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

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.

arXiv comment39 pages

References in corpus (5)

• Firedrake: automating the finite element method by composing abstractions
• Proximal methods for stationary Mean Field Games with local couplings
• Time-parallel iterative solvers for parabolic evolution equations
• Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions
• Second order local minimal-time Mean Field Games

Cited by in corpus (4)

• Regularization of Stationary Second-order Mean Field Game Partial Differential Inclusions
• A posteriori error bounds for finite element approximations of steady-state mean field games
• An introduction to the a posteriori error analysis of parabolic partial differential equations
• Rates of convergence of finite element approximations of second-order mean field games with nondifferentiable Hamiltonians