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

A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets