On Standard Finite Difference Discretizations of the Elliptic Monge–Ampère Equation

Abstract

Given an orthogonal lattice with mesh length h on a bounded two-dimensional convex domain $${\varOmega }$$Ω, we propose to approximate the Aleksandrov solution of the Monge---Ampere equation by regularizing the data and discretizing the equation in a subdomain using the standard finite difference method. The Dirichlet data is used to approximate the solution in the remaining part of the domain. We prove the uniform convergence on compact subsets of the solution of the discrete problems to an approximate problem on the subdomain. The result explains the behavior of methods based on the standard finite difference method and designed to numerically converge to non-smooth solutions.