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.

翻译：我们分析了二阶稳态平均场博弈的稳定化有限元离散格式的后验误差界。证明了误差的$H^1$-范数与残差对偶范数之间的局部等价性。随后针对一大类稳定化一阶有限元方法，推导出可靠且高效的误差估计子。进一步证明，对于仿射保持型稳定化格式，该估计子可简化为标准残差估计子。数值实验展示了在自适应方法框架下，采用该估计子对计算效率与精度的提升效果。