Tensor finite elements for smectic liquid crystals

TL;DR

This paper introduces a tensor-based finite element method for modeling smectic-A liquid crystals, providing theoretical convergence proofs and demonstrating effective numerical performance in two dimensions.

Contribution

It develops a novel finite element scheme for smectic-A liquid crystals, including a linear projection, boundary condition treatment, and a nonlinear Uzawa-based extension with convergence analysis.

Findings

The linear scheme achieves quasi-optimal convergence.

The nonlinear scheme exists and converges in two dimensions.

Numerical results confirm the scheme's effectiveness.

Abstract

We present a tensor-based finite element scheme for a smectic-A liquid crystal model. We propose a simple C\'ea-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.