向列型液晶简化埃里克森 - 莱斯利模型的增广拉格朗日预处理
Augmented Lagrangian preconditioning for a simplified Ericksen--Leslie model of nematic liquid crystals
AI总结：
针对向列型液晶简化埃里克森 - 莱斯利模型数值解的挑战，开发增广拉格朗日块预处理器，经有限元离散等处理，通过块对角近似得到相关舒尔补近似，测试表明该方法在多方面表现良好，如迭代次数与网格无关等。
AI中文摘要：
向列型液晶简化埃里克森 - 莱斯利模型的数值解具有挑战性，因其流动与指向矢方程强耦合，且需同时满足不可压缩性和单位长度条件。拉格朗日乘子公式避免了小的金兹堡 - 朗道参数，但牛顿系统具有双鞍点结构。我们开发了一种增广拉格朗日块预处理器，增强两个约束，离散执行仍基于乘子。经有限元离散和向后欧拉时间积分后，将牛顿增量分组为速度 - 指向矢和压力 - 乘子变量。耦合速度 - 指向矢块的块对角近似导致压力和指向矢 - 乘子舒尔补的单独物理尺度近似。制造解测试显示主要变量具有预期的空间精度和一阶时间收敛性；乘子误差在时间研究中使用的固定网格上达到空间误差下限。在报告的参数范围内，外部FGMRES迭代次数几乎与网格无关，在时间步长和粘度变化下保持稳定，并随增强参数增加而改善。一个平滑基准也显示计算总能量单调衰减。
英文摘要：
The numerical solution of the simplified Ericksen--Leslie model for nematic liquid crystals is challenging because the flow and director equations are strongly coupled and because incompressibility and the unit-length condition must be enforced simultaneously. A Lagrange multiplier formulation avoids a small Ginzburg--Landau parameter, but the Newton systems have a double saddle-point structure. We develop an augmented Lagrangian block preconditioner in which both constraints are augmented while their discrete enforcement remains multiplier based. After finite element discretization and backward Euler time integration, the Newton increments are grouped into velocity--director and pressure-multiplier variables. A block-diagonal approximation of the coupled velocity-director block then leads to separate, physically scaled approximations of the pressure and director-multiplier Schur complements. Manufactured-solution tests show the expected spatial accuracy and first-order temporal convergence for the primary variables; the multiplier error reaches a spatial-error floor on the fixed mesh used in the temporal study. In the reported parameter ranges, the outer FGMRES iteration counts are nearly mesh independent, remain stable under time-step and viscosity variation, and improve as the augmentation parameters increase. A smooth benchmark also exhibits monotone decay of the computed total energy.
