  We carry out an asymptotic analysis of a thin nematic liquid crystal in which
one elastic constant dominates over the others, namely \begin{align}
\label{energyab} \inf E_\varepsilon(u)\quad\mbox{where}\quad E_\varepsilon(u)
:= \frac{1}{2}\int_\Omega \left\{\varepsilon\,|\nabla u|^2 +
\frac{1}{\varepsilon} \,(|u|^2 - 1)^2 + L \,(\mathrm{div}\,u)^2\right\} \,dx.
\end{align} Here $u: \Omega \to \mathbb R^2$ is a vector field, $0 <
\varepsilon \ll 1 $ is a small parameter, and $L > 0$ is a fixed constant,
independent of $\varepsilon$. We derive the $\Gamma$-limit $E_0$, which is a
sum of a bulk term penalizing divergence and an Aviles-Giga type wall energy
involving the cube of the jump in the tangential component of the
$\mathbb{S}^1$-valued order parameter. We then derive criticality conditions
for $E_0$ and analyze minimization of $E_0$ both rigorously and numerically for
various domains $\Omega$ and a variety of Dirichlet boundary conditions.
