We address the numerical approximation of Mean Field Games with localcouplings. For power-like Hamiltonians, we consider both unconstrained andconstrained stationary systems with density constraints in order to model hardcongestion effects. For finite difference discretizations of the Mean FieldGame system, we follow a variational approach. We prove that the aforementionedschemes can be obtained as the optimality system of suitably definedoptimization problems. In order to prove the existence of solutions of thescheme with a variational argument, the monotonicity of the coupling term isnot used, which allow us to recover general existence results. Next, assumingnext that the coupling term is monotone, the variational problem is cast as aconvex optimization problem for which we study and compare several proximaltype methods. These algorithms have several interesting features, such asglobal convergence and stability with respect to the viscosity parameter, whichcan eventually be zero. We assess the performance of the methods via numericalexperiments.
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