  We develop a novel finite element method for a phase field model of nematic
liquid crystal droplets. The continuous model considers a free energy comprised
of three components: the Ericksen's energy for liquid crystals, the
Cahn-Hilliard energy representing the interfacial energy of the droplet, and a
weak anchoring energy representing the interaction of the liquid crystal
molecules with the surface tension on the interface (i.e. anisotropic surface
tension). Applications of the model are for finding minimizers of the free
energy and exploring gradient flow dynamics. We present a finite element method
that utilizes a special discretization of the liquid crystal elastic energy, as
well as mass-lumping to discretize the coupling terms for the anisotropic
surface tension part. Next, we present a discrete gradient flow method and show
that it is monotone energy decreasing. Furthermore, we show that global
discrete energy minimizers $\Gamma$-converge to global minimizers of the
continuous energy. We conclude with numerical experiments illustrating
different gradient flow dynamics, including droplet coalescence and break-up.
