On standard finite difference discretizations of the elliptic Monge-Ampere equation
This work provides theoretical justification and a practical algorithm for numerical solutions of the Monge-Ampère equation, benefiting computational mathematicians working on nonlinear PDEs.
The paper proposes a standard finite difference method for the elliptic Monge-Ampère equation, proving uniform convergence on compact subsets for an approximate problem. The algorithm is faster than Newton's method for smooth solutions and maintains high accuracy for non-smooth solutions, with convergence independent of initial guess quality.
Given an orthogonal lattice with mesh length h on a bounded convex domain, 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. We propose an algorithm which for smooth solutions appears faster than the popular Newton's method with a high accuracy for non smooth solutions. The convergence of the algorithm is independent of how close to the numerical solution the initial guess is, upon rescaling the equation and given a user's measure of the closeness of an initial guess.