Some error estimates for semidiscrete finite element approximations of stable solutions to mean field game systems

TL;DR

This paper establishes error estimates for finite element approximations of stable solutions to time-dependent mean field game systems, linking stability to invertibility and achieving quasi-optimal convergence under regularity assumptions.

Contribution

It introduces a novel approach using nonlinear mapping and the Brezzi-Rappaz-Raviart theorem to derive error bounds for semidiscrete MFG solutions.

Findings

Proved existence of solutions using approximation theorem and regularity estimates.

Derived quasi-optimal error bounds for sufficiently smooth solutions.

Linked solution stability to the invertibility of a nonlinear mapping.

Abstract

We derive a priori error estimates for semidiscrete finite element approximations of stable solutions to time-dependent mean field game systems with Dirichlet boundary conditions. Expressing solutions to the MFG system as zeros of a nonlinear abstract mapping, we show that the stability of solutions is equivalent to the invertibility of its differential. This characterization allows us to apply the Brezzi-Rappaz-Raviart approximation theorem in combination with discrete L p maximal regularity estimates to prove existence of solutions to the semidiscrete MFG system and to derive the error estimate. Finally, for solutions satisfying sufficient regularity assumptions, we establish quasi-optimal error bounds, meaning the approximation achieves the best possible convergence rate when the solution has sufficient smoothness.