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A posteriori error bounds for finite element approximations of steady-state mean field games

Yohance A. P. Osborne, Iain Smears, Harry Wells

TL;DR

This work develops computable a posteriori error bounds for stabilized finite element discretizations of steady-state mean field games, proving a local equivalence between the $H^1$-error and the dual residual norm. It introduces a broad class of stabilized FEM estimators that couple standard residuals with stabilization terms, and provides reliability and efficiency results, including data-oscillation terms. For affine-preserving stabilizations, the authors show stabilization effects are bounded by jump terms, yielding a fully localizable, standard residual estimator. Numerical experiments on adaptive meshes demonstrate improved efficiency and accuracy, validating the theory on nonconvex and complex geometries.

Abstract

We analyze a posteriori error bounds for stabilized finite element discretizations of second-order steady-state mean field games. We prove the local equivalence between the $H^1$-norm of the error and the dual norm of the residual. We then derive reliable and efficient estimators for a broad class of stabilized first-order finite element methods. We also show that in the case of affine-preserving stabilizations, the estimator can be further simplified to the standard residual estimator. Numerical experiments illustrate the computational gains in efficiency and accuracy from the estimators in the context of adaptive methods.

A posteriori error bounds for finite element approximations of steady-state mean field games

TL;DR

This work develops computable a posteriori error bounds for stabilized finite element discretizations of steady-state mean field games, proving a local equivalence between the -error and the dual residual norm. It introduces a broad class of stabilized FEM estimators that couple standard residuals with stabilization terms, and provides reliability and efficiency results, including data-oscillation terms. For affine-preserving stabilizations, the authors show stabilization effects are bounded by jump terms, yielding a fully localizable, standard residual estimator. Numerical experiments on adaptive meshes demonstrate improved efficiency and accuracy, validating the theory on nonconvex and complex geometries.

Abstract

We analyze a posteriori error bounds for stabilized finite element discretizations of second-order steady-state mean field games. We prove the local equivalence between the -norm of the error and the dual norm of the residual. We then derive reliable and efficient estimators for a broad class of stabilized first-order finite element methods. We also show that in the case of affine-preserving stabilizations, the estimator can be further simplified to the standard residual estimator. Numerical experiments illustrate the computational gains in efficiency and accuracy from the estimators in the context of adaptive methods.

Paper Structure

This paper contains 39 sections, 16 theorems, 113 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation and Setting
  3. Basic notation.
  4. Notation for inequalities.
  5. Problem data.
  6. Continuous problem and equivalence between error and residual
  7. Local equivalence between error and residual
  8. Stability of HJB and FP equations
  9. Proof of Theorem \ref{['thm:error-res-equiv']}
  10. Stabilized finite element discretizations
  11. Setting and notation
  12. The mesh.
  13. Finite element space.
  14. Quasi-interpolation operator.
  15. Family of numerical schemes
  16. ...and 24 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 (8)

  • Figure 1: Experiment 1: Comparisons between the $H^1$-norm of the error, the total estimator $\eta(u_\mathcal{T},m_\mathcal{T})$, and residual estimator.
  • Figure 2: Experiment 2: A sample of the adaptively refined meshes.
  • Figure 3: Experiment 2: Comparison between adaptive and uniform mesh refinement.
  • Figure 4: Experiment 3: A sample of the adaptively refined mesh.
  • Figure 5: Experiment 3: Numerical solutions computed on the adaptively refined mesh with 20 refinements.
  • ...and 3 more figures

Theorems & Definitions (40)

  • Lemma 2.1
  • Remark 3.1: Existence and uniqueness of the solution
  • Theorem 3.1: Local equivalence between error and residual
  • Remark 3.2
  • Lemma 3.2
  • proof
  • Lemma 3.3: Stability of continous HJB equation
  • proof
  • Lemma 3.4: Stability of continuous KFP equation
  • proof
  • ...and 30 more