Scaling equivalence between energy minimization by singular perturbation and finite elements on the Tartar square

Tuấn Tùng Nguyễn (University of Heidelberg) #

We study a variational problem arising in the theory of elasticity. On the one hand we consider energy minimization of the elastic energy singularly perturbed by a surface energy term with factor $\varepsilon$. On the other hand we consider energy minimization of the elastic energy over piecewise affine functions on a triangulation with grid size $\sqrt{\varepsilon}$. The density of the elastic energy has wells on a set of certain gradients. Lorent (2008) shows that for some finite set of matrices with gauge invariance both minimization problems scale equivalently for $\varepsilon \to 0$. In the thesis we adapt his proof to the case without gauge invariance where the set of matrices is the Tartar square.