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

We analyse a posteriori error bounds for stabilised finite element discretisations of second-order steady-state mean field games. We prove the local equivalence between the H¹-norm of the error and the dual norm of the residual. We then derive reliable and efficient estimators for a broad class of stabilised first-order finite element methods. We also show that in the case of affine-preserving stabilisations, 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.