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 ε u_\varepsilon and is accessible to the discretization with finite elements. This work establishes uniform convergence of u ε u_\varepsilon to the convex Alexandrov solution u u to the Monge–Ampère equation as the regularization parameter ε \varepsilon approaches 0 0 . A mixed finite element method for the approximation of u ε 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 L^1 norm. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions u u .