Numerical analysis for constrained and unconstrained Q-tensor energies for liquid crystals
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909639300349952 |
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| author | Gimperlein, Heiko Maity, Ruma R. |
| author_facet | Gimperlein, Heiko Maity, Ruma R. |
| contents | This paper introduces a comprehensive finite element approximation framework for three-dimensional Landau-de Gennes $Q$-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 $Q$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04880 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical analysis for constrained and unconstrained Q-tensor energies for liquid crystals Gimperlein, Heiko Maity, Ruma R. Numerical Analysis 65N12, 35J47, 65N30, 76A15 This paper introduces a comprehensive finite element approximation framework for three-dimensional Landau-de Gennes $Q$-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 $Q$-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. |
| title | Numerical analysis for constrained and unconstrained Q-tensor energies for liquid crystals |
| topic | Numerical Analysis 65N12, 35J47, 65N30, 76A15 |
| url | https://arxiv.org/abs/2506.04880 |