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Reduced relative entropy techniques for a priori analysis of multiphase problems in elastodynamics

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Abstract

We give an a priori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in Giesselmann (SIAM J Math Anal 46(5):3518–3539, 2014). The estimate we derive is optimal in the \(\hbox {L} _{\infty }(0,T;dG)\) norm for the strain and the \(\hbox {L} _{2}(0,T;dG)\) norm for the velocity, where dG is an appropriate mesh dependent \(\hbox {H} ^{1}\)-like space.

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Authors and Affiliations

  1. Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70563, Stuttgart, Germany

    Jan Giesselmann

  2. Department of Mathematics and Statistics, Whiteknights, PO Box 220, Reading, GB-RG6 6AX, England, UK

    Tristan Pryer

Authors
  1. Jan Giesselmann
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  2. Tristan Pryer
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Correspondence to Tristan Pryer.

Additional information

Communicated by Ralf Hiptmair.

J.G. was partially supported by the German Research Foundation (DFG) via SFB TRR 75 ‘Tropfendynamische Prozesse unter extremen Umgebungsbedingungen’. T.P. was partially supported by the EPSRC grant EP/H024018/1 and an LMS travel grant 41214.

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Giesselmann, J., Pryer, T. Reduced relative entropy techniques for a priori analysis of multiphase problems in elastodynamics. Bit Numer Math 56, 99–127 (2016). https://doi.org/10.1007/s10543-015-0560-2

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