Influence of gauges in the numerical simulation of the time-dependent Ginzburg-Landau model

TL;DR

This work unifies gauge choices in the time-dependent Ginzburg-Landau model via the $ω$-gauge and investigates how this parameter affects finite-element convergence in 2D and 3D. By combining manufactured solutions, nonconvex benchmarks, and 3D geometric configurations, the authors quantify how convergence orders for $ψ$, $\mathbf{A}$, $γ=\operatorname{curl}\mathbf{A}$, and related quantities depend on $ω$ and mesh resolution, using graphical and Richardson extrapolation methods. A key finding is the existence of a tipping point around $ω\in[10^{-2},10^{-3}]$ below which certain fields lose convergence accuracy, potentially causing artefacts, while larger $ω$ yields optimal convergence and faster energy decay. The study provides practical guidance for gauge selection in TDGL simulations of vortex dynamics, ensuring robust and artifact-free simulations in realistic 2D and 3D domains.

Abstract

The time-dependent Ginzburg-Landau (TDGL) model requires the choice of a gauge for the problem to be mathematically well-posed. In the literature, three gauges are commonly used: the Coulomb gauge, the Lorenz gauge and the temporal gauge. It has been noticed [J. Fleckinger-Pellé et al., Technical report, Argonne National Lab. (1997)] that these gauges can be continuously related by a single parameter considering the more general $ω$-gauge, where $ω$ is a non-negative real parameter. In this article, we study the influence of the gauge parameter $ω$ on the convergence of numerical simulations of the TDGL model using finite element schemes. A classical benchmark is first analysed for different values of $ω$ and artefacts are observed for lower values of $ω$. Then, we relate these observations with a systematic study of convergence orders in the unified $ω$-gauge framework. In particular, we show the existence of a tipping point value for $ω$, separating optimal convergence behaviour and a degenerate one. We find that numerical artefacts are correlated to the degeneracy of the convergence order of the method and we suggest strategies to avoid such undesirable effects. New 3D configurations are also investigated (the sphere with or without geometrical defect).

Figures (6)

• Figure 1: 2D benchmark of a disk with an entrant corner. Finite elements of order r = 1 (see definition \ref{['FEspacesMFETDGLLorenz']}). Contours of $|\psi|$ at $t = 5000$ for different values of the $\omega$ parameter.
• Figure 2: Same caption as Fig. \ref{['genGfig1']} with $r = 2$.
• Figure 3: 2D benchmark of a disk with an entrant corner. Finite elements of order r = 2. Relative energy difference $|{\cal G}_{n+1} -{\cal G}_n|/{\cal G}_n$ in logarithmic scale for $\omega = 1,10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}, 0$;
• Figure 4: 2D benchmark of a disk with an entrant corner. Finite elements of order r = 2. Configuration of lowest energy with 24 vortices (left). Minimizer of \ref{['serfatyWn']} corresponding to a system of 24 point vortices, taken from gueron1999discrete(right).
• Figure 5: 2D manufactured solutions benchmark. Finite elements of order r = 1. Orders for the vector potential $\mathbf{A}$ (left) and its divergence $\text{div}(\mathbf{A})$ (right).