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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model
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by V. Girault and F. Guillén-González PDF
Math. Comp. 80 (2011), 781-819 Request permission

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
  • MR Author ID: 326792
  • 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.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 781-819
  • MSC (2010): Primary 35Q35, 65M12, 65M15, 65M60
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02429-9
  • MathSciNet review: 2772096