Numerical simulations of the quantized vortices on a thin superconducting hollow sphere

Gamma-convergence and the emergence of vortices for Ginzburg–Landau on thin shells and manifolds

“…Exploiting the symmetry of the problem and with the intuition gained by observing the computations of [10,11], we seek a two-vortex critical point with vortices at the north and south pole in the form ψ(θ, φ) = f (φ)e iθ for some function f : [0, π] → R vanishing at the endpoints. Plugging this ansatz into (4.10), we are left with the task of finding a non-trivial critical point f of the functional…”

“…(The monopole assumption, which we do not invoke, leads to the condition that the magnetic field strength is uniform throughout the surface of the sphere.) Within the applied mathematics community, we note the computational work in [10] and [11] on superconducting spheres in the presence of a vertical magnetic field. Here the authors capture various vortex patterns on the surface of the sphere as the magnetic field strength is varied.…”

Centroidal Voronoi tessellation‐based finite element superconvergence

“…Two examples are shown in Figure 1(a) and (b) for complex domains with uniform and nonuniform sizing, respectively. In [24][25][26], finite volume schemes are constructed on CVT-based meshes and it is shown that the schemes have nice superconvergence properties. Hence, a question comes up naturally: does the superconvergence property of a finite element solution on equilateral triangular meshes hold for an almost equilateral triangular mesh?…”

Schrödinger eigenbasis on a class of superconducting surfaces: Ansatz, analysis, FEM approximations and computations

“…We refer to [6,7,16] for further details regarding the particulars and validity of this GL model realization. In [11], we developed a robust theoretical framework to characterize the space L 2 (S; C), and found in particular that a complete orthonormal system of eigenfunctions of the Schrödinger operator exists.…”

“…From standard gauge, scaling and approximation considerations we take this applied field as being equal to the system magnetic field, and given through the curl of the magnetic vector potential A 0 = Rθ , where R represents distance from the z-axis, and θ denotes the unit coordinate vector corresponding to the direction of increasing θ in a cylindrical-polar coordinate system. In this context, the GL system involves a single partial differential equation, which is nonlinear in the complex-valued order parameter, and is governed by the Schrödinger operator (i∇ + Rθ) 2 on S [6,7,16]. In a future work, we will seek an approximate solution to the nonlinear GL model as a linear combination of the eigenfunctions of (i∇ + Rθ) 2 on S. In [13], using eigenfunctions of the linear Stokes operator on a sphere, similar approximations were analyzed and implemented for the Navier-Stokes equations on the rotating sphere.…”