go to zero at the same time.Finally, we will show some numerical experiences for a phenomenon of annihilation ofsingularities.

Keywords

Liquid crystalNavier–Stokesstabilityconvergencefinite elementspenalization

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013, pp. 1433 - 1464
Copyright
© EDP Sciences, SMAI, 2013

References

Alouges, F., A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708–1726. Google Scholar
Azérad, P. and Guillen-González, F., Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics. SIAM J. Math. Anal. 33 (2001) 847–859. Google Scholar
Becker, R., Feng, X. and Prohl, A., Finite element approximations of the Ericksen–Leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46 (2008) 1704–1731. Google Scholar
B. Climent-Ezquerra, F. Guillén-González and M. Rojas-Medar, Reproductivity for a nematic liquid crystal model. Z. Angew. Math. Phys. (2006) 984–998.
Chen, Y.M., The weak solutions to the evolution problems of harmonic maps. Math. Z. 201 (1989) 69–74. Google Scholar
P.G. Ciarlet, The finite element method for elliptic problems. Amsterdam, North-Holland (1987).
Douglas, J.J., Dupont, T. and Wahlbin, L., The stability in L q of the L 2-projection into finite element function spaces. Numer. Math. 23 (1974/75) 193–197. Google Scholar
Girault, V. and Guillén-González, F., Mixed formulation, approximation and decoupling algorithm for a nematic liquid crystals model. Math. Comput. 80 (2011) 781–819. Google Scholar
Girault, V., Nochetto, N. and Scott, R., Estimates of the finite element Stokes projection in W 1,∞. C. R. Math. Acad. Sci. Paris 338 (2004) 957–962. Google Scholar
Girault, V. and Lions, J.L., Two-grid finite-element schemes for the transient Navier–Stokes problem. ESAIM: M2AN 35 (2001) 945–980. Google Scholar
V. Girault and P.A. Raviart. Finite element methods for Navier–Stokes equations: theory and algorithms. Springer-Verlag, Berlin (1986).
Guillén-González, F. and Gutiérrez-Santacreu, J.V., Unconditional stability and convergence of a fully discrete scheme for 2D viscous fluids models with mass diffusion. Math. Comput. 77 (2008) 1495–1524. Google Scholar
Lin, F.H., Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Comm. Pure Appl. Math. 42 (1989) 789–814. Google Scholar
Lin, F.H. and Liu, C., Non-parabolic dissipative systems modelling the flow of liquid crystals. Comm. Pure Appl. Math. 48 (1995) 501–537. Google Scholar
Lin, F. H. and Liu, C., Existence of solutions for the Ericksen–Leslie system. Arch. Rational. Mech. Anal. 154 (2000) 135–156. Google Scholar
Lin, P. and Liu, C., Simulations of singularity dynamics in liquid crystal flows: A C 0 finite element approach. J. Comput. Phys. 215 (2006) 1411–1427. Google Scholar
Lin, P., Liu, C. and Zhang, H., An energy law preserving C 0 finite element scheme for simulating the kinematic effects in liquid crystal flow dynamics. J. Comput. Phys. 227 (2007) 348–362. Google Scholar
Liu, C. and Walkington, N.J., Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37 (2000) 725–741. Google Scholar
Liu, C. and Walkington, N.J., Mixed methods for the approximation of liquid crystal flows. ESAIM: M2AN 36 (2002) 205–222. Google Scholar
A.J. Majda and A.L. Bertozzi, Vorticity and incompressible flows. Cambridge Texts in Applied Mathematics (2002).
Scott, L. R. and Zhang, S., Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. Google Scholar
Shen, J. and Yang, X., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Cont. Dyn. Sys. 28 (2010) 1669–1691. Google Scholar
Simon, J., Compact sets in the Space L p(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–97. Google Scholar