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A structure-preserving FEM for the uniaxially constrained \(\mathbf{Q}\)-tensor model of nematic liquid crystals

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Abstract

We consider the one-constant Landau-de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field \(\mathbf{Q}\), which is constrained to be uniaxial: \(\mathbf{Q}= s (\mathbf{n}\otimes \mathbf{n}- d^{-1}\mathbf{I})\) where \(\mathbf{n}\) is a director field, \(s\in \mathbb {R}\) is the degree of orientation, and \(d\ge 2\) is the dimension. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regularization, and \(\Gamma \)-convergence of the discrete energies towards the continuous one as the mesh size goes to zero. We design an alternating direction gradient flow algorithm for the solution of the discrete problems, and we show that such a scheme decreases the energy monotonically. Finally, we illustrate the method’s capabilities by presenting some numerical simulations in two and three dimensions including non-orientable line fields.

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Acknowledgements

The authors thank Wenbo Li for pointing out reference [7] and suggesting an idea for the proof of Lemma 4.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Juan Pablo Borthagaray

  2. Departamento de Matemática y Estadística del Litoral, Universidad de la República, Salto, Uruguay

    Juan Pablo Borthagaray

  3. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  4. Department of Mathematics and Center for Computation and Technology (CCT) Louisiana State University, Baton Rouge, LA, 70803, USA

    Shawn W. Walker

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  1. Juan Pablo Borthagaray
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Correspondence to Juan Pablo Borthagaray.

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JPB has been supported in part by NSF grant DMS-1411808 and an AMS-Simons Travel Grant. RHN has been supported in part by NSF Grants DMS-1411808 and DMS-1908267. SWW has been supported in part by NSF Grant DMS-1555222 (CAREER).

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Borthagaray, J.P., Nochetto, R.H. & Walker, S.W. A structure-preserving FEM for the uniaxially constrained \(\mathbf{Q}\)-tensor model of nematic liquid crystals. Numer. Math. 145, 837–881 (2020). https://doi.org/10.1007/s00211-020-01133-z

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