A finite difference method for the variational $p$-Laplacian

TL;DR

This paper introduces the first monotone finite difference discretization for the variational p-Laplacian, providing a convergent numerical scheme and demonstrating its effectiveness through simulations for nonhomogeneous problems.

Contribution

It presents the first monotone finite difference method for the variational p-Laplacian and develops a convergent scheme capable of handling nonhomogeneous Dirichlet problems.

Findings

First monotone finite difference discretization for the variational p-Laplacian.

Development of a convergent numerical scheme for related Dirichlet problems.

Numerical simulations support the theoretical convergence and applicability.

Abstract

We propose a new monotone finite difference discretization for the variational $p$-Laplace operator, \[ \Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u), \] and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational $p$-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.