Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: classical solutions

We propose a new variational formulation of the elliptic Monge–Ampère 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 ≥ n in dimensions n = 2 and n = 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 in two dimension a larger class of nonsmooth solutions than what is possible if one simply uses Newton's method.