Patrick Henning: Multiscale approximations for the stationary Ginzburg-Landau equation

Abstract:

In this presentation we discuss recent results on discrete minimizers of the Ginzburg-Landau energy in finite element and multiscale spaces. Special focus is given to the influence of the Ginzburg--Landau parameter \(\kappa\). This parameter is of physical interest as large values can trigger the appearance of vortex lattices in superconductors. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of \(\kappa\) into a mesh resolution condition, which can be done through error estimates that are explicit with respect to \(\kappa\) and the spatial mesh width \(h\). We present corresponding analytical results for Lagrange finite elements and identify a previously unknown numerical pollution effect. Furthermore, we show how the approximation properties can be enhanced with multiscale techniques.