AI 中文总结
本研究提出一种求解三维空间II型超导体伦敦方程的FEM-BEM耦合离散化策略，经收敛性测试验证后，应用于等通量问题发现细长椭球存在非唯一最大化器的退化旋转方向。
AI 中文摘要
在本研究中，我们提出了一种用于求解整个三维空间$\boldsymbol{\text{R}}^3$中II型超导体伦敦方程的通用离散化策略。为计算磁场$H_0$，我们将该问题重新表述为磁势的传输问题，并通过非标准有限元-边界元耦合（FEM-BEM耦合）对其进行离散化，该公式无需引入人工截断即可同时考虑有界内部域和无界外部域。随后我们计算矢量场$B_0$，该场源于超导样品中磁势的亥姆霍兹-霍奇分解，它进入等通量问题，该问题用于识别超导金兹堡-朗道模型中涡旋成核首次在能量上有利的曲线。我们使用菊池（Kikuchi）混合公式重构$B_0$的方程，其中散度自由约束以弱方式施加，并使用经典的$H(\text{curl})$协调有限元离散化对所得问题进行离散化。我们通过收敛性测试验证了该离散化策略，并以等通量问题的应用作为结尾：对于在恒定外加磁场下的球体，唯一最大化器是与磁场对齐的直径；对于在与长轴对齐的恒定外加磁场下的椭球体，我们的计算为足够细长的雪茄形几何结构提供了不同行为的数值证据：类似U形涡旋构型的离轴竞争者比长轴获得更大的等通量比。由于长轴因此不是最大化器，任何离轴最大化器通过旋转对称性会产生连续的等价构型族，这意味着等通量问题存在非唯一性和退化旋转方向。
英文摘要
In this work, we propose a general discretization strategy for solving the London equation for type-II superconductors in the whole space $\mathbb{R}^3$. To compute the magnetic field $H_0$, we reformulate the problem for the magnetic potential as a transmission problem and discretize it through a nonstandard FEM-BEM coupling. This formulation accounts for both the bounded interior domain and the unbounded exterior domain without introducing an artificial truncation. We then compute the vector field $B_0$, which arises from the Helmholtz-Hodge decomposition of the magnetic potential in the superconducting sample. This field enters the isoflux problem, which identifies the curves along which vortex nucleation first becomes energetically favorable in the Ginzburg--Landau model of superconductivity. We recast the equations for $B_0$ using the mixed formulation of Kikuchi, in which the divergence-free constraint is imposed weakly, and discretize the resulting problem using a classical $H(\operatorname{curl})$-conforming finite element discretization. We validate our discretization strategy through convergence tests and conclude with an application to the isoflux problem. For a ball under a constant applied magnetic field, the unique maximizer is the diameter aligned with the field. For ellipsoids under a constant applied magnetic field aligned with their major axis, our computations provide numerical evidence of a different behavior in sufficiently elongated, cigar-shaped geometries: off-axis competitors reminiscent of U-shaped vortex configurations attain a larger isoflux ratio than the major axis. Since the major axis is therefore not a maximizer, any off-axis maximizer generates, by rotational symmetry, a continuous family of equivalent configurations, implying non-uniqueness and the presence of a degenerate rotational direction in the isoflux problem.
Comments27 pages, 7 figures, 3 tables
