  In this article we study the numerical solution of the $L^1$-Optimal
Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation
introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the
Riemannian setting the DMK model and conjecture the equivalence with the
solution Monge-Kantorovich equations, a PDE-based formulation of the
$L^1$-Optimal Transport Problem.
  We generalize the numerical method proposed in
[FaccaCardinPutti:2018,FaccaDaneriCardinPutti:2020] to 2D surfaces embedded in
$\REAL^3$ using the Surface Finite Element Model approach to approximate the
Laplace-Beltrami equation arising from the model. We test the accuracy and
efficiency of the proposed numerical scheme, comparing our approximate solution
with respect to an exact solution on a 2D sphere. The results show that the
numerical scheme is efficient, robust, and more accurate with respect to other
numerical schemes presented in the literature for the solution of
ls$L^1$-Optimal Transport Problem on 2D surfaces.
