A high-order Lagrange-Galerkin scheme for a class of Fokker-Planck equations and applications to mean field games

In this paper we propose a high-order numerical scheme for linear Fokker-Planck equations with a constant diffusion term. The scheme, which is built by combining Lagrange-Galerkin and semi-Lagrangian techniques, is explicit, conservative, consistent, and stable for large time steps compared with the space steps. We provide a convergence analysis for the exactly integrated Lagrange-Galerkin scheme, and we propose an implementable version with inexact integration. Our main application is the construction of a high-order scheme to approximate solutions of time dependent mean field games systems.