A priori and a posteriori error analysis for the Nitsche's method of a reduced Landau-de Gennes problem

The equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device are modelled by a system of second order semi-linear elliptic partial differential equations with non-homogeneous boundary conditions. In this article, Nitsche's method is applied to approximate the solution of this non-linear model. A discrete inf-sup condition sufficient for the stability of a well-posed linear problem is established and this with a fixed point theorem allows the proof of local existence and uniqueness of a discrete solution to the semi-linear problems. {\it A priori} and {\it a posteriori} energy norm analysis is established for a sufficiently large penalization parameter and sufficiently fine triangulation. Optimal order {\it a priori} error estimates in $L^2$ norm is also established. Several numerical examples that confirm the theoretical results are presented.