论文标题
nematic和胆固醇液晶的Oseen-Frank模型的增强拉格朗日预处理
Augmented Lagrangian preconditioners for the Oseen-Frank model of nematic and cholesteric liquid crystals
论文作者
论文摘要
我们提出了一个强大而有效的增强拉格朗日型预处理，用于求解在胆固醇液体晶体中产生的Oseen-Frank模型的线性化。通过应用增强的拉格朗日方法，Lagrange乘数的加权质量矩阵可以更好地估算主任块的Schur补充，这是为了使增强的导演块更难解决。为了解决增强的导演块，我们开发了一种强大的跨部算法，其中包括一个添加剂Schwarz放松，该算法捕获了半明确术语的内核的刻度版本。此外，我们证明了增强的拉格朗日术语可以改善单位长度约束的离散执行。数值实验验证了与问题相关参数（弗兰克常数和胆固醇螺距）和网格大小相对于问题相关参数的效率及其鲁棒性。
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.
