Convergence of Fully Discrete Implicit and Semi-implicit Approximations of Singular Parabolic Equations

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. 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, which avoids the Aubin--Lions lemma, has to be carried out. The arguments presented in this article show that convergence holds under a moderate condition that relates the time-step size to the regularization parameter but which is independent of the spatial resolution.