The Monge-Amp\`{e}re equation

TL;DR

This paper reviews recent numerical methods for solving the Monge-Ampère equation, emphasizing stability, convergence, and practical performance across various computational techniques.

Contribution

It provides a comprehensive overview of recent advances in numerical analysis of the Monge-Ampère equation, including stability estimates and convergence rates for multiple methods.

Findings

Finite difference schemes achieve convergence with stability estimates.

Finite element and geometric methods demonstrate effective approximation.

Numerical experiments validate the theoretical convergence rates.

Abstract

We review recent advances in the numerical analysis of the Monge-Amp\`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods based on geometric considerations. Particular focus is the development of appropriate stability and consistency estimates which lead to rates of convergence of the discrete approximations. Finally we present numerical experiments which highlight each method for a variety of test problem with different levels of regularity.