  We propose a robust and efficient augmented Lagrangian-type preconditioner
for solving linearizations of the Oseen-Frank model arising in cholesteric
liquid crystals. By applying the augmented Lagrangian method, the Schur
complement of the director block can be better approximated by the weighted
mass matrix of the Lagrange multiplier, at the cost of making the augmented
director block harder to solve. In order to solve the augmented director block,
we develop a robust multigrid algorithm which includes an additive Schwarz
relaxation that captures a pointwise version of the kernel of the semi-definite
term. Furthermore, we prove that the augmented Lagrangian term improves the
discrete enforcement of the unit-length constraint. Numerical experiments
verify the efficiency of the algorithm and its robustness with respect to
problem-related parameters (Frank constants and cholesteric pitch) and the mesh
size.
