电耦合液晶系统中的误差估计器与标记策略
Error Estimators and Marking Strategies for Electrically Coupled Liquid Crystal Systems
AI总结：
本文推导了非线性一阶最优条件的后验误差估计器，用于电耦合和柔性电耦合Frank-Oseen液晶模型。提出两种方法的估计器，证明了惩罚法估计器的可靠性与效率，通过数值实验验证了适应性细化的准确性和效率。
摘要：
本文推导了非线性一阶最优条件的后验误差估计器，用于电耦合和柔性电耦合Frank-Oseen液晶模型，基于[14]对弹性系统的结果。提出了拉格朗日和惩罚方法用于满足模型所需的单位长度约束。此外，理论证明了惩罚法估计器作为全局近似误差的可靠估计和局部误差的高效度量，适用于自适应细化。本文的数值实验显示，使用所提估计器指导的自适应细化在准确性和效率上均有显著提升。数值结果还扩展了[14]的模拟，包括具有已知解析解的系统，验证了理论结果并允许对几种已知标记策略的性能进行比较。在每种情况下，适应网格均能显著减少计算工作量，实现可比或更优的物理特性，并提供更均匀分布的误差。
英文摘要：
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.
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