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

Osborne, Yohance A. P.; Smears, Iain

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 L2(H1) of the value function approximations, and strong convergence in Lp(L2) of the density function approximations, together with strong L2-convergence of the value function approximations at the initial time.

Citation

Osborne, Y. A. P., & Smears, I. (2025). Finite element approximation of time-dependent mean field games with nondifferentiable Hamiltonians. Numerische Mathematik, 157(1), 165-211. https://doi.org/10.1007/s00211-024-01447-2

• Journal Article Type: Article
• Acceptance Date: Nov 27, 2024
• Online Publication Date: Dec 11, 2024
• Publication Date: Feb 1, 2025
• Deposit Date: Jan 8, 2025
• Publicly Available Date: Jan 8, 2025
• Journal: Numerische Mathematik
• Print ISSN: 0029-599X
• Electronic ISSN: 0945-3245
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 157
• Issue: 1
• Pages: 165-211
• DOI: https://doi.org/10.1007/s00211-024-01447-2
• Keywords: 65M60, 65M12
• Public URL: https://durham-repository.worktribe.com/output/3327341