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能量的离散极小化子，特别关注Ginzburg-Landau参数$\kappa$的影响。该参数具有重要物理意义，因为其大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进行求解，因此需要将$\kappa$的尺度转化为网格分辨率条件，这可通过显式依赖于$\kappa$和空间网格宽度$h$的误差估计实现。为此，我们首先在一般离散空间的抽象框架中开展工作，提出了问题自适应$\kappa$加权范数下的收敛性结果。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。在数值实验中，我们进一步探究了所导出的$L^2$和$H^1$误差估计关于$\kappa$和$h$的渐近最优性，并观察到当网格尺寸$h$较大时出现的预渐近效应。