Volume 25, 2019
|Number of page(s)
|25 October 2019
Convergence analysis of time-discretisation schemes for rate-independent systems*
Institute of Mathematics, University of Kassel,
Heinrich-Plett Str. 40,
** Corresponding author: firstname.lastname@example.org
Accepted: 9 September 2018
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 minimisation, a relaxed version of it and an alternate minimisation 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 reparametrisation argument. We illustrate the different schemes with a toy example.
Mathematics Subject Classification: 49J27 / 49J40 / 35Q74 / 65M12 / 74C05 / 74H15
Key words: Rate-independent system / local minimisation scheme / alternate minimisation scheme / convergence analysis of time-discrete schemes / parametrised BV-solution
This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through the Priority Program SPP 1962 Nonsmooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimisation, Project P09 Optimal Control of Dissipative Solids: Viscosity Limits and Non-Smooth Algorithms. The author is grateful to the unknown referees for their careful reading and the valuable remarks.
© EDP Sciences, SMAI 2019
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