A Lagrange-Galerkin scheme for first order mean field games systems

TL;DR

This paper introduces a novel Lagrange-Galerkin numerical scheme for solving first order mean field games systems with non-local couplings, demonstrating convergence and practical implementation in multiple dimensions.

Contribution

It develops a combined Lagrange-Galerkin and semi-Lagrangian scheme for mean field games and proves its convergence in arbitrary dimensions.

Findings

Scheme successfully approximates mean field games in 1D and 2D

Convergence of the scheme is rigorously established

Numerical experiments validate the method's effectiveness

Abstract

In this work, we consider a first order mean field games system with non-local couplings. A Lagrange-Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton-Jacobi-Bellman equation, is proposed to discretize the mean field games system. The convergence of solutions to the scheme towards a solution to the mean field game system is established in arbitrary space dimensions. The scheme is implemented to approximate two mean field games systems in dimension one and two.