  In this paper, we propose and analyze a finite element discretization for the
computation of fractional minimal graphs of order~$s \in (0,1/2)$ on a bounded
domain $\Omega$. Such a Plateau problem of order $s$ can be reinterpreted as a
Dirichlet problem for a nonlocal, nonlinear, degenerate operator of order $s +
1/2$. We prove that our numerical scheme converges in $W^{2r}_1(\Omega)$ for
all $r<s$, where $W^{2s}_1(\Omega)$ is closely related to the natural energy
space. Moreover, we introduce a geometric notion of error that, for any pair of
$H^1$ functions, in the limit $s \to 1/2$ recovers a weighted $L^2$-discrepancy
between the normal vectors to their graphs. We derive error bounds with respect
to this novel geometric quantity as well. In spite of performing approximations
with continuous, piecewise linear, Lagrangian finite elements, the so-called
{\em stickiness} phenomenon becomes apparent in the numerical experiments we
present.
