Influence of Gauges in the Numerical Simulation of the Time-Dependent Ginzburg-Landau Model

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. in Dynamics of the Ginzburg-Landau equations of superconductivity, Technical report, Argonne National Lab. (ANL), Argonne, IL, United States, 1997) that these gauges can be continuously related by a single parameter considering the more general \(\omega \)-gauge, where \(\omega \) is a non-negative real parameter. In this article, we study the influence of the gauge parameter \(\omega \) on the convergence of numerical simulations of the TDGL model using finite element schemes. A classical benchmark is first analysed for different values of \(\omega \) and artefacts are observed for lower values of \(\omega \). Then, we relate these observations with a systematic study of convergence orders in the unified \(\omega \)-gauge framework. In particular, we show the existence of a tipping point value for \(\omega \), 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).