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. Rat. 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 L p -theory bounds, and provides an error estimate for the approximation before the formation of shocks.keywords: nonlinear elasticity, polyconvexity, variational approximation scheme.

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

“…In this article we establish the stability of numerical solutions and derive the relative entropy identity which is central to establishing the convergence and providing an error estimate. Our stability analysis follows in spirit the work of Miroshnikov and Tzavaras [15] where the authors established the direct convergence of iterates produced by the time-discrete scheme (1.9). Specifically, following [15], we consider the relative entropy η r = η r (x, t)…”

“…Our stability analysis follows in spirit the work of Miroshnikov and Tzavaras [15] where the authors established the direct convergence of iterates produced by the time-discrete scheme (1.9). Specifically, following [15], we consider the relative entropy η r = η r (x, t)…”

“…and give rise to time-continuous approximates converging to the solution of elastodynamics before shock formation (see [15]).…”

A discrete variational scheme for isentropic processes in polyconvex thermoelasticity

“…We organize this paper as follows: In section 2 we extend system (1.1),(1.2) into a symmetrizable, hyperbolic system, exploiting the polyconvex structure of the problem. In section 3 we give an outline of the variational scheme and its main properties (for analogous studies of the isothermal problem see for instance [14,22], while other related work includes [6,31]). We state and prove the minimization theorem 1 in section 4 and as a consequence of that, in lemma 3 we show that the scheme dissipates the energy which in turn, leads to the stability estimate (4.15).…”

Relative entropy in hyperbolic relaxation for balance laws

“…The relative entropy method relies on the "weak-strong" uniqueness principle established by Dafermos for systems of conservation laws admitting convex entropy functional [9], see also DiPerna [13]. In addition to the pioneer work of Dafermos and DiPerna, the relative entropy method has been successfully used to study hydrodynamic limits of particle systems [4,14,22,21,25], hydrodynamic limits from kinetic equations to multidimensional macroscopic models [1,3,17], as well as the convergence of numerical schemes in the context of three-dimensional polyconvex elasticity [18,20].…”