The numerical approximation of the minimum problem: ${min}_{A\subseteq \mathrm{\Omega }}\tilde{\mathcal{F}}\left(A\right)$, is considered, where $\tilde{\mathcal{F}}\left(A\right)=P_{\mathrm{\Omega }}\left(A\right)+\cos \left(\theta \right){\mathcal{H}}^{n-1}\left(\partial A\cap \partial \mathrm{\Omega }\right)-{\int }_A\kappa$. The solution to this problem is a set $A\subseteq \mathrm{\Omega }\subset {\mathbb{R}}^n$ with prescribed mean curvature $\kappa$ and contact angle $\theta$ at the intersection of $\partial A$ with $\partial \mathrm{\Omega }$. The functional $\tilde{\mathcal{F}}$ is first relaxed with a sequence of nonconvex functionals defined in $H^1\left(\mathrm{\Omega }\right)$ which, in turn, are discretized by finite elements. The $\mathrm{\Gamma }$-convergence of the discrete functionals to $\tilde{\mathcal{F}}$ as well as the compactness of any sequence of discrete absolute minimizers are proven.

We compute the Γ-limit of a sequence of non-local integral functionals depending on a regularization of the gradient term by means of a convolution kernel. In particular, as Γ-limit, we obtain free discontinuity functionals with linear growth and with anisotropic surface energy density.