Pseudo transient continuation and time marching methods for Monge-Ampere type equations

TL;DR

This paper introduces two novel numerical methods for solving the fully nonlinear elliptic Monge-Ampere equation, demonstrating convergence to convex solutions and providing theoretical and numerical support.

Contribution

The paper proposes a pseudo transient continuation and a pure pseudo time marching method with proven convergence for Monge-Ampere equations, including for non-smooth solutions.

Findings

Methods converge to strictly convex solutions

Theoretical proof of convergence for smooth solutions

Numerical evidence supports effectiveness for non-smooth solutions

Abstract

We present two numerical methods for the fully nonlinear elliptic Monge-Ampere equation. The first is a pseudo transient continuation method and the second is a pure pseudo time marching method. The methods are proven to converge to a strictly convex solution of a natural discrete variational formulation with $C^1$ conforming approximations. The assumption of existence of a strictly convex solution to the discrete problem is proven for smooth solutions of the continuous problem and supported by numerical evidence for non smooth solutions.