Projection Methods in the Context of Nematic Crystal Flow

TL;DR

This work develops and analyzes predictor-corrector Finite Element schemes for the Ericksen--Leslie nematic flow with a nonconvex unit-norm constraint on the director. It introduces both a continuous (CG) and a discontinuous (DG) director discretization, each followed by an explicit projection to enforce |d|=1, and proves discrete energy-dissipation laws and convergence to energy-variational solutions under a time-step condition k = o(h^{1+N/6}). The analysis combines Lax--Milgram well-posedness, energy-dissipation inequalities, and compactness arguments (Aubin--Lions, BV bounds) to pass to the limit and establish convergence of subsequences; it also demonstrates that the projection step yields improved stability and L∞ bounds, aiding the convergence proofs. Computational experiments (Magical Spiral and Defects) show the projection method enhances accuracy and efficiency, with CG often outperforming DG in time efficiency while preserving accuracy across scenarios.

Abstract

We present a continuous and a discontinuous linear Finite Element method based on a predictor-corrector scheme for the numerical approximation of the Ericksen-Leslie equations, a model for nematic liquid crystal flow including a non-convex unit-sphere constraint. As predictor step we propose a linear semi-implicit Finite Element discretization which naturally offers a local orthogonality relation between the approximate director field and its time derivative. Afterwards an explicit discrete projection onto the unit-sphere constraint is applied without increasing the modeled energy. For the Finite Element approximation of the director field, we compare the usage of a discrete inner product, usually referred to as mass-lumping, for a globally continuous, piecewise linear discretization to a piecewise constant, discontinuous Galerkin approach. Discrete well-posedness results and energy laws are established. Conditional convergence of the approximate solutions to energy-variational solutions of the Ericksen-Leslie equations is shown for a time-step restriction, see Theorems 1 and 2. Computational studies indicate the efficiency of the proposed linearization and the improved accuracy by including a projection step in the algorithm.

Figures (5)

• Figure 1: Visualization of the functions $g(x)= \left \lvert \left \lvert x \right \rvert-1 \right \rvert$ (blue, dashed) and $f(x) = \lvert {\left \lvert x \right \rvert^2-1} \rvert$ (red).
• Figure 2: Experiment \ref{['sec:spiral']}: Evolution of Algorithm \ref{['algo:dg']} (left) and Algorithm \ref{['algo:cg']} (right) for $h=0.1$ and $k=0.01$
• Figure 3: Error comparison for Experiment \ref{['sec:spiral']}. The error is computed as difference of the angles of the director field in the $L^2(\Omega)$-norm. We set $h=0.1$ and $k=0.01$ if not mentioned otherwise.
• Figure 4: $L^2$-error of the approximate solution compared to the exact stationary solution at $t=1.5$. Visualization of Table \ref{['tab:error-magical-spiral']}.
• Figure 5: Experiment \ref{['sec:defects']}: Evolution of the director and velocity fields of the approximate solutions of Algorithm \ref{['algo:dg']} (left) and Algorithm \ref{['algo:cg']} (right) in the plane $z=0$ for $k=0.001, h=1/16$. The arrows show only the first two components of the vector fields.

Theorems & Definitions (59)

• Definition 2.1: Energy-variational solutions
• Remark 2.2: Strong continuity of initial values, cf. eiter-lasarzik24lasarzik21lasarzik-reiter23
• Remark 2.3: Properties
• Definition 2.4: Non-Obtuse Mesh
• Definition 2.5: Weakly Acute Mesh, see bartels09
• Remark 2.6: On the restrictiveness of non-obtuse and weakly acute meshes
• Definition 2.7: Admissible Meshes
• Remark 2.8: On the restrictiveness of the admissibility condition
• Proposition 2.9
• proof