# 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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