Convergence of a Finite Element-Based Space-Time Discretization in Elastodynamics
Andreas Prohl · SIAM Journal on Numerical Analysis · 2008
We study a finite element-based space-time discretization of the elastodynamics equation $u^{\varepsilon}_{tt} - {\rm div} \sigma( abla u^{\varepsilon}) - \varepsilon \Delta u_t^{\varepsilon} = 0$ for $\varepsilon \geq 0$, where $\sigma = D\phi$ and $\phi$ is a nonconvex function. The convergence for regularization parameters $\varepsilon > 0$ of iterates to weak solutions and for the limiting problem $\varepsilon = 0$ of iterates towards generalized solutions is shown in a general setting of data. Computational experiments are included to motivate formation and propagation of two-dimensional microstructures for decreasing values of the regularization parameter.