  We address the numerical approximation of Mean Field Games with local
couplings. For power-like Hamiltonians, we consider both unconstrained and
constrained stationary systems with density constraints in order to model hard
congestion effects. For finite difference discretizations of the Mean Field
Game system, we follow a variational approach. We prove that the aforementioned
schemes can be obtained as the optimality system of suitably defined
optimization problems. In order to prove the existence of solutions of the
scheme with a variational argument, the monotonicity of the coupling term is
not used, which allow us to recover general existence results. Next, assuming
next that the coupling term is monotone, the variational problem is cast as a
convex optimization problem for which we study and compare several proximal
type methods. These algorithms have several interesting features, such as
global convergence and stability with respect to the viscosity parameter, which
can eventually be zero. We assess the performance of the methods via numerical
experiments.
