Tensor finite elements for smectic liquid crystals

TL;DR

The paper develops a tensor-based finite element framework for the reduced PSS smectic-A model, utilizing an energy-space formulation in H(dDiv,Ω;S) and a Céa-type projection for the linear problem to achieve quasi-optimal convergence. It extends naturally to a nonlinear two-dimensional setting by coupling a postprocessed density variation u with a director-field representation ν via φ, solved through a simple Uzawa-type algorithm that decouples M and φ using a Poisson solve. Key contributions include a rigorous trace-duality treatment for boundary conditions, a conforming H(dDiv)-element construction with stable interpolation, and proven existence/approximate convergence results alongside comprehensive numerical experiments illustrating optimal rates and the effects of boundary data and regularity. The approach yields symmetric positive definite linear systems and demonstrates robustness to boundary conditions and director-field singularities, offering a practical pathway for simulating smectic-A liquid crystals within an energy-consistent FE framework.

Abstract

We present a tensor-based finite element scheme for a smectic-A liquid crystal model. We propose a simple Céa-type finite element projection in the linear case and prove its quasi-optimal convergence. Special emphasis is put on the formulation and treatment of appropriate boundary conditions. For the nonlinear case we present a formulation in two space dimensions and prove the existence of a solution. We propose a discretization that extends the linear case in Uzawa-fashion to the nonlinear case by an additional Poisson solver. Numerical results illustrate the performance and convergence of our schemes.

Figures (7)

• Figure 1: Error contributions for the linear problem from Section \ref{['sec_numeric_linKnown']} for $q=1,20,40,60$.
• Figure 2: Vectors $\boldsymbol{\nu}^{(1)}$, $\boldsymbol{\nu}^{(2)}$, $\boldsymbol{\nu}^{(3)}$ (left column, top to bottom) with corresponding approximations $u_h$ (right column) for the linear problem from Section \ref{['sec_numeric_linUnknown']}.
• Figure 3: Experimental orders of convergence for the setup of Section \ref{['sec_numeric_linUnknown']} using the vector fields $\boldsymbol{\nu}^{(j)}$, $j=1,2,3$.
• Figure 4: Error contributions for the nonlinear problem with known solution for $q=1,20,40,60$.
• Figure 5: Experimental orders of convergence for the setup of Section \ref{['sec_numeric_nonlinUnknown']} using the Dirichlet data $\eta_j$, $j=1,2$.

Theorems & Definitions (12)

• Lemma 1
• proof
• Remark 2
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof
• Remark 6