Convergence of fully discrete implicit and semi-implicit approximations of nonlinear parabolic equations

TL;DR

This paper proves the convergence of fully discrete implicit and semi-implicit schemes for nonlinear parabolic equations, particularly those involving the p-Laplace operator, under practical conditions relating step size and regularization.

Contribution

It provides a rigorous convergence analysis for semi-implicit and implicit discretizations of nonlinear parabolic problems, relaxing previous restrictive conditions.

Findings

Convergence holds under a moderate relation between step size and regularization.

The analysis is independent of spatial resolution.

Applicable to schemes involving the p-Laplace operator.

Abstract

The article addresses the convergence of implicit and semi-implicit, fully discrete approximations of a class of nonlinear parabolic evolution problems. Such schemes are popular in the numerical solution of evolutions defined with the $p$-Laplace operator since the latter lead to linear systems of equations in the time steps. The semi-implicit treatment of the operator requires introducing a regularization parameter that has to be suitably related to other discretization parameters. To avoid restrictive, unpractical conditions, a careful convergence analysis has to be carried out. The arguments presented in this article show that convergence holds under a moderate condition that relates the step size to the regularization parameter but which is independent of the spatial resolution.