Monotone and consistent discretization of the Monge-Ampère operator

Abstract

We introduce a novel discretization of the Monge-Ampère operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method’s efficiency.