Error Estimators and Marking Strategies for Electrically Coupled Liquid Crystal Systems

TL;DR

This paper develops and validates a posteriori error estimators for nonlinear liquid crystal models, enabling adaptive refinement that improves accuracy and efficiency in simulations with known solutions.

Contribution

It introduces reliable and efficient error estimators for both Lagrangian and penalty methods in liquid crystal modeling, extending previous elastic system results.

Findings

Adaptive refinement reduces computational work significantly.

Estimators improve accuracy and physical property preservation.

Numerical results confirm theoretical reliability and efficiency.

Abstract

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.