Convergence of Variational Approximation Schemes for Elastodynamics with Polyconvex Energy

Abstract

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.