Augmented Lagrangian preconditioners for the Oseen-Frank model of nematic and cholesteric liquid crystals

TL;DR

This paper introduces a robust augmented Lagrangian preconditioner for linearized Oseen-Frank models in liquid crystals, improving solver efficiency and constraint enforcement through a multigrid approach and numerical validation.

Contribution

It develops a novel augmented Lagrangian preconditioner with a multigrid solver for the Oseen-Frank model, enhancing robustness and efficiency in liquid crystal simulations.

Findings

The preconditioner improves solver convergence across parameters.

The multigrid method effectively handles the augmented director block.

Numerical results demonstrate robustness against mesh size and material parameters.

Abstract

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.