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Zeitschrift für Analysis und ihre Anwendungen


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Volume 33, Issue 1, 2014, pp. 43–64
DOI: 10.4171/ZAA/1498

Published online: 2013-12-27

Convergence of Variational Approximation Schemes for Elastodynamics with Polyconvex Energy

Alexey Miroshnikov[1] and Athanasios E. Tzavaras[2]

(1) University of Maryland, College Park, USA
(2) University of Crete, Heraklion, Greece

We consider a variational scheme developed by S. Demoulini, D. M. A. Stuart and A. E. Tzavaras [Arch. Ration. Mech. Anal. 157 (2001), 325–344] that approximates the equations of three dimensional elastodynamics with polyconvex stored energy. We establish the convergence of the time-continuous interpolates constructed in the scheme to a solution of polyconvex elastodynamics before shock formation. The proof is based on a relative entropy estimation for the time-discrete approximants in an environment of $L^p$-theory bounds, and provides an error estimate for the approximation before the formation of shocks.

Keywords: Nonlinear elasticity, polyconvexity, variational approximation scheme

Miroshnikov Alexey, Tzavaras Athanasios: Convergence of Variational Approximation Schemes for Elastodynamics with Polyconvex Energy. Z. Anal. Anwend. 33 (2014), 43-64. doi: 10.4171/ZAA/1498