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

Ruma Rani Maity, Apala Majumdar, Neela Nataraj

Research output: Contribution to journalArticle

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
Number of pages26
JournalIMA Journal of Numerical Analysis
Publication statusAccepted/In press - 11 Feb 2020

Keywords

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

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