In this talk, we present our results on discrete minimizers of the Ginzburg--Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg--Landau parameter. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of the parameter into a mesh resolution condition, which can be done through error estimates that are explicit with respect to the Ginzburg--Landau parameter and the spatial mesh width.
For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted weighted norm. Afterwards we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. We further present numerical experiments that confirm that our derived L^2- and H^1-error estimates are indeed optimal with respect to the in Ginzburg--Landau parameter and the spatial mesh width.
The talk is based on joint work with P. Henning (RU Bochum) in [1].
[1] B. Dörich and P. Henning. Error bounds for discrete minimizers of the Ginzburg-Landau energy in the high-κ regime. CRC 1173 Preprint 2023/11, Karlsruhe Institute of Technology, March 2023.