This paper considers minimizers of the Ginzburg-Landau energy functional in particular multiscale spaces which are based on finite elements. The spaces are constructed by localized orthogonal decomposition techniques and their usage for solving the Ginzburg-Landau equation was first suggested in [D\"orich, Henning, SINUM 2024]. In this work we further explore their approximation properties and give an analytical explanation for why vortex structures of energy minimizers can be captured more accurately in these spaces. We quantify the necessary mesh resolution in terms of the Ginzburg-Landau parameter $\kappa$ and a stabilization parameter $\beta \ge 0$ that is used in the construction of the multiscale spaces. Furthermore, we analyze how $\kappa$ affects the necessary locality of the multiscale basis functions and we prove that the choice $\beta=0$ yields typically the highest accuracy. Our findings are supported by numerical experiments.

翻译：本文研究了基于有限元的特定多尺度空间中Ginzburg-Landau能量泛函极小化问题。该空间通过局部正交分解技术构建，其用于求解Ginzburg-Landau方程的方法首次在[D\"orich, Henning, SINUM 2024]中被提出。本工作进一步探究了这些空间的逼近特性，并从理论上解释了为何能量极小化子的涡旋结构能在这些空间中更精确地捕获。我们依据Ginzburg-Landau参数$\kappa$与多尺度空间构造中使用的稳定化参数$\beta \ge 0$量化了必要的网格分辨率。此外，我们分析了$\kappa$如何影响多尺度基函数所需的局部性，并证明选择$\beta=0$通常能获得最高精度。数值实验支持了我们的研究结论。