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Near and full quasi-optimality of finite element approximations of stationary second-order mean field games

Yohance A. P. Osborne, Iain Smears

TL;DR

The paper develops quantitative, a priori error bounds for stabilized piecewise affine finite element discretizations of stationary second-order mean field games on Lipschitz polytopal domains. By leveraging a discrete maximum principle and two stabilization frameworks (XZ and strict acuteness), the authors prove that the $H^1$-norm error in the density and value function is near quasi-optimal, with additional stabilization terms that are of optimal order in the mesh size. They further show that, on strictly acute meshes or under vanishing stabilization, full asymptotic quasi-optimality is achievable, enabling optimal convergence rates for sufficiently regular solutions. A complementary result on convex domains demonstrates that the $H^1$-norm error in the value function can converge optimally even when the density has minimal $H^1$ regularity, via a mixed $L^2$-$H^1$ bound. Numerical experiments corroborate the theoretical rates, including cases with nonsmooth value functions and densities, and illustrate the predicted optimal convergence behavior of the stabilized FEM for MFG systems.

Abstract

We establish a priori error bounds for monotone stabilized finite element discretizations of stationary second-order mean field games (MFG) on Lipschitz polytopal domains. Under suitable hypotheses, we prove that the approximation is asymptotically nearly quasi-optimal in the $H^1$-norm in the sense that, on sufficiently fine meshes, the error between exact and computed solutions is bounded by the best approximation error of the corresponding finite element space, plus possibly an additional term, due to the stabilization, that is of optimal order with respect to the mesh size. We thereby deduce optimal rates of convergence of the error with respect to the mesh-size for solutions with sufficient regularity. We further show full asymptotic quasi-optimality of the approximation error in the more restricted case of sequences of strictly acute meshes. Our third main contribution is to further show, in the case where the domain is convex, that the convergence rate for the $H^1$-norm error of the value function approximation remains optimal even if the density function only has minimal regularity in $H^1$.

Near and full quasi-optimality of finite element approximations of stationary second-order mean field games

TL;DR

The paper develops quantitative, a priori error bounds for stabilized piecewise affine finite element discretizations of stationary second-order mean field games on Lipschitz polytopal domains. By leveraging a discrete maximum principle and two stabilization frameworks (XZ and strict acuteness), the authors prove that the -norm error in the density and value function is near quasi-optimal, with additional stabilization terms that are of optimal order in the mesh size. They further show that, on strictly acute meshes or under vanishing stabilization, full asymptotic quasi-optimality is achievable, enabling optimal convergence rates for sufficiently regular solutions. A complementary result on convex domains demonstrates that the -norm error in the value function can converge optimally even when the density has minimal regularity, via a mixed - bound. Numerical experiments corroborate the theoretical rates, including cases with nonsmooth value functions and densities, and illustrate the predicted optimal convergence behavior of the stabilized FEM for MFG systems.

Abstract

We establish a priori error bounds for monotone stabilized finite element discretizations of stationary second-order mean field games (MFG) on Lipschitz polytopal domains. Under suitable hypotheses, we prove that the approximation is asymptotically nearly quasi-optimal in the -norm in the sense that, on sufficiently fine meshes, the error between exact and computed solutions is bounded by the best approximation error of the corresponding finite element space, plus possibly an additional term, due to the stabilization, that is of optimal order with respect to the mesh size. We thereby deduce optimal rates of convergence of the error with respect to the mesh-size for solutions with sufficient regularity. We further show full asymptotic quasi-optimality of the approximation error in the more restricted case of sequences of strictly acute meshes. Our third main contribution is to further show, in the case where the domain is convex, that the convergence rate for the -norm error of the value function approximation remains optimal even if the density function only has minimal regularity in .
Paper Structure (31 sections, 15 theorems, 112 equations, 2 figures)

This paper contains 31 sections, 15 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and Notation
  3. Notation for inequalities.
  4. General notation.
  5. Problem data.
  6. Continuous Problem
  7. Finite Element Discretization
  8. Meshes.
  9. Vertices and edges.
  10. Approximation space.
  11. General stabilized methods
  12. Vanishing stabilization.
  13. Xu--Zikatanov (XZ) condition xu1999monotone
  14. Formulation of the condition.
  15. Construction of the stabilization.
  16. ...and 16 more sections

Key Result

Lemma 2.1

For any $\epsilon>0$, there exists a $R>0$, depending only on $\epsilon$, $\Omega$, $L_H$ and $L_{H_p}$, such that whenever $v,w\in H^1(\Omega)$ satisfy $\|v- w\|_{H^1(\Omega)}\leq R$.

Figures (2)

  • Figure 1: Experiment 1 -- convergence plots for approximations of the value function and density function. The asymptotic rate of convergence for the total error in the $H^1$-norm is close to order $3/10$. This is due to the observed asymptotic rate of convergence in the $H^1$-norm for the approximations of the value function being close to the optimal value of $3/10$. The rate of convergence in the $H^1$-norm of the density function approximations is of optimal order $1$.
  • Figure 2: Experiment 2 -- convergence plots for approximations of the value function and density function. The asymptotic rate of convergence in the total error $\|m-m_k\|_{\Omega}+\|u-u_k\|_{H^1(\Omega)}$ is of optimal order 1. The convergence rate in the $H^1$-norm for the approximations of the density function is of order $1/2$, which is also optimal given the lower regularity of the density function.

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Remark 2.1: Semismoothness of the Hamiltonian
  • Remark 2.2: Existence and uniqueness of weak solutions
  • Remark 2.3: Essential boundedness of the density function
  • Lemma 3.1
  • proof
  • Remark 3.1: Dependencies of $C_{\mathrm{stab}}$
  • Remark 3.2: Well-posedness of discrete approximations and convergence
  • Lemma 3.2: osborne2023finite
  • ...and 26 more