Stability and guaranteed error control of approximations to the Monge–Ampère equation

This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the \(L^\infty \) norm from the theory of viscosity solutions which are independent of the regularization parameter \(\varepsilon \). They allow for the uniform convergence of the solution \(u_\varepsilon \) to the regularized problem towards the Alexandrov solution u to the Monge–Ampère equation for any nonnegative \(L^n\) right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the \(L^\infty \) norm for continuously differentiable finite element approximations of u or \(u_\varepsilon \).