Finite elements for Wasserstein W_pgradient flows

Wasserstein Wp gradient flows for nonlinear integral functionals of the density yield degenerate parabolic equations involving diffusion operators of q-Laplacian type, with q being p's conjugate exponent. We propose a finite element scheme building on conformal P1 Lagrange elements with mass lumping and a backward Euler time discretization strategy. Our scheme preserves mass and positivity while energy decays in time. Building on the theory of gradient flows in metric spaces, we further prove convergence towards a weak solution of the PDE that satisfies the energy dissipation equality. The analytical results are illustrated by numerical simulations.