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

TL;DR

This paper develops reliable a posteriori error bounds for stabilized finite element methods applied to steady-state mean field games, enabling more efficient and accurate adaptive numerical solutions.

Contribution

It introduces new error estimators for stabilized finite element discretizations of mean field games, including simplified forms for affine-preserving stabilizations.

Findings

Error bounds are locally equivalent to the dual residual norm.

Estimators are reliable and efficient for a broad class of methods.

Numerical experiments demonstrate improved efficiency and accuracy.

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.