  The $L^1$ optimal transport density $\mu^*$ is the unique $L^\infty$ solution
of the Monge-Kantorovich equations. It has been recently characterized also as
the unique minimizer of the $L^1$ -transport energy functional E. In the
present work we develop and we prove convergence of a numerical approxi- mation
scheme for $\mu^*$ . Our approach relies upon the combination of a FEM-
inspired variational approximation of E with a minimization algorithm based on
a gradient flow method.
