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arXiv · 1907.06847

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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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 estimate in the energy norm is derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of Newton's iterates along with complementary numerical experiments.

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Ruma Rani Maity, Apala Majumdar, Neela Nataraj. 2020-05-28. Discontinuous Galerkin Finite Element Methods for the Landau-de Gennes Minimization Problem of Liquid Crystals. https://arxiv.org/abs/1907.06847

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