  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 $d\ge2$ 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.
