Applied PDE Group
Nonlinear Partial Differential Equations in Fluid and Solid Mechanics

MiroTzav

Convergence of variational approximation schemes for three dimensional elastodynamims with polyconvex energy

A. Miroshnikov, A.E. Tzavaras, Convergence of variational approximation schemes for three dimensional elastodynamims with polyconvex energy,  J. Analysis Appl. (ZAA)  33(1):43-64, 2013​​
A. Miroshnikov, A.E. Tzavaras
Nonlinear elasticity, Polyconvexity, Variational approximation scheme
2013
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 poly- convex stored energy. We establish the convergence of the time-continuous interpolates constructed in the scheme to a solution of polyconvex elas- todynamics 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 approxima- tion before the formation of shocks.​