双折射和双折射液晶Oseen-Frank模型的增广拉格朗日预条件子
Augmented Lagrangian preconditioners for the Oseen-Frank model of nematic and cholesteric liquid crystals
摘要：
我们提出了一个强大的和有效的增广拉格朗日型预处理器解决线性化的Oseen-Frank模型中产生的液晶。通过应用增广拉格朗日方法，指向块的舒尔补可以通过拉格朗日乘子的加权质量矩阵更好地近似，代价是使增广指向块更难求解。为了解决增广董事块，我们开发了一个强大的多重网格算法，其中包括一个添加剂Schwarz松弛，捕捉点态版本的内核的半定项。此外，我们证明了增广拉格朗日项改善了单位长度约束的离散执行。数值实验验证了该算法的有效性及其对问题相关参数（Frank常数和螺距）和网格步长的鲁棒性。
英文摘要：
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.
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