Stability of fully discrete variational schemes for elastodynamics with a polyconvex stored energy

TL;DR

This paper introduces a fully discrete variational scheme for 3D elastodynamics with polyconvex energy, demonstrating its unconditional stability through a relative entropy approach.

Contribution

It extends a time-discrete variational scheme to a fully discrete setting and proves its unconditional stability for elastodynamics with polyconvex stored energy.

Findings

The scheme is unconditionally stable.

Stability is proven via relative entropy estimates.

Applicable to three-dimensional elastodynamics.

Abstract

In this article we develop a fully discrete variational scheme that approximates the equations of three dimensional elastodynamics with polyconvex stored energy. The fully discrete scheme is based on a time-discrete variational scheme developed by S.~Demoulini, D.~M.~A.~Stuart and A.~E.~Tzavaras (2001). We show that the fully discrete scheme is unconditionally stable. The proof of stability is based on a relative entropy estimation for the fully discrete approximates.