On second-order optimality in the high-$\kappa$ regime of the Ginzburg-Landau model

TL;DR

This paper investigates the spectral gap of the second derivative of the Ginzburg-Landau energy in the high-$ppa$ regime, providing numerical evidence for a conjectured polynomial dependence that impacts vortex structure approximations.

Contribution

It offers the first numerical computation of the spectral gap across various GL parameters, supporting the conjecture of polynomial dependence in the high-$ppa$ regime.

Findings

Spectral gap decreases rapidly with increasing GL parameter.

Numerical evidence supports polynomial dependence of the spectral gap.

Implications for finite element approximations of vortex structures.

Abstract

We study energy minimizers of the Ginzburg-Landau (GL) free energy, a fundamental model of superconductivity. We address the high-$\kappa$ regime, the regime of a large GL parameter, in which energy minimizers exhibit vortex structures whose finite element approximations require a fine mesh resolution. This difficulty is reflected in the error analysis of discrete minimizers, which relies on a second-order optimality condition. The spectrum of the energy's second Fr\'echet derivative must be bounded away from zero up to symmetry. In practice, the associated spectral gap decreases rapidly with the GL parameter. This degrades the quality of the approximations because the GL parameter directly enters as an additional factor in the error estimates. Although a polynomial dependence of the spectral gap on the GL parameter has been conjectured, its precise behavior remains unclear. As a first step toward addressing this issue, we compute the spectral gap based on a finite element approximation for a range of GL parameters, providing numerical evidence for the conjectured polynomial dependence.