Zeitschrift für Analysis und ihre Anwendungen
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Convergence of Variational Approximation Schemes for Elastodynamics with Polyconvex EnergyAlexey Miroshnikov and Athanasios E. Tzavaras (1) Department of Mathematics, University of Maryland, MD 20742, College Park, USA
(2) Department of Applied Mathematics, University of Crete, Knossou Ave. (P.O. Box 2208), 71409, 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