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

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 uε and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of uε to the convex Alexandrov solution u to the Monge-Ampère equation as the regularization parameter ε approaches 0. A mixed finite element method for the approximation of uε 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.

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

“…The proof is carried out in any space dimension n and does not rely on the concept of strong solutions in two space dimensions from [18,19]. It departs from a main result of [12]. Theorem 3.1 (convergence of regularization for smooth data).…”

“…with the following two applications. First, this paper establishes, in extension to [12], uniform convergence of (generalized) viscosity solutions u ε of the regularized PDE (1.2) to the Alexandrov solution u ∈ C(Ω) of the Monge-Ampère equation (1.2) under the (essentially) minimal assumptions 0 ≤ f ∈ L n (Ω) and g ∈ C(∂Ω) on the data. Second, (1.4) provides guaranteed error control in the L ∞ norm (even for inexact solve) for H 2 conforming FEM.…”

“…In addition, it was shown [4] that if f ∈ C 0,α (Ω), 0 < λ ≤ f ≤ Λ, and g ∈ C 1,β (∂Ω) with positive constants 0 < α, β < 1 and 0 < λ ≤ Λ, then u ∈ C(Ω) ∩ C 2,α loc (Ω). It is known [14,10] that (1.1) can be equivalently formulated as a Hamilton-Jacobi-Bellman (HJB) equation, a property that turned out useful for the numerical solution of (1.1) [10,12]; one of the reasons being that the latter is elliptic on the whole space of symmetric matrices S ⊂ R n×n and, therefore, the convexity constraint is automatically enforced by the HJB formulation. For nonnegative continuous right-hand sides 0 ≤ f ∈ C(Ω), the Monge-Ampère equation (1.1) is equivalent to F 0 (f ; x, D 2 u) = 0 in Ω and u = g on ∂Ω with F 0 (f ; x, M ) := sup A∈S(0) (−A : M + f n √ det A) for any x ∈ Ω and M ∈ R n×n .…”

“…Here, S(0) := {A ∈ S : A ≥ 0 and tr A = 1} denotes the set of positive semidefinite symmetric matrices A with unit trace tr A = 1. Since F 0 is only degenerate elliptic, the regularization scheme proposed in [12] replaces S(0) by a compact subset S(ε) := {A ∈ S(0) : A ≥ ε} ⊂ S(0) of matrices with eigenvalues bounded from below by the regularization parameter 0 < ε ≤ 1/n. The solution u ε to the regularized PDE solves…”

“…in Ω [18,19]. The paper [12] establishes uniform convergence of u ε towards the generalized solution u to the Monge-Ampère equation (1.1) as ε 0 under the assumption g ∈ H 2 (Ω) ∩ C 1,α (Ω) and that 0 ≤ f ∈ L 2 (Ω) can be approximated from below by a pointwise monotone sequence of positive continuous functions.…”