A Gauge-Invariant Finite Element Method for the Time-Dependent Ginzburg--Landau Equations

Abstract

It is well known that the Ginzburg--Landau equations are gauge invariant. The most popular gauge choice in practical simulations is the temporal gauge, also known as the zero electric potential gauge. However, the commonly used Nédélec approximations may produce spurious giant vortices on domains with re-entrant corners. To address this issue while retaining the Nédélec approximation, we propose a finite element scheme that preserves the gauge invariance of the time-dependent Ginzburg--Landau equations at the discrete level. Under the temporal gauge, the method employs lowest-order Nédélec edge elements for the magnetic potential and continuous piecewise-linear elements for the order parameter on triangular and tetrahedral meshes, and yields a linear, decoupled formulation that requires only the solution of linear systems at each time step. We prove that the scheme is gauge invariant, preserves a discrete maximum bound principle for the order parameter, $\|ψ_h^n\|_{L^\infty(Ω)}\le 1$, and satisfies a discrete energy law under mild step-size and mesh conditions. Numerical experiments confirm these theoretical properties. In particular, on domains with re-entrant corners, the scheme computes the correct physical solution, whereas conventional temporal-gauge discretizations with Nédélec elements produce an unphysical solution.