In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen-Leslie nematic liquid crystal model by means of a Ginzburg-Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen-Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.
Mots-clés : liquid crystal, Navier-Stokes, stability, convergence, finite elements, penalization
@article{M2AN_2013__47_5_1433_0,
author = {Guill\'en-Gonz\'alez, F. M. and Guti\'errez-Santacreu, J. V.},
title = {A linear mixed finite element scheme for a nematic {Ericksen-Leslie} liquid crystal model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1433--1464},
publisher = {EDP-Sciences},
volume = {47},
number = {5},
year = {2013},
doi = {10.1051/m2an/2013076},
mrnumber = {3100770},
zbl = {1290.82031},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013076/}
}
TY - JOUR AU - Guillén-González, F. M. AU - Gutiérrez-Santacreu, J. V. TI - A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1433 EP - 1464 VL - 47 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013076/ DO - 10.1051/m2an/2013076 LA - en ID - M2AN_2013__47_5_1433_0 ER -
%0 Journal Article %A Guillén-González, F. M. %A Gutiérrez-Santacreu, J. V. %T A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1433-1464 %V 47 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013076/ %R 10.1051/m2an/2013076 %G en %F M2AN_2013__47_5_1433_0
Guillén-González, F. M.; Gutiérrez-Santacreu, J. V. A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1433-1464. doi : 10.1051/m2an/2013076. http://archive.numdam.org/articles/10.1051/m2an/2013076/
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