i will present the dynamical formulation of optimal transport (a.k.a benamou-brenier formulation): it consists in writing the optimal transport problem as the minimization of a convex functional under a pde constraint, and can handle a priori a vast class of cost functions and geometries. it is one of the oldest numerical method to solve the problem, and it is also the basis for a lot of extensions and generalizations of the optimal transport problem. the optimization problem is then discretized to end up with a finite dimensional convex optimization problem. i will illustrate this method by presenting a discretization when the ground space is a surface. although much effort has been devoted to solve efficently the discretized problem, the study of convergence under mesh refinement of the solution of the approximate problems has only been tackled recently. i will present an abstract framework guaranteeing convergence under mesh refinement, with no condition on the relative scale of the spacial and temporal mesh sizes, and even if the densities are very singular.