A Newton Div-Curl Least-Squares Finite Element Method for the Elliptic Monge–Ampère Equation

Abstract

Abstract This paper develops a new finite element approach for the efficient approximation of classical solutions of the elliptic Monge–Ampère equation. We use an outer Newton-like linearization and a first-order system least-squares reformulation at the continuous level to define a sequence of first-order div-curl systems. For problems on convex domains with smooth and appropriately bounded data, this framework gives robust results: convergence of the nonlinear iteration in a small number of steps, and optimal finite element convergence rates with respect to the meshsize. Numerical results using standard piecewise quadratic or cubic elements for all unknowns illustrate convergence results.