  Given a bounded open set $\Omega \subset \mathbb{R}^2$, we study the
relaxation of the nonparametric area functional in the strict topology in
$BV(\Omega;\mathbb{R}^2)$, and compute it for vortex-type maps, and more
generally for maps in $W^{1,1}(\Omega;\mathcal{S}^1)$ having a finite number of
topological singularities. We also extend the analysis to some specific
piecewise constant maps in $BV(\Omega;\mathcal{S}^1)$, including the symmetric
triple junction map.
