Two-Scale Method for the Monge-Amp\`ere Equation: Convergence to the Viscosity Solution

TL;DR

This paper introduces a two-scale finite element method for solving the Monge-Ampère equation, ensuring convergence to the viscosity solution on unstructured grids by using dual scales and discrete comparison principles.

Contribution

It develops a novel two-scale finite element approach that extends previous finite difference methods to unstructured grids with proven convergence.

Findings

Method converges uniformly to the viscosity solution.

Applicable to unstructured grids in dimensions ≥ 2.

Uses discrete comparison principles and barrier functions.

Abstract

We propose a two-scale finite element method for the Monge-Amp\`ere equation with Dirichlet boundary condition in dimension $d\ge2$ and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite difference method of Froese and Oberman, but is defined on unstructured grids and relies on two separate scales: the first one is the mesh size $h$ and the second one is a larger scale that controls appropriate directions and substitutes the need of a wide stencil. The main tools for the analysis are a discrete comparison principle and discrete barrier functions that control the behavior of the discrete solution, which is continuous piecewise linear, both close to the boundary and in the interior of the domain.