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.