  This paper investigates energy-minimization finite-element approaches for the
computation of nematic liquid crystal equilibrium configurations. We compare
the performance of these methods when the necessary unit-length constraint is
enforced by either continuous Lagrange multipliers or a penalty functional.
Building on previous work in [1,2], the penalty method is derived and the
linearizations within the nonlinear iteration are shown to be well-posed under
certain assumptions. In addition, the paper discusses the effects of tailored
trust-region methods and nested iteration for both formulations. Such methods
are aimed at increasing the efficiency and robustness of each algorithms'
nonlinear iterations. Three representative, free-elastic, equilibrium problems
are considered to examine each method's performance. The first two
configurations have analytical solutions and, therefore, convergence to the
true solution is considered. The third problem considers more complicated
boundary conditions, relevant in ongoing research, simulating surface
nano-patterning. A multigrid approach is introduced and tested for a
flexoelectrically coupled model to establish scalability for highly complicated
applications. The Lagrange multiplier method is found to outperform the penalty
method in a number of measures, trust regions are shown to improve robustness,
and nested iteration proves highly effective at reducing computational costs.
