Vortex-capturing multiscale spaces for the Ginzburg-Landau equation

TL;DR

This paper investigates specialized multiscale finite element spaces for the Ginzburg-Landau equation, demonstrating improved vortex structure approximation and analyzing the influence of parameters on accuracy through theoretical and numerical methods.

Contribution

It introduces and analyzes multiscale spaces based on localized orthogonal decomposition for better vortex capturing in the Ginzburg-Landau equation.

Findings

Multiscale spaces improve vortex structure approximation.

Mesh resolution depends on the Ginzburg-Landau parameter and stabilization parameter.

Choosing zero stabilization parameter yields higher accuracy.

Abstract

This paper considers minimizers of the Ginzburg-Landau energy functional in special multiscale spaces that are based on finite elements. The spaces are constructed by localized orthogonal decomposition techniques and their usage for solving the Ginzburg-Landau equation was first suggested in [D\"orich, Henning, SINUM 2024]. In this work we further explore their approximation properties and give an analytical explanation for why vortex structures of energy minimizers can be captured more accurately in these spaces. We quantify the necessary mesh resolution in terms of the Ginzburg-Landau parameter $\kappa$ and a stabilization parameter $\beta \ge 0$ that is used in the construction of the multiscale spaces. Furthermore, we analyze how $\kappa$ affects the necessary locality of the multiscale basis functions and we prove that the choice $\beta=0$ yields typically the highest accuracy. Our findings are supported by numerical experiments.