Stability and guaranteed error control of approximations to the Monge--Amp\`ere equation

This paper analyzes a regularization scheme of the Monge--Amp\`ere 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--Amp\`ere 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 $u_\varepsilon$.