Convergence of a regularized finite element discretization of the two-dimensional Monge-Amp\`ere equation

Abstract

This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Amp\`ere equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.