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The Ginzburg-Landau equations: Vortex states and numerical multiscale approximations

Christian Döding, Patrick Henning

TL;DR

This work develops a multiscale numerical framework for approximating minimizers of the Ginzburg–Landau energy in type-II superconductors by leveraging Localized Orthogonal Decomposition (LOD). It establishes κ-dependent error estimates under weak and stronger mesh-resolution conditions, proving energy convergence and near-optimal $H^1_\kappa$ and $L^2$ accuracies for discrete minimizers in the LOD space, aided by a Ritz projection analysis of the second variation. A fixed-point strategy guarantees discrete minimizers close to the continuous ones, and a nonlinear conjugate Sobolev gradient descent (CSG) algorithm enables practical computation of GL minimizers within the LOD framework. Numerical experiments in two dimensions demonstrate vortex lattices and multiple local minimizers for $\kappa$ up to 100, while illustrating the efficiency of the LOD approach in resolving intricate vortex patterns with relatively few degrees of freedom. The results offer a robust, κ-explicit pathway to accurate and efficient simulations of vortex states in superconductors and potentially extend to other multiscale nonlinear PDE minimization problems.

Abstract

In this review article, we provide an overview of recent advances in the numerical approximation of minimizers of the Ginzburg-Landau energy in multiscale spaces. Such minimizers represent the most stable states of type-II superconductors and, for large material parameters $κ$, capture the formation of lattices of quantized vortices. As the vortex cores shrink with increasing $κ$, while their number grows, it is essential to understand how $κ$ should couple to the mesh size in order to correctly resolve the vortex patterns in numerical simulations. We summarize and discuss recent developments based on LOD (Localized Orthogonal Decomposition) multiscale methods and review the corresponding error estimates that explicitly reflect the $κ$-dependence and the observed superconvergence. In addition, we include several minor refinements and extensions of existing results by incorporating techniques from recent contributions to the field. Finally, numerical experiments are presented to illustrate and support the theoretical findings.

The Ginzburg-Landau equations: Vortex states and numerical multiscale approximations

TL;DR

This work develops a multiscale numerical framework for approximating minimizers of the Ginzburg–Landau energy in type-II superconductors by leveraging Localized Orthogonal Decomposition (LOD). It establishes κ-dependent error estimates under weak and stronger mesh-resolution conditions, proving energy convergence and near-optimal and accuracies for discrete minimizers in the LOD space, aided by a Ritz projection analysis of the second variation. A fixed-point strategy guarantees discrete minimizers close to the continuous ones, and a nonlinear conjugate Sobolev gradient descent (CSG) algorithm enables practical computation of GL minimizers within the LOD framework. Numerical experiments in two dimensions demonstrate vortex lattices and multiple local minimizers for up to 100, while illustrating the efficiency of the LOD approach in resolving intricate vortex patterns with relatively few degrees of freedom. The results offer a robust, κ-explicit pathway to accurate and efficient simulations of vortex states in superconductors and potentially extend to other multiscale nonlinear PDE minimization problems.

Abstract

In this review article, we provide an overview of recent advances in the numerical approximation of minimizers of the Ginzburg-Landau energy in multiscale spaces. Such minimizers represent the most stable states of type-II superconductors and, for large material parameters , capture the formation of lattices of quantized vortices. As the vortex cores shrink with increasing , while their number grows, it is essential to understand how should couple to the mesh size in order to correctly resolve the vortex patterns in numerical simulations. We summarize and discuss recent developments based on LOD (Localized Orthogonal Decomposition) multiscale methods and review the corresponding error estimates that explicitly reflect the -dependence and the observed superconvergence. In addition, we include several minor refinements and extensions of existing results by incorporating techniques from recent contributions to the field. Finally, numerical experiments are presented to illustrate and support the theoretical findings.

Paper Structure

This paper contains 15 sections, 18 theorems, 155 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Ginzburg-Landau energy minimization problem
  3. Well-posedness and first order conditions
  4. Gauge invariance and second order conditions
  5. Minimizers in multiscale approximation spaces
  6. Main results
  7. Proof of the main results
  8. Approximation in the LOD space
  9. Proof of energy error estimates
  10. The operator $E"(u)$ and its Ritz-projection
  11. Discrete minimizers close to $u$
  12. Proof of $H^1_\kappa$ and $L^2$ error estimates
  13. Computation of GL minimizers
  14. A nonlinear conjugate Sobolev gradient descent method
  15. Numerical results

Key Result

Lemma 2.1

Let $\Omega \subset \mathbb{R}^{d}$ ($d=2,3$) be a bounded Lipschitz-domain and let A1-A2 be fulfilled. Then there exists a global minimizer $u \in H^1(\Omega)$ of $E$, i.e. minimizer-energy-def holds.

Figures (2)

  • Figure 1: In a fixed configuration, vortex patterns in the minimizing density $|u|^2$ for $\kappa=10, 25, 50, 75, 100$.
  • Figure 2: Real part (top line), imaginary part (middle line), and density (bottom line) of discrete minimizers $u_h^{{\text{\normalfont ms}}}$ for $\kappa=10, 25, 50, 75, 100$ (left to right).

Theorems & Definitions (33)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3: Quasi-isolated minimizer
  • Lemma 2.4
  • proof
  • Lemma 2.5: Regularity of solutions to $E^{\prime\prime}(u)z = {f}$ in $( \mathrm{i} u)^{\perp}$
  • proof
  • Lemma 2.6: Gårding inequality for $\langle E^{\prime\prime}(u) \,\cdot\,,\cdot \rangle$
  • proof
  • Lemma 3.1
  • ...and 23 more