Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation
John W. Barrett; Xiaobing Feng; Andreas Prohl
ESAIM: Mathematical Modelling and Numerical Analysis (2006)
- Volume: 40, Issue: 1, page 175-199
- ISSN: 0764-583X
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topBarrett, John W., Feng, Xiaobing, and 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 40.1 (2006): 175-199. <http://eudml.org/doc/249687>.
@article{Barrett2006,
abstract = { 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.
},
author = {Barrett, John W., Feng, Xiaobing, Prohl, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nematic liquid crystal; degenerate parabolic system;
existence;
finite element method;
convergence.; nematic liquid crystal; fegenerate parabolic system; existence; finite element method; convergence; Dirichlet boundary condition; dissipative energy law; regularized system},
language = {eng},
month = {2},
number = {1},
pages = {175-199},
publisher = {EDP Sciences},
title = {Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation},
url = {http://eudml.org/doc/249687},
volume = {40},
year = {2006},
}
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
DA - 2006/2//
PB - EDP Sciences
VL - 40
IS - 1
SP - 175
EP - 199
AB - 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.
LA - eng
KW - Nematic liquid crystal; degenerate parabolic system;
existence;
finite element method;
convergence.; nematic liquid crystal; fegenerate parabolic system; existence; finite element method; convergence; Dirichlet boundary condition; dissipative energy law; regularized system
UR - http://eudml.org/doc/249687
ER -
References
top- R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
- 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.
- C.M. Elliott and S. Larsson, A finite element model for the time-dependent joule heating problem. Math. Comp.64 (1995) 1433–1453.
- J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal.113 (1991) 97–120.
- R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin (1984).
- N.G. Meyers, An Lp estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa17 (1963) 189–206.
- X. Xu, Existence for a model arising from the in situ vitrification process. J. Math. Anal. Appl.271 (2002) 333–342.
- E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer, New York (1990).
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