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

TL;DR

This review discusses recent numerical methods for approximating vortex states in type-II superconductors using multiscale techniques, emphasizing error estimates and the coupling of material parameters with mesh resolution.

Contribution

It introduces and reviews localized orthogonal decomposition (LOD) multiscale methods for Ginzburg-Landau energy minimizers, including error analysis and superconvergence results.

Findings

Error estimates explicitly depend on the Ginzburg-Landau parameter 

Superconvergence phenomena observed in numerical approximations

Numerical experiments support theoretical error bounds

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 $\kappa$, capture the formation of lattices of quantized vortices. As the vortex cores shrink with increasing $\kappa$, while their number grows, it is essential to understand how $\kappa$ 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 $\kappa$-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.