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

TL;DR

This paper introduces a multiscale finite element method for approximating solutions to the Ginzburg-Landau equations in superconductivity, effectively handling large Ginzburg-Landau parameters with reduced computational constraints.

Contribution

It proposes a novel discretization combining mixed meshes and localized orthogonal decomposition, with rigorous error analysis accounting for the Ginzburg-Landau parameter influence.

Findings

Reduced mesh resolution requirements for large  using the proposed method

Rigorous a-priori error estimates in L^2 and H^1 norms

Numerical experiments validating theoretical results

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 $\kappa$. In particular, $\kappa$ 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 $\kappa$ in all error contributions. This allows us to conclude $\kappa$-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.