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

01/17/2023

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by   Dietmar Gallistl, et al.

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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^∞ norm from the theory of viscosity solutions which are independent of the regularization parameter ε. They allow for the uniform convergence of the solution u_ε 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^∞ norm for continuously differentiable finite element approximations of u or u_ε.