Convergence of the fully discrete JKO scheme

The JKO scheme provides the discrete-in-time approximation for the solutions of evolutionary equations with Wasserstein gradient structure. We study a natural space-discretization of this scheme by restricting the minimization to the measures supported on the nodes of a regular grid. The study of the fully discrete JKO scheme is motivated by the applications to developing numerical schemes for the nonlinear diffusion equation with drift and the crowd motion model. The main result of this paper is the convergence of the scheme as both the time and space discretization parameters tend to zero in a suitable regime.