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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.08571 |
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Table of Contents:
- 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.