Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Amp\`{e}re equation

TL;DR

This paper analyzes the Oliker-Prussner method for the Monge-Ampère equation, establishing stability, consistency, and convergence rates by leveraging geometric properties and inequalities.

Contribution

It introduces a new stability and consistency framework for the Oliker-Prussner method, enabling pointwise convergence rates for classical and non-classical solutions.

Findings

Method is exact for convex quadratic polynomials with translation-invariant nodes.

Derived discrete stability and continuous dependence estimates in max-norm.

Established pointwise convergence rates for solutions of the Monge-Ampère equation.

Abstract

We study the Oliker-Prussner method exploiting its geometric nature. We derive discrete stability and continuous dependence estimates in the max-norm by using a discrete Alexandroff estimate and the Brunn-Minkowski inequality. We show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain; nodes still conform to the domain boundary. This gives a suitable notion of operator consistency which, combined with stability, leads to pointwise rates of convergence for classical and non-classical solutions of the Monge-Amp\`{e}re equation.