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The numerical methods are presented for solving economic problems formulated in the Mean-Field Game (MFG) form. The mean-field equilibrium is a solution of the coupled system of two parabolic partial differential equations: the Fokker–Planck–Kolmogorov equation and the Hamilton–Jacobi–Bellman one. The description focuses on the discrete approximation of these equations and on the application of the MFG theory directly at discrete level. This approach results in an efficient algorithm for finding the corresponding grid control functions. Contrary to other difference schemes, here the semi-Lagrangian approximation is applied, which improves properties of a discrete problem of this type. This implies the fast convergence of an iterative algorithm for the minimization of the cost functional. The constructed algorithms are implemented to the problem of carbon dioxide pollution. • Mean Field Games describe many events in economics and physics. • They generalize Nash's equilibrium for infinite number of participants (particles). • Special difference schemes inherit properties of Mean Field Game at discrete level. • With these schemes, iterative process for finding the equilibrium is more effective. [ABSTRACT FROM AUTHOR]

An adaptive mesh refinement algorithm based on a continuous adjoint approach is developed. Both the primal equation and the adjoint equation are approximated with the discontinuous Galerkin (DG) method. The proposed adaptive algorithm is used in compressible Euler equations. Numerical tests are made to show the superiority of the proposed adaptive algorithm. [ABSTRACT FROM AUTHOR]