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.
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: 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},
mrnumber = {2223509},
zbl = {1097.35082},
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: Modélisation mathématique et analyse numérique PY - 2006 SP - 175 EP - 199 VL - 40 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006005/ DO - 10.1051/m2an:2006005 LA - en ID - M2AN_2006__40_1_175_0 ER -
%0 Journal Article %A Barrett, John W. %A Feng, Xiaobing %A Prohl, Andreas %T Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 175-199 %V 40 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006005/ %R 10.1051/m2an:2006005 %G en %F M2AN_2006__40_1_175_0
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://www.numdam.org/articles/10.1051/m2an:2006005/
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