Rates of convergence in $W^2_p$-norm for the Monge-Amp\`ere equation

TL;DR

This paper establishes convergence rates in the $W^2_p$-norm for a numerical method solving the Monge-Ampère equation, extending discrete estimates and confirming sharpness through numerical experiments.

Contribution

It introduces new discrete $W^2_p$-norm error estimates for the Oliker-Prussner method, linking contact set properties to second order differences.

Findings

Error estimates depend on $p$ and dimension $d$

Numerical examples confirm the sharpness of the estimates

Extended discrete Alexandroff estimates for the method

Abstract

We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner method applied to the Monge-Amp\`ere equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $\|u - u_h\|_{W^2_p} \leq C h^{1/p}$ if $p > d$ and $\|u - u_h\|_{W^2_p} \leq C h^{1/d} \big(\ln\left(\frac 1 h \right)\big)^{1/d} $ if $p \leq d$. Here the constant $C$ depends on $\|{u}\|_{C^{3,1}(\bar\Omega)}$, the dimension $d$, and the constant $p$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases.