  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.
