The relaxed area of ${\mathcal{S}}^1$-valued singular maps in the strict $BV$-convergence

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