🇬🇧

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

Knees, Dorothee

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 analyze the convergence of solutions of three time-discretization schemes, namely an approach based on local minimization, a relaxed version of it and an alternate minimization scheme. For all three cases we show that under suitable conditions on the discretization 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.

Classification / Keywords
Sponsor
Deutsche Forschungsgemeinschaft (DFG), Priority Program SPP 1962 Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization, Project P09 Optimal Control of Dissipative Solids: Viscosity Limits and Non-Smooth Algorithms
Citation
In: Mathematische Schriften Kassel 17, 03 / (2017-12-21) , S. ; 
@article{urn:nbn:de:hebis:34-2017122154061,
  author    ={Knees, Dorothee},
  title    ={Convergence analysis of time-discretization schemes for rate-independent systems},
  copyright  ={https://rightsstatements.org/page/InC/1.0/},
  language ={en},
  year   ={2017-12-21}
}