Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions

Abstract

Abstract. We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampère equation using finite dimensional spaces of piecewise polynomial C 0 or C 1 functions. Standard discretizations of the type considered in this paper have been previous analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge-Ampère equation.

The Monge–Ampère equation

“…While this method may be convergent (cf. [19,84,6,3,33,10]), the appearance of global secondorder derivatives in the method necessitates the use of C 1 finite element spaces which can be arduous to implement and are not found in most finite element software packages. In addition, C 1 finite element generally require high-degree polynomial bases, resulting in a relatively large algebraic system.…”

A convexity enforcing $${C}^{{0}}$$ interior penalty method for the Monge–Ampère equation on convex polygonal domains

“…There are many numerical approaches to the Dirichlet boundary value problem of the Monge-Ampère equation (and related equations) in 2 and 3 spatial dimensions, with respect to different solution classes (classical solutions, Aleksandrov solutions [2] and viscosity solutions [54]). They include (i) geometric finite difference methods [63,66,68,69], (ii) monotone finite difference methods [7][8][9][39][40][41]48,50,67], (iii) augmented Lagrangian and least-squares finite element methods [19,[28][29][30][31], (iv) finite element methods based on the vanishing moment approach [3,[35][36][37]57], (v) finite element methods based on L 2 projection [4,5,10,11,13,15,27,51,[58][59][60], (vi) finite element methods based on a reformulation of the Monge-Ampère equation as a Hamilton-Jacobi-Bellman equation [14,34], and (v) two-scale methods [53,64,65]. Comprehensive reviews of the literature can be found in [33,61].…”

On Standard Finite Difference Discretizations of the Elliptic Monge–Ampère Equation

“…A sequence u m of convex functions is locally equicontinuous by [26, Lemma 3.2.1], c.f. [3] for details. If the sequence is also locally uniformly bounded, the result follows from the Arzela-Ascoli theorem [35, p. 179].…”

“…By [11], the problem (1.3) has a unique convex solution u ms ∈ C ∞ (Ω s ). As s → ∞, the sequence u ms converges uniformly on compact subsets of Ω to the unique convex solution u m ∈ C( Ω) of the problem (1.4) [3].…”

“…It is known, c.f. [3] or [36,Proposition 2.4 ], that the Aleksandrov solution of (1.2) det D 2 u m = f m in Ω, u m = g m on ∂Ω, converges uniformly on compact subsets of Ω to the Aleksandrov solution u of (1.1).…”