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：本文提出了一种新的一般性方法，用于对二阶平均场博弈系统中稳定均衡解进行数值分析，尤其适用于解的唯一性可能失效的情形。为简化讨论，我们聚焦于简单的平稳情况。我们构建了一个抽象框架，通过将平均场博弈系统重新表述为巴拿赫空间中的抽象方程来研究这些解。在此背景下，稳定均衡解恰好是该方程的正则解，即线性化系统是适定的。我们给出了该性质的三个应用：研究稳定解的敏感性分析，建立其有限元近似的误差估计，并证明了牛顿法在无穷维空间中的局部收敛性。