  In this article we aim at defining the discrete Gaussian free field (DGFF) on
a compact manifold. Since there is no canonical grid approximation of a
manifold, we construct a random graph that suitably replaces the square lattice
$\mathbb{Z}^d$ in Euclidean space, and prove that the scaling limit of the DGFF
is given by the manifold continuum Gaussian free field (GFF). Furthermore using
Voronoi tessellations we can interpret the DGFF as element of a Sobolev space
and show convergence to the GFF in law with respect to the strong Sobolev
topology.
