The Monge–Ampère equation \[\begin{aligned} \det D^2 u = f \text{ in } \Omega \quad\text{and}\quad u = g \text{ on } \partial \Omega \end{aligned}\] in a convex domain \(\Omega\) with suitable data \(f\), \(g\) admits a unique generalized solution in the cone of convex functions. The use of high-order methods or local mesh refinement is very desirable for the discretization of the above problem. On the other hand, a stable algorithmic realization of the finite element method is difficult to achieve due to the strong nonlinearity and the convexity constraint.

This talk discusses a regularization approach 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. The contribution establishes locally uniform convergence of \(u_\varepsilon\) to the convex Alexandrov solution \(u\) to the Monge–Ampère equation as the regularization parameter \(\varepsilon\) approaches \(0\). A finite element method for the approximation of \(u_\varepsilon\) is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Based on Alexandrov’s estimate, some a posteriori error estimates are also shown.

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