Liquid crystals are anisotropic electro-optical materials with characteristics of both liquid and crystalline phases. Widely known for their use in display technology, recent advances have expanded their applications to include biological sensors, soft robotics, smart windows, and advanced optical devices. Mathematically, the classical Landau-de Gennes model encodes the properties of a liquid crystal in terms of the Q-tensor order parameter that satisfies a semilinear elliptic boundary value problem involving a nonlinear nonconvex potential function. In this talk, we discuss finite element approximation framework for the Landau-de Gennes Q-tensor energies for nematic liquid crystals, incorporating 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 discuss a priori error analysis to guarantee discrete local solutions near a regular root of the nonlinear elliptic partial differential equations with non-homogeneous boundary conditions associated with Landau-de Gennes energies, and address two-sided energy control to capture the complicated energy landscape. Numerical experiments support the optimal convergence estimates and validate the energy bounds.
