Numerical analysis for constrained and unconstrained Q-tensor energies for liquid crystals
Abstract
This paper introduces a comprehensive finite element approximation framework for three-dimensional Landau-de Gennes -tensor energies for nematic liquid crystals, with a particular focus on the anisotropy of the elastic energy and the Ball-Majumdar singular potential. This potential imposes essential physical constraints on the eigenvalues of the -tensor, ensuring realistic modeling. We address the approximation of regular solutions to nonlinear elliptic partial differential equations with non-homogeneous boundary conditions associated with Landau-de Gennes energies. The well-posedness of the discrete linearized problem is rigorously demonstrated. The existence and local uniqueness of the discrete solution is derived using the Newton-Kantorovich theorem. Furthermore, we demonstrate an optimal order convergence rate in the energy norm and discuss the impact of eigenvalue constraints on the a priori error analysis.
Cite
@article{arxiv.2506.04880,
title = {Numerical analysis for constrained and unconstrained Q-tensor energies for liquid crystals},
author = {Heiko Gimperlein and Ruma R. Maity},
journal= {arXiv preprint arXiv:2506.04880},
year = {2025}
}