Convergence of finite difference schemes to the Aleksandrov solution of the Monge-Ampere equation

TL;DR

This paper introduces a method to prove convergence of finite difference schemes to the Aleksandrov solution of the Monge-Ampere equation, emphasizing stability, consistency, and discrete convexity.

Contribution

It provides a novel convergence proof framework for finite difference schemes solving the Monge-Ampere equation using discrete convexity and stability conditions.

Findings

Established convergence of schemes to Aleksandrov solutions.

Defined discrete convexity with stability and equicontinuity properties.

Applicable to stable and consistent finite difference methods.

Abstract

We present a technique for proving convergence to the Aleksandrov solution of the Monge-Ampere equation of a stable and consistent finite difference scheme. We also require a notion of discrete convexity with a stability property and a local equicontinuity property for bounded sequences.