Convergence analysis of time-discretisation schemes for rate-independent systems

TL;DR

This paper investigates the convergence of three time-discretisation schemes for rate-independent systems with nonconvex energies, demonstrating that under certain conditions, solutions converge to BV-solutions using reparametrization techniques.

Contribution

It provides a rigorous convergence analysis of three different numerical schemes for approximating BV-solutions in rate-independent systems.

Findings

Discrete solutions converge to BV-solutions under suitable discretisation conditions.

Reparametrization arguments are effective in proving convergence.

Numerical schemes are validated with a toy example.

Abstract

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimization, a penalized version of it and an alternate minimization scheme. For all three cases we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrization argument. We illustrate the different schemes with a toy example.