  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.
