Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

Abstract

We propose a two-scale finite element method for the Monge-Ampère equation with Dirichlet boundary condition in dimension d ≥ 2 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.

Convergent Two-Scale Filtered Scheme for the Monge--Ampère Equation

“…Since Ω is polygonal, the computational domain Ω h = Ω and the isoparametric maps of T 2 2h for boundary elements are simply affine. This choice simplifies the numerics and allows us to compare with earlier experiments from [27]. We indeed…”

“…Monotone Operator [27,28]: We discretize the domain Ω by a shape regular and quasi-uniform mesh T 1 h with spacing h, the fine scale, and construct a space V 1 h of continuous piecewise linear functions over T 1 h . The superscript 1 of V 1 h indicates the use of linear polynomials whereas that of T 1 h entails the use of straight (affine equivalent) simplices.…”

The Monge–Ampère equation

“…Having defined all the discretization ingredients, which are parametrized by the triple ε = (h, δ, θ), following [89] we introduce the two scale discrete Monge-Ampère operator by defining, for w h ∈ X h , and…”

“…Let us now provide, following [89,90,76], an analysis of (2.54). We will first introduce a discrete notion of convexity, based on the positivity of the second differences defined in (2.51).…”

“…On the other hand, discrete convexity implies nonnegativity of the two scale discrete Monge-Ampère operator; see [89,Lemma 2.2].…”