  This paper outlines an energy-minimization finite-element approach to the
modeling of equilibrium configurations for nematic liquid crystals in the
presence of internal and external electric fields. The method targets
minimization of system free energy based on the electrically and
flexoelectrically augmented Frank-Oseen free energy models. The Hessian,
resulting from the linearization of the first-order optimality conditions, is
shown to be invertible for both models when discretized by a mixed
finite-element method under certain assumptions. This implies that the
intermediate discrete linearizations are well-posed. A coupled multigrid solver
with Vanka-type relaxation is proposed and numerically vetted for approximation
of the solution to the linear systems arising in the linearizations. Two
electric model numerical experiments are performed with the proposed iterative
solver. The first compares the algorithm's solution of a classical Freedericksz
transition problem to the known analytical solution and demonstrates the
convergence of the algorithm to the true solution. The second experiment
targets a problem with more complicated boundary conditions, simulating a
nano-patterned surface. In addition, numerical simulations incorporating these
nano-patterned boundaries for a flexoelectric model are run with the iterative
solver. These simulations verify expected physical behavior predicted by a
perturbation model. The algorithm accurately handles heterogeneous coefficients
and efficiently resolves configurations resulting from classical and
complicated boundary conditions relevant in ongoing research.
