In this work, we study the numerical approximation of minimizers of the Ginzburg-Landau free energy, a common model to describe the behavior of superconductors under magnetic fields. The unknowns are the order parameter, which characterizes the density of superconducting charge carriers, and the magnetic vector potential, which allows to deduce the magnetic field that penetrates the superconductor. Physically important and numerically challenging are especially settings which involve lattices of quantized vortices which can be formed in materials with a large Ginzburg-Landau parameter $\kappa$. In particular, $\kappa$ introduces a severe mesh resolution condition for numerical approximations. In order to reduce these computational restrictions, we investigate a particular discretization which is based on mixed meshes where we apply a Lagrange finite element approach for the vector potential and a localized orthogonal decomposition (LOD) approach for the order parameter. We justify the proposed method by a rigorous a-priori error analysis (in $L^2$ and $H^1$) in which we keep track of the influence of $\kappa$ in all error contributions. This allows us to conclude $\kappa$-dependent resolution conditions for the various meshes and which only impose moderate practical constraints compared to a conventional finite element discretization. Finally, our theoretical findings are illustrated by numerical experiments.

翻译：本文研究金兹堡-朗道自由能极小化子的数值逼近问题，该模型常用于描述超导体在磁场中的行为。未知量包括表征超导载流子密度的序参数，以及用于推导穿透超导体的磁场的磁矢势。物理上重要且数值计算具有挑战性的情形尤其涉及具有大金兹堡-朗道参数$\kappa$的材料中可能形成的量子化涡旋晶格。特别地，$\kappa$对数值逼近提出了严格的网格分辨率要求。为降低这些计算限制，我们研究一种基于混合网格的特殊离散化方案：对磁矢势采用拉格朗日有限元方法，对序参数采用局部正交分解（LOD）方法。我们通过严格的先验误差分析（在$L^2$和$H^1$范数下）论证所提方法的合理性，在分析中精确追踪了$\kappa$对所有误差项的影响。这使得我们能够推导出各网格与$\kappa$相关的分辨率条件，与传统有限元离散化相比，这些条件仅施加了适度的实际限制。最后，我们通过数值实验验证理论结果。