A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

Abstract

We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in R 2 {\mathbb {R}}^2 . It is based on an isoparametric C 0 C^0 finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented.