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Convergence of a fully discrete finite element methodfor a degenerate parabolic system modelling nematic liquid crystals withvariable degree of orientation

Published online by Cambridge University Press:  23 February 2006

John W. Barrett
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2AZ, UK. [email protected]
Xiaobing Feng
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA.
Andreas Prohl
Affiliation:
Department of Mathematics, ETH, 8092 Zürich, Switzerland.
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Abstract

We consider a degenerate parabolic system which modelsthe evolution of nematic liquid crystal with variable degree of orientation.The systemis a slight modificationto 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; andto 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

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References

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