Discontinuous Galerkin finite element methods for the Landau-de Gennes minimization problem of liquid crystals

Ruma Rani Maity, Apala Majumdar, Neela Nataraj

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5 Citations (Scopus)
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Abstract

We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element methods are used to approximate the solutions of this nonlinear problem with non-homogeneous Dirichlet boundary conditions. A discrete inf-sup condition demonstrates the stability of the discontinuous Galerkin discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the non-linear problem. An a priori error estimates in the energy and L2 norm are derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of Newton’s iterates along with complementary numerical experiments.
Original languageEnglish
Pages (from-to)1130–1163
Number of pages34
JournalIMA Journal of Numerical Analysis
Volume41
Issue number2
Early online date19 Jun 2020
DOIs
Publication statusPublished - 30 Apr 2021

Keywords

  • nematic liquid crystals
  • energy optimization
  • Landau-de Gennes energy functional
  • discontinuous Galerkin finite element methods
  • error analysis
  • convergence rate

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