Error bounds for discrete minimizers of the Ginzburg-Landau energy in the high-$κ$ regime

Abstract: In this work, we study discrete minimizers of the Ginzburg-Landau energy in finite element 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. 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 $h$. 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 $\kappa$-weighted norm. Afterwards we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we further explore the asymptotic optimality of our derived $L^2$- and $H^1$-error estimates with respect to $\kappa$ and $h$. Preasymptotic effects are observed for large mesh sizes $h$.