Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A multiscale approach to the stationary Ginzburg-Landau equations of superconductivity

Christian Döding, Benjamin Dörich, Patrick Henning

TL;DR

This work analyzes a multiscale discretization for the stationary Ginzburg–Landau equations in superconductivity, combining a Localized Orthogonal Decomposition (LOD) space for the complex order parameter $u$ with a standard finite-element space for the magnetic vector potential $oldsymbol{A}$. The authors derive κ-dependent a priori error estimates that reveal how κ influences resolution requirements, and show that the LOD approach significantly relaxes these constraints compared with conventional FE discretizations. They establish an abstract error framework and then tailor it to the LOD setting, obtaining concrete convergence rates in the $H^1_kappa$ and $L^2$ norms, as well as energy errors, with explicit κ-scaling and a κ-dependent stabilization framework. Numerical experiments in 2D corroborate the theoretical rates, demonstrate robust vortex-pattern resolution on coarse meshes, and explore the influence of the auxiliary potential $oldsymbol{A}_igstar$ on accuracy, confirming the practical viability of the proposed multiscale discretization for type-II superconductors under strong magnetic fields.

Abstract

In this work, we study the numerical approximation of minimizers of the Ginzburg-Landau free energy, a common model to describe the behavior of superconductors under magnetic fields. The unknowns are the order parameter, which characterizes the density of superconducting charge carriers, and the magnetic vector potential, which allows to deduce the magnetic field that penetrates the superconductor. Physically important and numerically challenging are especially settings which involve lattices of quantized vortices which can be formed in materials with a large Ginzburg-Landau parameter $κ$. In particular, $κ$ introduces a severe mesh resolution condition for numerical approximations. In order to reduce these computational restrictions, we investigate a particular discretization which is based on mixed meshes where we apply a Lagrange finite element approach for the vector potential and a localized orthogonal decomposition (LOD) approach for the order parameter. We justify the proposed method by a rigorous a-priori error analysis (in $L^2$ and $H^1$) in which we keep track of the influence of $κ$ in all error contributions. This allows us to conclude $κ$-dependent resolution conditions for the various meshes and which only impose moderate practical constraints compared to a conventional finite element discretization. Finally, our theoretical findings are illustrated by numerical experiments.

A multiscale approach to the stationary Ginzburg-Landau equations of superconductivity

TL;DR

This work analyzes a multiscale discretization for the stationary Ginzburg–Landau equations in superconductivity, combining a Localized Orthogonal Decomposition (LOD) space for the complex order parameter with a standard finite-element space for the magnetic vector potential . The authors derive κ-dependent a priori error estimates that reveal how κ influences resolution requirements, and show that the LOD approach significantly relaxes these constraints compared with conventional FE discretizations. They establish an abstract error framework and then tailor it to the LOD setting, obtaining concrete convergence rates in the and norms, as well as energy errors, with explicit κ-scaling and a κ-dependent stabilization framework. Numerical experiments in 2D corroborate the theoretical rates, demonstrate robust vortex-pattern resolution on coarse meshes, and explore the influence of the auxiliary potential on accuracy, confirming the practical viability of the proposed multiscale discretization for type-II superconductors under strong magnetic fields.

Abstract

In this work, we study the numerical approximation of minimizers of the Ginzburg-Landau free energy, a common model to describe the behavior of superconductors under magnetic fields. The unknowns are the order parameter, which characterizes the density of superconducting charge carriers, and the magnetic vector potential, which allows to deduce the magnetic field that penetrates the superconductor. Physically important and numerically challenging are especially settings which involve lattices of quantized vortices which can be formed in materials with a large Ginzburg-Landau parameter . In particular, introduces a severe mesh resolution condition for numerical approximations. In order to reduce these computational restrictions, we investigate a particular discretization which is based on mixed meshes where we apply a Lagrange finite element approach for the vector potential and a localized orthogonal decomposition (LOD) approach for the order parameter. We justify the proposed method by a rigorous a-priori error analysis (in and ) in which we keep track of the influence of in all error contributions. This allows us to conclude -dependent resolution conditions for the various meshes and which only impose moderate practical constraints compared to a conventional finite element discretization. Finally, our theoretical findings are illustrated by numerical experiments.
Paper Structure (18 sections, 34 theorems, 247 equations, 8 figures, 1 table)

This paper contains 18 sections, 34 theorems, 247 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analytical framework
  3. Fréchet derivatives and stability bounds
  4. Higher order regularity of the minimizers
  5. Kernel in the second Fréchet derivative $E"$
  6. LOD discretization and main results
  7. Analytical preparations
  8. Abstract error analysis
  9. Error analysis in the LOD space
  10. Numerical experiments
  11. Gradient descent method
  12. Model problem and discrete minimizers
  13. Convergence rates
  14. Choice of $\mathbf{A}_{\star}$
  15. Proofs of the higher regularity results
  16. ...and 3 more sections

Key Result

Theorem 2.1

There exists at least one minimizer of the energy eq:energy_functional_stab, i.e., there is $(u,\mathbf{A}) \in H^1(\Omega) \times \mathbf{H}^1_\mathrm{n}(\Omega)$ such that In particular, for any minimizer $(u,\mathbf{A})$ the vector potential $\mathbf{A}$ satisfies $\operatorname{div} \mathbf{A} = 0$, and thus also minimizes eq:energy_functional. The result remains valid for external fields $\m

Figures (8)

  • Figure 1: Real part $\mathop{\mathrm{Re}}\nolimits u$ (top row), imaginary part $\mathop{\mathrm{Im}}\nolimits u$ (middle row), and density $|u|^2$ (bottom row) of the order parameter component $u$ of the GL energy minimizer $(u,\mathbf{A})$ for $\kappa = 5, 10, 15, 20$ (left to right).
  • Figure 2: Real part $\mathop{\mathrm{Re}}\nolimits u$ (top row), imaginary part $\mathop{\mathrm{Im}}\nolimits u$ (middle row), and density $|u|^2$ (bottom row) of the order parameter component $u$ of the GL energy minimizer $(u,\mathbf{A})$ for $\kappa = 25, 30, 50, 100$ (left to right).
  • Figure 3: The vector potential, $\mathbf{A}$ (left, plotted on a coarse mesh), and $\operatorname{curl} \mathbf{A}$ (middle), of the GL energy minimizer $(u,\mathbf{A})$ for $\kappa = 5$, representative of all values of $\kappa = 5, 10, 15, 20, 25, 30, 50, 100$.
  • Figure 4: Difference $\operatorname{curl} \mathbf{A} - \mathbf{H}$ of the GL energy minimizer $(u,\mathbf{A})$ for $\kappa = 5, 10, 15, 20, 25, 30, 50, 100$ (left to right and top to bottom).
  • Figure 5: Error of the order parameter $u$ for the mesh sizes $H = 2^{-\{ 2,3,4,5,6 \}}$, $h = 2^{-6}$ and LOD parameters $h_{\mathrm{fine}} = 2^{-9}$ and $\ell = 10$. Left: $\kappa$-scaled $H^1_\kappa$-error $\kappa^{-3}\| u - u^{{\hbox{\normalfont\tiny LOD}}}_H\|_{H^1_\kappa}$. Right: $\kappa$-scaled $L^2$-error $\kappa^{-4}\| u - u^{{\hbox{\normalfont\tiny LOD}}}_H\|_{L^2}$.
  • ...and 3 more figures

Theorems & Definitions (66)

  • Theorem 2.1: DuGP92
  • Lemma 2.2
  • Lemma 2.3: Ginzburg--Landau equations
  • Lemma 2.4
  • Lemma 2.5: Stability bounds
  • proof
  • Theorem 2.6
  • Remark 2.7
  • Lemma 2.8
  • Corollary 2.9
  • ...and 56 more