Numerical approach to the London Equation of superconductivity

Abstract

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.