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