Convergence of Variational Approximation Schemes for Elastodynamics with Polyconvex Energy

TL;DR

This paper proves the convergence of a variational approximation scheme for 3D elastodynamics with polyconvex energy, providing error estimates before shock formation using relative entropy methods.

Contribution

It establishes the convergence and error bounds of a specific variational scheme for elastodynamics with polyconvex energy prior to shock development.

Findings

Convergence of the scheme before shock formation.

Error estimates for the approximation.

Use of relative entropy in the proof.

Abstract

We consider a variational scheme developed by S. Demoulini, D. M. A. Stuart and A. E. Tzavaras [Arch. Ration. Mech. Anal. 157 (2001)] 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 Lp-theory bounds, and provides an error estimate for the approximation before the formation of shocks.