A convergent finite element method for a model of liquid crystal dynamics subjected to an electric field

Abstract We present a convergent and constraint-preserving fully-discrete finite element discretization of a mathematical model for the dynamics of a liquid crystal subjected to an electric field. This model can be derived from the Oseen-Frank director field theory, assuming that the dynamics of the electric field are governed by the electrostatics equations with a suitable constitutive relation for the electric displacement field that describes the coupling with the liquid crystal director field. The resulting system of partial differential equations consists of an elliptic equation that is coupled to the wave map equations through a quadratic source term. We show that the discretization preserves the unit length constraint of the director field at every node of the mesh, is energy-stable and convergent. In numerical experiments, we show that the method is stable even when singularities develop. Moreover, predictions about the alignment of the director field with the electric field are confirmed.