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

Jules Berry

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.

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

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.

Paper Structure

This paper contains 19 sections, 21 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Notations and assumptions
  3. Notations.
  4. Assumptions.
  5. General properties and stable solutions
  6. Preliminary results on parabolic equations
  7. Well-posedness
  8. Viscous Hamilton-Jacobi equations on nonsmooth domains
  9. Well-posedness of the MFG system
  10. Stable solutions
  11. Semi-discrete error estimates for finite element approximations of stable solutions
  12. Some results on finite element approximations
  13. Estimates on linear semidiscrete problems
  14. Semidiscrete approximations of stable solutions to the MFG system
  15. Nemytskii operators
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $b \in L^{d+2}(Q_T)$, $f,\, g \in L^2(Q_T)$, and $\rho_0 \in L^2(\Omega)$. Then there exists a unique weak solution $\rho \in \mathcal{H}^1_2(Q_T)$ to and there exists $C = C( \left \lVert {b} \right \rVert_{L^{d+2}}, T, d)$ such that

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1: LSU1968
  • Proposition 3.2: De Giorgi-Nash-Moser, LSU1968
  • Proposition 3.3: Maximal $L^p$ regularity, HP1997
  • Lemma 3.4
  • proof
  • Definition 3.5: Weak solution
  • Remark 3.6
  • Proposition 3.7: Comparison principle
  • ...and 29 more