BIT Numerical Mathematics

, Volume 56, Issue 1, pp 99–127 | Cite as

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

Article

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.

Keywords

Discontinuous Galerkin finite element method A priori error analysis Multiphase elastodynamics Relative entropy  Reduced relative entropy 

Mathematics Subject Classification

65M60 65M12 65M15 74B20 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institute of Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany
  2. 2.Department of Mathematics and StatisticsWhiteknightsReadingEngland, UK

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