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 language | English |
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Number of pages | 26 |
Journal | IMA Journal of Numerical Analysis |
Publication status | Accepted/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