Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T23:27:19.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of a fully discrete finite element methodfor a degenerate parabolic system modelling nematic liquid crystals withvariable degree of orientation

Published online by Cambridge University Press:  23 February 2006

John W. Barrett ,
Xiaobing Feng  and
Andreas Prohl
John W. Barrett
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2AZ, UK. [email protected]
Xiaobing Feng
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA.
Andreas Prohl
Affiliation:
Department of Mathematics, ETH, 8092 Zürich, Switzerland.
Get access

Abstract

We consider a degenerate parabolic system which modelsthe evolution of nematic liquid crystal with variable degree of orientation.The systemis a slight modificationto that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical fully discrete finite element method for this regularized system, and we establish the (subsequence) convergence of this finite element approximation to the solution of the regularized system as the mesh parameters tend to zero; andto a solution of the original degenerate parabolic system when the the mesh and regularization parameters all approach zero. Finally, numerical experiments are included which show the formation, annihilation and evolution of line singularities/defects in such models.

Keywords

Nematic liquid crystaldegenerate parabolic systemexistencefinite element methodconvergence.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 1 , January 2006 , pp. 175 - 199
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Calderer, M.C., Golovaty, D., Lin, F.-H. and Liu, C., Time evolution of nematic liquid crystals with variable degree of orientation. SIAM J. Math. Anal. 33 (2002) 1033–1047. CrossRef
Elliott, C.M. and Larsson, S., A finite element model for the time-dependent joule heating problem. Math. Comp. 64 (1995) 1433–1453. CrossRef
Ericksen, J.L., Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113 (1991) 97–120. CrossRef
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin (1984).
Meyers, N.G., An L p estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189–206.
Existence, X. Xu for a model arising from the in situ vitrification process. J. Math. Anal. Appl. 271 (2002) 333–342.
E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer, New York (1990).