Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 175-199.

We consider a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation. The system is a slight modification to 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; and to 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.

DOI : 10.1051/m2an:2006005
Classification : 35K55, 35K65, 35Q35, 65M12, 65M60, 76A15
Mots-clés : nematic liquid crystal, degenerate parabolic system, existence, finite element method, convergence
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     title = {Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {175--199},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
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     url = {http://archive.numdam.org/articles/10.1051/m2an:2006005/}
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Barrett, John W.; Feng, Xiaobing; Prohl, Andreas. Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 175-199. doi : 10.1051/m2an:2006005. http://archive.numdam.org/articles/10.1051/m2an:2006005/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] M.C. Calderer, D. Golovaty, F.-H. Lin and C. Liu, Time evolution of nematic liquid crystals with variable degree of orientation. SIAM J. Math. Anal. 33 (2002) 1033-1047. | Zbl

[3] C.M. Elliott and S. Larsson, A finite element model for the time-dependent joule heating problem. Math. Comp. 64 (1995) 1433-1453. | Zbl

[4] J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113 (1991) 97-120. | Zbl

[5] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin (1984). | MR | Zbl

[6] N.G. Meyers, An L p estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | Zbl

[7] X. Xu, Existence for a model arising from the in situ vitrification process. J. Math. Anal. Appl. 271 (2002) 333-342. | Zbl

[8] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer, New York (1990). | Zbl

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