Finite element approximation of time-dependent mean field games with nondifferentiable Hamiltonians

TL;DR

This paper extends the finite element approximation framework to time-dependent mean field games with nondifferentiable Hamiltonians by formulating a PDE inclusion, proving existence and uniqueness of solutions, and demonstrating strong convergence of discretizations.

Contribution

It introduces a novel PDE inclusion approach for nondifferentiable Hamiltonians in MFGs, along with a finite element discretization and convergence analysis.

Findings

Proved existence and uniqueness of weak solutions.

Established strong convergence of the finite element scheme.

Demonstrated convergence of value function and density approximations.

Abstract

The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonian. However in many cases, the structure of the underlying optimal problem leads to a convex but nondifferentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusion (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniqueness of weak solutions to the resulting MFG PDI system under standard assumptions in the literature. We propose a monotone stabilized finite element discretization of the problem, using conforming affine elements in space and an implicit Euler discretization in time with mass-lumping. We prove the strong convergence in $L^2(H^1)$ of the value function approximations, and strong convergence in $L^p(L^2)$ of the density function approximations, together with strong $L^2$-convergence of the value function approximations at the initial time.