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 confirm that our derived $L^2$- and $H^1$-error estimates are indeed optimal in $\kappa$ and $h$.

翻译：本文研究了有限元空间中金茨堡-朗道能量的离散极小值问题，重点关注金茨堡-朗道参数 $\kappa$ 的影响。该参数具有物理意义，因为其大值可能触发涡旋晶格的出现。由于涡旋必须在足够精细的计算网格上分辨，因此需将 $\kappa$ 的大小转化为网格分辨率条件，这可通过关于 $\kappa$ 和空间网格宽度 $h$ 显式的误差估计实现。为此，我们首先在抽象框架下针对一般离散空间类，在问题自适应的 $\kappa$ 加权范数中给出收敛性结果，随后将结论应用于拉格朗日有限元和特定的广义有限元构造。数值实验证实，我们推导的 $L^2$ 和 $H^1$ 误差估计在 $\kappa$ 和 $h$ 意义上确实是最优的。