, 56:6 | Cite as

Finite element discretization of local minimization schemes for rate-independent evolutions

  • Christian Meyer
  • Michael Sievers


This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in Efendiev and Mielke (J Convex Anal 13(1):151–167, 2006), which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions for mesh size tending to zero exist and are so-called parametrized solutions of the continuous problem. The discrete problems are solved by means of a mass lumping scheme for the non-smooth dissipation functional in combination with a semi-smooth Newton method. A numerical test indicates the efficiency of this approach. In addition, we compared the local minimization scheme with a time stepping scheme for global energetic solutions, which shows that both schemes yield different solutions with differing time discontinuities.


Rate independent evolutions Parametrized solutions Finite elements Semi-smooth Newton methods 

Mathematics Subject Classification

65J08 65J15 65M60 



The authors are very grateful to Dorothee Knees (University of Kassel) for various helpful discussions.


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Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

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