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.
|Number of pages||26|
|Journal||IMA Journal of Numerical Analysis|
|Publication status||Accepted/In press - 11 Feb 2020|
- nematic liquid crystals
- energy optimization
- Landau-de Gennes energy functional
- discontinuous Galerkin finite element methods
- error analysis
- convergence rate
Maity, R. R., Majumdar, A., & Nataraj, N. (Accepted/In press). Discontinuous Galerkin finite element methods for the Landau-de Gennes minimization problem of liquid crystals. IMA Journal of Numerical Analysis .