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2017-12-21Author
Knees, DorotheeSubject
510 MathematicsMetadata
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Preprint
Convergence analysis of time-discretization schemes for rate-independent systems
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 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.
Sponsorship
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 AlgorithmsCitation
@article{urn:nbn:de:hebis:34-2017122154061,
author={Knees, Dorothee},
title={Convergence analysis of time-discretization schemes for rate-independent systems},
year={2017}
}
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2017-12-21T09:43:27Z 2017-12-21T09:43:27Z 2017-12-21 urn:nbn:de:hebis:34-2017122154061 http://hdl.handle.net/123456789/2017122154061 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 eng Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ rate-independent system local minimization scheme alternate minimization scheme convergence analysis of time-discrete schemes parametrized BV-solution 510 Convergence analysis of time-discretization schemes for rate-independent systems Preprint 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. open access Knees, Dorothee Mathematische Schriften Kassel ;; 17, 03 74C05 49J40 65M12 49J27 35Q74 74H15 Mathematische Schriften Kassel 17, 03
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