On standard finite difference discretizations of the elliptic Monge-Ampère equation

Given an orthogonal lattice with mesh length $h$ on a bounded convex domain $\Omega$, we show that the Aleksandrov solution of the Monge-Ampère equation is the pointwise limit of mesh functions $u_h$ which solve a discrete Monge-Ampère equation with the Hessian discretized using the standard finite difference method. The result provides the mathematical foundation of methods based on the standard finite difference method and designed to numerically converge to non-smooth solutions.