# Two-scale methods for convex envelopes

## TL;DR

This paper introduces two-scale numerical methods for efficiently computing convex envelopes of continuous functions, providing convergence proofs and error estimates based on a nonlinear obstacle problem formulation.

## Contribution

It develops and analyzes two-scale methods for convex envelope computation, extending previous finite difference approaches with rigorous convergence and error analysis.

## Key findings

- Methods converge to the true convex envelope
- Error estimates are established in the max norm
- Approach extends to modified wide stencil finite difference methods

## Abstract

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension.This hinges on a fully nonlinear obstacle formulation [A. M. Oberman, "The convex envelope is the solution of a nonlinear obstacle problem", Proc. Amer. Math. Soc. 135(6):1689--1694, 2007]. We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method of [A. M. Oberman, "Computing the convex envelope using a nonlinear partial differential equation", Math. Models Meth. Appl. Sci, 18(05):759--780, 2008].