A convergent finite element scheme for a fourth-order liquid crystal model

In this manuscript we propose and analyse a fully discrete, unconditionally stable finite element scheme for a recently developed director model for liquid crystalline flows (Metzger, S. (2020) On a novel approach for modeling liquid crystalline flows. Commun. Math. Sci., 18, 359-378). The model consists of nonlinear fourth-order partial differential equations describing the evolution of the director field and Navier-Stokes equations governing the velocity field. We employ a stable splitting approach to reduce the computational complexity by decoupling the update of the director field from the update of the velocity field. We also perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of finite element approximations converges towards a weak solution of the original model. Passing to the limit in the nonlinear terms requires us to derive the strong convergence of the gradient of the director field from uniform bounds for its discrete Laplacian. Furthermore, we present simulations underlining the practicability of the proposed scheme, investigate its convergence properties and discuss the differences between the underlying model and already established Ericksen-Leslie-type models.