超导静态金兹堡-朗道方程的多尺度方法

Christian Döding, Benjamin Dörich, Patrick Henning
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摘要

在这项工作中,我们研究了金兹堡-朗道自由能最小化的数值近似,这是一种描述超导体在磁场下行为的常见模型。未知数包括阶次参数(表征超导电荷载流子的密度)和磁矢量势(推导出渗透超导体的磁场)。物理上重要、数值上具有挑战性的设置尤其涉及量子化涡旋晶格,这些涡旋晶格可以在具有较大金兹堡-朗道参数 $\kappa$ 的材料中形成。特别是,$\kappa$ 为数值近似引入了苛刻的网格分辨率条件。为了减少这些计算限制,我们研究了一种基于混合网格的特殊离散化方法,即对矢量势采用拉格朗日有限元方法,对阶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$H^1$中)来证明所提出的方法,其中我们跟踪了所有误差贡献中$\kappa$的影响。这使我们能够为各种网格总结出依赖于 $\kappa$ 的分辨率条件,与传统的有限元离散化相比,这些条件只施加了适度的实际限制。最后,我们通过数值实验来说明我们的理论发现。
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A multiscale approach to the stationary Ginzburg-Landau equations of superconductivity
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.
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