Uniform limit of discrete convex functions

TL;DR

This paper proves that uniformly bounded sequences of discrete convex functions on meshes converge uniformly on compact sets to a continuous convex function, with applications to numerical methods for the Monge-Ampère equation.

Contribution

It establishes the convergence of discrete convex functions to continuous convex functions under broad conditions, including non-uniformly convex domains.

Findings

Uniform limits of discrete convex functions are continuous convex functions.

Limit functions satisfy boundary conditions strongly when interpolating boundary data.

Results apply to convergence analysis of numerical methods for the Monge-Ampère equation.

Abstract

We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform limit on compact subsets of discrete convex mesh functions which are uniformly bounded is a continuous convex function. Furthermore, if the discrete convex mesh functions interpolate boundary data of a continuous convex function and their Monge-Ampere masses are uniformly bounded, the limit function satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. The result is applied to the convergence of some numerical methods for the Monge-Ampere equation.