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, Volume 40 (2006) no. 1, p. 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
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
     year = {2006},
     pages = {175-199},
     doi = {10.1051/m2an:2006005},
     zbl = {1097.35082},
     mrnumber = {2223509},
     language = {en},
     url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, pp. 175-199. doi : 10.1051/m2an:2006005. http://www.numdam.org/item/M2AN_2006__40_1_175_0/

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