Monotone and consistent discretization of the Monge-Ampère operator

Jean-David Benamou, Francis Collino, Jean-Marie Mirebeau
2016 Mathematics of Computation  
We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic
more » ... an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency. * 1 Assuming the solution hessian condition number is uniformly bounded which is symmetric with respect to the origin (i.e. −e ∈ V for each e ∈ V ). Definition 1.3. (DE2 scheme) A numerical scheme −D is Degenerate Elliptic, with stencil V , iff for each x ∈ X the quantity Du(x) is a non-decreasing, locally Lipschitz function of the second order differences ∆ e u(x), e ∈ V . Observing that the second order difference ∆ e u(x) can be expressed as a non-negative weighted sum of first order differences (3), we immediately find that a DE2 scheme is degenerate elliptic in the sense of [Obe06] . DE2 schemes are also positive difference operators in the sense of [KT92] in this paper schemes are indeed built using directional second order finite differences. In particular, for any ε > 0, the slightly perturbed operator −D ε , defined by D ε u(x) := Du(x) − εu(x), is proper degenerate elliptic [FO13] . This in turn implies that the discrete system (5) associated with D ε has a unique solution, which can be computed with a geometric convergence rate using an iterative Euler scheme. We refer to [FO13] and references
doi:10.1090/mcom/3080 fatcat:tqinxtmr3fdqtimyysmn3a7yhu