  We present a convergent and constraint-preserving numerical 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, 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.
