  This paper proposes a multiscale method for solving the numerical solution of
mean field games which accelerates the convergence and addresses the problem of
determining the initial guess. Starting from an approximate solution at the
coarsest level, the method constructs approximations on successively finer
grids via alternating sweeping, which not only allows for the use of classical
time marching numerical schemes but also enables applications to both local and
nonlocal problems. At each level, numerical relaxation is used to stabilize the
iterative process. A second-order discretization scheme is derived for
higher-order convergence. Numerical examples are provided to demonstrate the
efficiency of the proposed method in both local and nonlocal, 1-dimensional and
2-dimensional cases.
