Finite difference methods for the Infinity Laplace and p-Laplace equations

TL;DR

This paper introduces convergent finite difference discretizations and efficient semi-implicit solvers for the Infinity Laplacian and p-Laplacian equations, improving computational speed and generalizing previous methods.

Contribution

It presents new discretizations that simplify and extend earlier approaches, along with a semi-implicit solver that is faster and independent of problem size.

Findings

Proved convergence of the schemes to viscosity solutions.

Developed a fast semi-implicit solver with iteration count independent of problem size.

Generalized earlier discretization methods for these equations.

Abstract

We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical $p$-Laplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosity solution of the underlying equation. We build a semi-implicit solver, which solves the Laplace equation as each step. It is fast in the sense that the number of iterations is independent of the problem size. This is an improvement over previous explicit solvers, which are slow due to the CFL-condition.