Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation
ESAIM: Mathematical Modelling and Numerical Analysis - 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 : https://doi.org/10.1051/m2an:2006005
Classification : 35K55,  35K65,  35Q35,  65M12,  65M60,  76A15
Mots clés : nematic liquid crystal, degenerate parabolic system, existence, finite element method, convergence
@article{M2AN_2006__40_1_175_0,
author = {Barrett, John W. and Feng, Xiaobing and Prohl, Andreas},
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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {175--199},
publisher = {EDP-Sciences},
volume = {40},
number = {1},
year = {2006},
doi = {10.1051/m2an:2006005},
zbl = {1097.35082},
mrnumber = {2223509},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006005/}
}
TY  - JOUR
AU  - Barrett, John W.
AU  - Feng, Xiaobing
AU  - Prohl, Andreas
TI  - Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2006
DA  - 2006///
SP  - 175
EP  - 199
VL  - 40
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2006005/
UR  - https://zbmath.org/?q=an%3A1097.35082
UR  - https://www.ams.org/mathscinet-getitem?mr=2223509
UR  - https://doi.org/10.1051/m2an:2006005
DO  - 10.1051/m2an:2006005
LA  - en
ID  - M2AN_2006__40_1_175_0
ER  - 
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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 1, pp. 175-199. doi : 10.1051/m2an:2006005. http://www.numdam.org/articles/10.1051/m2an:2006005/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[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 1003.35108

[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 0846.65047

[4] J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113 (1991) 97-120. | Zbl 0729.76008

[5] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin (1984). | MR 737005 | Zbl 0536.65054

[6] N.G. Meyers, An ${L}^{\mathrm{p}}$ estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | Zbl 0127.31904

[7] X. Xu, Existence for a model arising from the in situ vitrification process. J. Math. Anal. Appl. 271 (2002) 333-342. | Zbl 1011.35135

[8] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer, New York (1990). | Zbl 0684.47029

Cité par Sources :