Error Bounds for Discrete Minimizers of the Ginzburg–Landau Energy in the High-\(\boldsymbol{\kappa }\) Regime
Benjamin Dörich, Patrick Henning · SIAM Journal on Numerical Analysis · 2024
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 [Formula: see text]. 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 [Formula: see text] into a mesh resolution condition, which can be done through error estimates that are explicit with respect to [Formula: see text] and the spatial mesh width [Formula: see text]. 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 [Formula: see text]-weighted norm. Afterward we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived [Formula: see text]- and [Formula: see text]-error estimates are indeed optimal in [Formula: see text] and [Formula: see text].