Convergent, with rates, methods for normalized infinity Laplace, and related, equations

TL;DR

This paper introduces a monotone, consistent numerical scheme for approximating the normalized Infinity Laplacian Dirichlet problem, establishing convergence and convergence rates influenced by solution regularity and right-hand side conditions.

Contribution

It presents a novel two-scale numerical method for the normalized Infinity Laplacian with proven convergence and explicit rates, extending to obstacle problems and Finsler norms.

Findings

The scheme is proven to be convergent.

Convergence rates depend on solution regularity and right-hand side.

Extensions include obstacle problems and symmetric Finsler norms.

Abstract

We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so--called two--scale methods. We show that this method is convergent, and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms are also considered.