Two-scale method for the Monge-Ampère Equation: Pointwise Error Estimates
Provides rigorous error analysis for numerical solutions of the Monge-Ampère equation, a challenging fully nonlinear PDE, benefiting computational mathematicians.
The paper proves pointwise error estimates and convergence rates for the two-scale method applied to the Monge-Ampère equation, establishing continuous dependence on data via a discrete Alexandroff estimate.
In this paper we continue the analysis of the two-scale method for the Monge-Ampère equation for dimension $d \geq 2$ introduced in [10]. 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.