Finite Element Analysis of a Nematic Liquid Crystal Landau–de Gennes Model with Quartic Elastic Terms

Golovaty et al. [ 15 ] present a [Formula: see text]-tensor model for liquid crystal dynamics which reduces to the well-known Oseen–Frank director field model in uniaxial states. This model has been shown to capture phase transitions in nematic liquid crystals. We study a closely related model and present an energy stable scheme for the corresponding gradient flow. We prove well-posedness of the scheme via the Leray-Schauder fixed point theorem and rigorously show the [Formula: see text]-convergence of discrete minimizers as the mesh size approaches zero. In the numerical experiments, we successfully simulate isotropic-to-nematic phase transitions as expected. We also compare simulations for elastic energies with identical and with highly disparate elastic constants and observe different defect dynamics. A comparison with simulations for the standard Landau–de Gennes model with quadratic energy shows that the quartic model is able to capture richer dynamics than the quadratic model.