发表机构
University of Maryland; University of California, Berkeley（马里兰大学; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对液晶动力学Beris-Edwards系统，提出全离散有限元格式，结合线性隐式BDF2、增量Chorin投影及能量二次化，证明无条件稳定且收敛到弱解，数值实验验证二阶精度及缺陷分裂和skyrmion输运。
AI 中文摘要
我们提出并分析了一个用于向列型液晶动力学Beris-Edwards系统的全离散有限元格式，其中不可压缩Navier-Stokes方程与Landau-de Gennes Q-张量的梯度流耦合。该格式在时间上结合了线性隐式、形式上二阶精度的后向微分公式，在不可压缩约束上采用了增量Chorin投影步骤，在空间上使用协调有限元，并对非线性体势能采用带质量集中的不变能量二次化方法。每个时间步需要求解一个线性系统和一个Poisson问题。我们证明了该格式是唯一可解的，它保持了离散Q-张量和分子场的对称性及无迹结构，并且在不限制时间步长的情况下满足离散能量律。我们的主要结果是：当网格尺寸$h$和时间步长$\Delta t$趋于零且满足$h^{2}=o(\Delta t)$时，近似解沿子序列收敛到Beris-Edwards系统的一个弱解。收敛性证明处理了两个困难：投影方法产生两个速度近似，其中只有一个在$L^2(0,T;H^1_0(\Omega))$中一致有界；耦合项$\mathcal{H}\nabla Q$需要$\nabla Q$的强收敛，我们通过$\mathcal{H}$方程的结构而非任何离散$H^2$界来获得该强收敛。二维数值实验展示了空间和时间上近似二阶收敛，并再现了$+1$点缺陷分裂为两个$+1/2$缺陷，以及由恒定压力梯度引起的skyrmion的输运和变形。
英文摘要
We propose and analyze a fully discrete finite element scheme for the Beris-Edwards system of nematic liquid crystal dynamics, in which the incompressible Navier-Stokes equations are coupled to a gradient flow for the Landau-de Gennes Q-tensor. The scheme combines a linearly implicit formally second-order accurate backward differentiation formula in time with an incremental Chorin projection step for the incompressibility constraint, conforming finite elements in space, and the invariant energy quadratization approach with mass lumping for the nonlinear bulk potential. Each time step requires the solution of one linear system and one Poisson problem. We show that the scheme is uniquely solvable, that it preserves the symmetry and trace-free structure of the discrete Q-tensor and molecular field, and that it satisfies a discrete energy law without any restriction on the time step. Our main result is that, as the mesh size $h$ and the time step $Δt$ tend to zero subject to $h^{2} = o(Δt)$, the approximations converge along a subsequence to a weak solution of the Beris-Edwards system. The convergence proof addresses two difficulties: the projection method produces two velocity approximations, only one of which is uniformly bounded in $L^2(0,T;H^1_0(Ω))$, and the coupling term $\mathcal{H}\nabla Q$ requires strong convergence of $\nabla Q$, which we obtain from the structure of the equation for $\mathcal{H}$ rather than from any discrete $H^2$-bound. Numerical experiments in two dimensions exhibit approximately second-order convergence in space and time, and reproduce the splitting of a $+1$ point defect into two $+1/2$ defects and the transport and deformation of a skyrmion induced by a constant pressure gradient.
Comments51 pages. 16 figures
