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

Abstract

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 $\epsilon$. They allow for the uniform convergence of the solution $u_{\epsilon }$ 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 $u_{\epsilon }$.

References in corpus (1)

• Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation