Rates of convergence of finite element approximations of second-order mean field games with nondifferentiable Hamiltonians
This provides theoretical convergence guarantees for numerical solutions of mean field games with nondifferentiable Hamiltonians, which are important for applications in economics and engineering.
The paper proves convergence rates for finite element approximations of stationary second-order mean field games with nondifferentiable Hamiltonians in bounded polytopal Lipschitz domains, achieving rates in H^1-norm for value function and L^2-norm for density approximations. It also establishes convergence rates for the error between the exact solution and discretizations using a regularized Hamiltonian.
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.