Unconditional stability of semi-implicit discretizations of singular flows

TL;DR

This paper proves that a semi-implicit discretization method for singular flows involving the p-Laplace operator is unconditionally energy stable, with error estimates influenced by regularization, and confirms these findings through numerical experiments.

Contribution

It establishes unconditional energy stability for a semi-implicit discretization of singular flows and analyzes the impact of regularization on convergence rates.

Findings

Unconditional energy stability is proven for the discretization method.

Error estimates depend critically on the regularization parameter.

Numerical experiments show reduced convergence rates with smaller regularization parameters.

Abstract

A popular and efficient discretization of evolutions involving the singular $p$-Laplace operator is based on a factorization of the differential operator into a linear part which is treated implicitly and a regularized singular factor which is treated explicitly. It is shown that an unconditional energy stability property for this semi-implicit time stepping strategy holds. Related error estimates depend critically on a required regularization parameter. Numerical experiments reveal reduced experimental convergence rates for smaller regularization parameters and thereby confirm that this dependence cannot be avoided in general.