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 $h$.

翻译：本文研究了有限元空间中Ginzburg-Landau能量的离散极小值问题，重点探讨Ginzburg-Landau参数κ的影响。该参数具有重要的物理意义，因其大数值会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解析，将κ的大小转化为网格分辨率条件至关重要——这可通过显式依赖于κ和空间网格宽度h的误差估计实现。为此，我们首先在抽象框架中构建一般离散空间类别，给出在问题适配的κ加权范数下的收敛结果；随后将结论应用于拉格朗日有限元和特定广义有限元构造。通过数值实验，我们进一步探究所推导的L²和H¹误差估计关于κ和h的渐近最优性，并观察到在大网格尺寸h情况下的预渐近效应。