A Finite Difference Method for the Variational p-Laplacian

Abstract

We propose a new monotone finite difference discretization for the variational p-Laplace operator, $$\Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u),$$ Δ p u = div ( | ∇ u | p - 2 ∇ 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.

Finite difference schemes for the parabolic p-Laplace equation

“…A proof of consistency (A c ) can be found in Theorem 1.1 in [9]. Assumption (A ω ) trivially holds for h = o(r α ) for some α > 0 according to (5.1) since…”

“…When p ∈ N, a more explicit value of this constant is given in [9]. In general, the explicit value is given by…”

“…Recently, a new monotone finite difference discretization of the p-Laplacian was introduced by the authors in [9]. It is based on the mean value property presented in [4,8].…”