Finite difference schemes for the parabolic p-Laplace equation

Abstract

We propose a new finite difference scheme for the degenerate parabolic equation ∂tu-div(|∇u|p-2∇u)=f,p≥2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u - \ ext{ div }(|\ abla u|^{p-2}\ abla u) =f, \\quad p\\ge 2. \\end{aligned}$$\\end{document}Under the assumption that the data is Hölder 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.