  This paper derives a posteriori error estimators for the nonlinear
first-order optimality conditions associated with the electrically and
flexoelectrically coupled Frank-Oseen model of liquid crystals, building on the
results of [14] for elastic systems. Estimators are proposed for both
Lagrangian and penalty approaches to imposing the unit-length constraint
required by the model. Moreover, theory is proven establishing the penalty
method estimator as a reliable estimate of global approximation error and an
efficient measure of local error, suitable for use in adaptive refinement.
Numerical experiments conducted herein demonstrate significant improvements in
both accuracy and efficiency with adaptive refinement guided by the proposed
estimators for both constraint formulations. The numerical results also extend
the simulations of [14] to include systems with known analytical solutions,
confirming the theoretical results and enabling performance comparisons for a
selection of established marking strategies. In each case, the adapted grids
successfully yield substantial reductions in computational work, comparable or
better physical properties, and deliver more uniformly distributed error.
