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

TL;DR

This work addresses the numerical approximation of the two-dimensional Monge-Ampère equation by regularizing it into a uniformly elliptic Hamilton-Jacobi-Bellman problem and discretizing the regularized system with a mixed finite element method. It proves existence, uniqueness, and stability of the regularized strong solution $u_ε$, establishes convergence to the Alexandrov solution as $ε→0$, and provides a sublinear rate $∥u−u_ε∥_{L^∞}≲ε^{1/32}$ under smooth data. The paper then develops and analyzes a mixed FEM discretization for $u_ε$, proving convergence on arbitrary coarse meshes and offering a practical pathway to approximate singular Monge-Ampère solutions. Numerical experiments on the unit square confirm robust convergence behavior for various singular solutions and illustrate the method’s effectiveness when $f$ is non-smooth or vanishes in regions, highlighting the practical impact of the regularized-HJB approach for fully nonlinear PDEs.

Abstract

This paper proposes a regularization of the Monge-Ampère equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Ampère equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.

Figures (5)

• Figure 1: Convergence history for the first experiment.
• Figure 2: Convergence history for the second experiment.
• Figure 3: Convergence history for the third experiment.
• Figure 4: Graph of the computed approximation in the third experiment; mesh size $h=1/64$ and $\varepsilon=10^{-3}$.
• Figure 5: Convergence history for the fourth experiment.

Theorems & Definitions (33)

• Definition 2.1: Alexandrov solution
• Definition 2.2: convex viscosity solution
• Theorem 2.3: existence and uniqueness
• proof
• Definition 2.4: viscosity solution
• Remark 2.5: convexity
• Theorem 2.6: equivalence
• Lemma 3.1: convexity and uniform ellipticity of $F_\varepsilon$
• proof
• Lemma 3.2: comparison principle