Overview
A numerical approach has been developed for solving the London equation, which describes type-II superconductors in the entirety of $\mathbb{R}^3$. The strategy addresses the magnetic potential by reformulating it as a transmission problem and employs a nonstandard Finite Element Method-Boundary Element Method (FEM-BEM) coupling for its discretization. This formulation accounts for both bounded interior and unbounded exterior domains without requiring artificial truncation. The method proceeds to compute the vector field $B_0$, derived from the Helmholtz-Hodge decomposition of the magnetic potential within the superconducting sample. This $B_0$ field is then integrated into the isoflux problem, which aims to identify current paths where vortex nucleation becomes energetically favorable according to the Ginzburg-Landau model of superconductivity. The equations governing $B_0$ are recast using Kikuchi's mixed formulation, where the divergence-free constraint is weakly imposed, and subsequently discretized using a classical $H(\operatorname{curl})$-conforming finite element method.
Approach
The proposed discretization strategy for the London equation begins with the computation of the magnetic field $H_0$. This involves reformulating the problem for the magnetic potential into a transmission problem. The discretization of this transmission problem is achieved through a nonstandard FEM-BEM coupling. This specific coupling allows for the consideration of both the interior (bounded) domain and the exterior (unbounded) domain without the need for an artificial truncation of the external space.
Following the calculation of $H_0$, the vector field $B_0$ is computed. This field is a result of the Helmholtz-Hodge decomposition of the magnetic potential within the superconducting sample. The field $B_0$ is critical for the subsequent isoflux problem, which seeks to identify curves where vortex nucleation first becomes energetically favorable within the Ginzburg-Landau model of superconductivity. To handle the equations for $B_0$, the researchers adopted the mixed formulation proposed by Kikuchi. In this formulation, the constraint that the field must be divergence-free is imposed in a weak sense. The resulting problem is then discretized using a classical finite element method that conforms to the $H(\operatorname{curl})$ space.
Findings
The proposed discretization strategy was validated through convergence tests. An application of this strategy to the isoflux problem yielded specific observations.
- For a spherical geometry (a ball) subjected to a constant applied magnetic field, the unique maximizer for the isoflux problem was found to be the diameter of the ball that aligns with the applied field.
- For ellipsoidal geometries under a constant applied magnetic field, with the field aligned with the major axis of the ellipsoid, numerical computations indicated a different behavior in sufficiently elongated, cigar-shaped geometries.
- In these elongated ellipsoids, off-axis competitors, described as reminiscent of U-shaped vortex configurations, achieved a larger isoflux ratio compared to the major axis.
- This observation implies that the major axis in such elongated geometries is not the maximizer.
- The presence of an off-axis maximizer, coupled with rotational symmetry, suggests that there exists a continuous family of equivalent configurations. This indicates non-uniqueness and the presence of a degenerate rotational direction within the isoflux problem for these specific geometries.