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An A Posteriori Error Estimator for Electrically Coupled Liquid Crystal Equilibrium Configurations

J. H. Adler, D. B. Emerson

TL;DR

The paper develops a reliable and locally efficient a posteriori error estimator for the nonlinear first-order optimality conditions of the electrically and flexoelectrically coupled Frank-Oseen model under a penalty constraint enforcing unit-length of the director. It extends previous elastic estimators to the coupled system and proves global reliability and local efficiency, enabling effective adaptive mesh refinement. Numerical experiments within a multilevel nested-iteration and adaptive refinement framework show substantial reductions in degrees of freedom, work units, and runtime while preserving energy and constraint conformance. The methods enable efficient, accurate simulation of strongly coupled liquid-crystal configurations under challenging boundary conditions and electric-field patterns, with clear pathways for further methodological enhancements.

Abstract

This paper derives an a posteriori error estimator for the nonlinear first-order optimality conditions associated with the electrically and flexoelectrically coupled Frank-Oseen model of liquid crystals, building on previous results for elastic systems. The estimator is proposed for a penalty approach to imposing the unit-length constraint required by the model. Moreover, theory is proven establishing that the estimator provides a reliable estimate of global approximation error and an efficient measure of local error, suitable for use in adaptive refinement. Numerical experiments demonstrate significant improvements in efficiency with adaptive refinement guided by the proposed estimator in a multilevel, nested-iteration framework and superior physical properties for challenging electrically coupled systems.

An A Posteriori Error Estimator for Electrically Coupled Liquid Crystal Equilibrium Configurations

TL;DR

The paper develops a reliable and locally efficient a posteriori error estimator for the nonlinear first-order optimality conditions of the electrically and flexoelectrically coupled Frank-Oseen model under a penalty constraint enforcing unit-length of the director. It extends previous elastic estimators to the coupled system and proves global reliability and local efficiency, enabling effective adaptive mesh refinement. Numerical experiments within a multilevel nested-iteration and adaptive refinement framework show substantial reductions in degrees of freedom, work units, and runtime while preserving energy and constraint conformance. The methods enable efficient, accurate simulation of strongly coupled liquid-crystal configurations under challenging boundary conditions and electric-field patterns, with clear pathways for further methodological enhancements.

Abstract

This paper derives an a posteriori error estimator for the nonlinear first-order optimality conditions associated with the electrically and flexoelectrically coupled Frank-Oseen model of liquid crystals, building on previous results for elastic systems. The estimator is proposed for a penalty approach to imposing the unit-length constraint required by the model. Moreover, theory is proven establishing that the estimator provides a reliable estimate of global approximation error and an efficient measure of local error, suitable for use in adaptive refinement. Numerical experiments demonstrate significant improvements in efficiency with adaptive refinement guided by the proposed estimator in a multilevel, nested-iteration framework and superior physical properties for challenging electrically coupled systems.
Paper Structure (8 sections, 9 theorems, 53 equations, 2 figures, 4 tables)

This paper contains 8 sections, 9 theorems, 53 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Free-Energy Model and Optimality Conditions
  3. Preliminary Theory and Notation
  4. A Reliable and Efficient Coupled Error Estimator
  5. Numerical Results
  6. A Flexoelectric Field with Patterned Boundaries
  7. Electric Field with Sharp Transition
  8. Conclusion and Future Work

Key Result

Lemma 3.1

For any $T \in \mathcal{T}_h$ and $E \in \mathcal{E}_h$ where $C_1$ and $C_2$ depend only on the quasi-uniformity condition in quasiuniform.

Figures (2)

  • Figure 5.1: (\ref{['2DPatternRefinement:left1']}) Fine-mesh computed solution (restricted for visualization) using Dorfler AMR with $\nu = 0.1$. (\ref{['2DPatternRefinement:center1']}) Resulting mesh patterns after four levels of adaptive refinement overlaid on the value of $n_3$. (\ref{['2DPatternRefinement:right1']}) Electric potential produced by the patterned surfaces and flexoelectric coupling.
  • Figure 5.2: (\ref{['2DSquareRefinement:left1']}) Fine-mesh computed solution (restricted for visualization) using Dorfler AMR with $\nu = 0.25$. (\ref{['2DSquareRefinement:center1']}) Resulting mesh patterns after five levels of adaptive refinement overlaid on the value of $n_1$. (\ref{['2DSquareRefinement:right1']}) Electric potential produced by the patterned surfaces and flexoelectric coupling.

Theorems & Definitions (14)

  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Proposition 3.4: Verfurth2
  • Proposition 3.5: Verfurth2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 4 more