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

Abstract

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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.

How to cite

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Barrett, 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

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  1. R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).  
  2. 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.  Zbl1003.35108
  3. C.M. Elliott and S. Larsson, A finite element model for the time-dependent joule heating problem. Math. Comp.64 (1995) 1433–1453.  Zbl0846.65047
  4. J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal.113 (1991) 97–120.  Zbl0729.76008
  5. R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin (1984).  Zbl0536.65054
  6. 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.  Zbl0127.31904
  7. X. Xu, Existence for a model arising from the in situ vitrification process. J. Math. Anal. Appl.271 (2002) 333–342.  Zbl1011.35135
  8. E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer, New York (1990).  Zbl0684.47029

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