This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple stationary case. We propose an abstract framework to study these solutions by reformulating the mean field game system as an abstract equation in a Banach space. In this context, stable equilibria turn out to be regular solutions to this equation, meaning that the linearized system is well-posed. We provide three applications of this property: we study the sensitivity analysis of stable solutions, establish error estimates for their finite element approximations, and prove the local converge of Newton's method in infinite dimensions.

翻译：本文针对二阶平均场博弈系统中稳定平衡态（可能不唯一）的数值分析，提出了一种全新的一般性方法。为简化讨论，我们聚焦于简单的平稳情形。通过将平均场博弈系统重述为巴拿赫空间中的抽象方程，我们构建了一个研究此类解的抽象框架。在此框架下，稳定平衡态恰为该方程的正则解，即其线性化系统是适定的。我们给出了该性质的三个应用：分析稳定解的敏感性、建立其有限元逼近的误差估计，并证明无穷维牛顿法的局部收敛性。