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

TL;DR

This work develops an $L^\\infty$-stable regularization of the Monge–Ampère equation via uniformly elliptic HJB equations, yielding convergence of the regularized solutions $u_\\varepsilon$ to the Alexandrov solution $u$ under minimal assumptions on $f$ and $g$. A key contribution is an $L^\\infty$-based stability estimate that is independent of the regularization parameter $\\varepsilon$, enabling guaranteed a posteriori error control for $H^2$-conforming FEM approximations of either $u$ or $u_\\varepsilon$. The authors derive a computable upper bound $RHS_0$, depending on boundary residuals and the Monge–Ampère measures of the convex envelope $\\Gamma_{v_h}$, and demonstrate its use in adaptive mesh refinement. Numerical experiments in 2D illustrate the method on regular and nonsmooth solutions, highlighting how adaptivity and boundary handling influence convergence and the reliability of the error bound. Overall, the paper provides a rigorous framework for reliable, provably bounded discretizations of the Monge–Ampère equation via HJB regularization, with practical guidance for implementing and validating adaptive FEM schemes.

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 $\varepsilon$. 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 $u_\varepsilon$.

Figures (7)

• Figure 1: Convergence history for the first experiment with $\varepsilon = 10^{-3}$.
• Figure 2: Convergence history for the second experiment with $\varepsilon = 10^{-4}$.
• Figure 3: Discrete solution on a uniform mesh with 4225 nodes.
• Figure 4: Adaptive mesh with 1907 nodes for the second experiment.
• Figure 5: Convergence history for the third experiment with $\varepsilon = 10^{-4}$.

Theorems & Definitions (34)

• Lemma 1.1: Alexandrov maximum principle
• proof
• Definition 2.1: viscosity solution
• Lemma 2.2: classical comparison principle
• proof
• Lemma 2.3: comparison principle
• proof
• Proposition 2.4: properties of HJB equation
• proof
• Theorem 2.5: $L^\infty$ stability