Two-scale method for the Monge–Ampère equation: pointwise error estimates

Abstract

In this paper we continue the analysis of the two-scale method for the Monge-Ampère equation for dimension d ≥ 2 introduced in [12]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete version of the Alexandroff estimate. They are both instrumental to prove pointwise error estimates for classical solutions with Hölder and Sobolev regularity. We also derive convergence rates for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.

The Monge–Ampère equation

“…The techniques used in this reference were very similar to those that we will describe in Section 3 and so, to avoid repetition, we shall not elaborate on them here. This is further justified by that fact that, although [90] was the first work to provide rates of convergence for wide stencil-type methods, the rates of convergence obtained in this work were suboptimal. Let us here, instead, present the results obtained in [76], where optimal rates of convergence have been obtained.…”

Convergent Two-Scale Filtered Scheme for the Monge--Ampère Equation

“…Upon examining where the error is larger, we realize that it appears on the boundary layer that arises from the definition of T ε,m . As shown in [28,Theorem 5.3], this error does not obey operator consistency, but is instead bounded by C u W 2 ∞ (Ω) δ m through a barrier argument.…”

“…In this work we extend our two-scale method from [27,28], where we use continuous piecewise linear polynomials on a quasi-uniform mesh of size h and an approximation of the determinant that hinges on a second coarser scale δ, in order to solve the Monge-Ampère equation numerically. In [27] we introduce the two-scale method and prove uniform convergence to the viscosity solution of (1.1), whereas in [28] we derive rates of converges in L ∞ for classical and viscosity solutions that belong to certain Hölder and Sobolev spaces. The idea of a filtered scheme that we employ here is motivated by the work of Froese and Oberman in [20], but follows a different approach; we refer to [32] and [7] for stationary and time depentent Hamilton-Jacobi equations.…”

“…These choices lead to schemes with different theoretical properties and levels of accuracy. We first briefly recall the discretization used in [27,28] and then introduce a more accurate approach. Combining the two leads to the main contribution of this work, which we call the filtered scheme, due to the use of a filter function that allows us to appropriately combine the two discretizations.…”

“…Monotone Operator [27,28]: We discretize the domain Ω by a shape regular and quasi-uniform mesh T 1 h with spacing h, the fine scale, and construct a space V 1 h of continuous piecewise linear functions over T 1 h . The superscript 1 of V 1 h indicates the use of linear polynomials whereas that of T 1 h entails the use of straight (affine equivalent) simplices.…”