WIAS Preprint No. 1295, (2008)

A discrete variational principle for rate-independent evolution



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Stefanelli, Ulisse

2010 Mathematics Subject Classification

  • 35K55

Keywords

  • Variational Principle, rate-independent process, time-discretization

Abstract

We develop a global-in-time variational approach to the time-discretization of rate-independent processes. In particular, we investigate a discrete version of the variational principle based on the weighted energy-dissipation functional introduced by A. Mielke and M. Ortiz in ESAIM Control Optim. Calc. Var., 2008. We prove the conditional convergence of time-discrete approximate minimizers to energetic solutions of the time-continuous problem. Moreover, the convergence result is combined with approximation and relaxation. For a fixed partition the functional is shown to have an asymptotic development by Gamma convergence, cf. G. Anzellotti and S. Baldo (Appl. Math. Optim., 1993), in the limit of vanishing viscosity.

Appeared in

  • Adv. Calculus Variations, 1 (2008) pp. 399--431.

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