Finite difference schemes for the parabolic $p$-Laplace equation

TL;DR

This paper introduces a new finite difference scheme for the degenerate parabolic p-Laplace equation, proving convergence under certain conditions and highlighting computational efficiency improvements related to the CFL-condition.

Contribution

A novel explicit-in-time finite difference scheme for the p-Laplace equation with proven convergence and a CFL-condition that leverages regularity to reduce computational cost.

Findings

Convergence of the scheme under a CFL-condition for H"older continuous data.

CFL-condition similar to the heat equation for Lipschitz data, independent of p.

Reduced computational cost due to regularity-based CFL-condition.

Abstract

We propose a new finite difference scheme for the degenerate parabolic equation \[ \partial_t u - \mbox{div}(|\nabla u|^{p-2}\nabla u) =f, \quad p\geq 2. \] Under the assumption that the data is H\"older continuous, we establish the convergence of the explicit-in-time scheme for the Cauchy problem provided a suitable stability type CFL-condition. An important advantage of our approach, is that the CFL-condition makes use of the regularity provided by the scheme to reduce the computational cost. In particular, for Lipschitz data, the CFL-condition is of the same order as for the heat equation and independent of $p$.