Two-Scale Method for the Monge-Ampère Equation: Convergence to the Viscosity Solution
It provides a convergent numerical method for the Monge-Ampère equation on unstructured grids, addressing a known bottleneck in computational geometry and optimal transport.
The paper proposes a two-scale finite element method for the Monge-Ampère equation and proves it converges uniformly to the viscosity solution, extending a finite difference method to unstructured grids.
We propose a two-scale finite element method for the Monge-Ampère 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.