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Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model


Authors: V. Girault and F. Guillén-González
Journal: Math. Comp. 80 (2011), 781-819
MSC (2010): Primary 35Q35, 65M12, 65M15, 65M60
Published electronically: December 8, 2010
MathSciNet review: 2772096
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Abstract: A linear fully discrete mixed scheme, using $ C^0$ finite elements in space and a semi-implicit Euler scheme in time, is considered for solving a penalized nematic liquid crystal model (of the Ginzburg-Landau type). We prove: 1) unconditional stability and convergence towards weak solutions, and 2) first-order optimal error estimates for regular solutions (but without imposing the well-known global compatibility condition for the initial pressure in the Navier-Stokes framework). These results are valid in a general connected polygon or in a Lipschitz polyhedral domain (without any constraints on its angles).

Finally, since the scheme couples the unknowns, we propose several algorithms for decoupling the computation of these unknowns and establish their rates of convergence in convex domains when the mesh size is sufficiently small compared to the time step.


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Additional Information

V. Girault
Affiliation: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris cedex 05, France

F. Guillén-González
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

DOI: https://doi.org/10.1090/S0025-5718-2010-02429-9
Keywords: Nematic liquid crystal, Ginzburg-Landau penalization, mixed formulation, finite element method, convergence, stability, error estimates, decoupling algorithm
Received by editor(s): February 11, 2009
Received by editor(s) in revised form: March 1, 2010
Published electronically: December 8, 2010
Additional Notes: The second author was partially supported by DGI-MEC (Spain), Grant MTM2006–07932 and by Junta de Andalucía project P06-FQM-02373.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.