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

Christian Döding

Abstract

We study energy minimizers of the Ginzburg-Landau (GL) free energy, a fundamental model of superconductivity. We address the high-$κ$ 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 Fréchet 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.

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

Abstract

We study energy minimizers of the Ginzburg-Landau (GL) free energy, a fundamental model of superconductivity. We address the high- 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 Fréchet 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.
Paper Structure (6 sections, 2 theorems, 43 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 2 theorems, 43 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Ginzburg-Landau model
  3. First- and second-order optimality condition
  4. Computation of GL minimizers
  5. A gradient descent method
  6. Numerical results

Key Result

Lemma 1

Under the general assumptions, there exists at least on minimizer $u \in H^1$ of $E$ such that minimizer holds and If in addition $\partial \Omega$ is of class $C^2$ or $\Omega$ is a convex polytope, then $u \in H^2$ with

Figures (3)

  • Figure 1: Densities $|u|^2$ of the computed discrete GL energy minimizers with $\boldsymbol{A} = \boldsymbol{A}_1$ and $\kappa = 5j$, $j =2,\dots,20$.
  • Figure 2: Densities $|u|^2$ of the computed discrete GL energy minimizers with $\boldsymbol{A} = \boldsymbol{A}_2$ and $\kappa = 5j$, $j =2,\dots,20$.
  • Figure 3: Energy levels $E(u)$ (left) and spectral gap $\lambda_2(\kappa)$ of $E"(u)$ for the computed discrete minimizers $u$ for both model problem $\boldsymbol{A} = \boldsymbol{A}_1$ and $\boldsymbol{A} = \boldsymbol{A}_2$.

Theorems & Definitions (5)

  • Lemma 1: existence and stability bounds
  • proof
  • Lemma 2
  • proof
  • Conjecture 3