Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: classical solutions
This provides a simpler finite element method for solving the elliptic Monge-Ampere equation, which is important for applications in optimal transport and differential geometry, but the approach is incremental as it builds on existing variational techniques.
The authors propose a new variational formulation for the elliptic Monge-Ampere equation that allows standard Lagrange finite elements to solve classical solutions without jump terms, providing error estimates for degree ≥2 in 2D/3D and numerical evidence of weak solution approximation in 2D.
We propose a new variational formulation of the elliptic Monge-Ampere equation and show how classical Lagrange elements can be used for the numerical resolution of classical solutions of the equation. Error estimates are given for Lagrange elements of degree d >= 2 in dimensions 2 and 3. No jump term is used in the variational formulation. We propose to solve the discrete nonlinear system of equations by a time marching method and numerical evidence is given which indicates that one approximates weak solutions in two dimensions.