Convergence of a regularized finite element discretization of the two-dimensional Monge–Ampère equation

Abstract:

This paper proposes a regularization of the Monge–Ampère 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 uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge–Ampère 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 uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the $L^1$ norm. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.