Convergent approaches for the Dirichlet Monge amp\`ere problem

TL;DR

This paper introduces three convergent numerical methods for solving the Dirichlet Monge-Ampère equation in two dimensions, utilizing new equivalent problems, finite difference discretization, and monotone schemes, with convergence proven and numerical results demonstrated.

Contribution

The paper presents three novel numerical approaches for the Dirichlet Monge-Ampère problem, including new problem formulations and convergence analysis.

Findings

Convergent finite difference schemes for the Dirichlet Monge-Ampère equation.

Application of Barles-Souganidis theory to prove scheme convergence.

Numerical results demonstrating the effectiveness of the proposed methods.

Abstract

In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite difference discretization and monotone schemes are obtained. Hence, we apply the Barles-Souganidis theory to prove the convergence of the schemes and the Damped Newtons method is used to compute the solutions of the schemes. Finally, some numerical results are illustrated.