Rates of convergence of finite element approximations of second-order mean field games with nondifferentiable Hamiltonians

TL;DR

This paper establishes convergence rates for finite element methods applied to second-order mean field games with nondifferentiable Hamiltonians, providing theoretical guarantees in various norms.

Contribution

It introduces the first convergence rate results for finite element approximations of MFGs with nondifferentiable Hamiltonians in general domains.

Findings

Convergence rate in $H^1$-norm for value function approximations.

Convergence rate in $L^2$-norm for density approximations.

Error bounds for regularized Hamiltonian discretizations.

Abstract

We prove a rate of convergence for finite element approximations of stationary, second-order mean field games with nondifferentiable Hamiltonians posed in general bounded polytopal Lipschitz domains with strongly monotone running costs. In particular, we obtain a rate of convergence in the $H^1$-norm for the value function approximations and in the $L^2$-norm for the approximations of the density. We also establish a rate of convergence for the error between the exact solution of the MFG system with a nondifferentiable Hamiltonian and the finite element discretizations of the corresponding MFG system with a regularized Hamiltonian.