A New Mixed Formulation and Efficient Numerical Solution of Ginzburg--Landau Equations Under the Temporal Gauge

Abstract

In this paper, we present a new numerical approach to the time-dependent Ginzburg--Landau (GL) equations under the temporal gauge (zero electric potential gauge). The approach is based on a mixed formulation of the GL equations, which consists of two parabolic equations for the order parameter $\psi$ and the magnetic field $\sigma = \mathrm{curl} \, \mathbf{A}$, respectively, and a vector ordinary differential equation for the magnetic potential $\mathbf{A}$. A fully linearized Galerkin finite element method is presented for solving the mixed GL system. The new approach offers many advantages on both accuracy and efficiency over existing methods. In particular, the equations for $\psi$ and $\sigma$ are uniformly parabolic and, therefore, the method provides optimal-order accuracy for the two physical components $\psi$ and $\sigma$. Since in the temporal direction, a fully linearized backward Euler scheme is used for $\psi$ and $\sigma$ and a forward Euler scheme is used for $\mathbf{A}$, respectively, the system is fully decoupled and at each time step, the three variables $\psi$, $\sigma$, and $\mathbf{A}$ can be solved simultaneously. Moreover, we present numerical comparisons with two commonly used Galerkin methods for the GL equations under the temporal gauge and the Lorentz gauge, respectively. Our numerical results show that the new approach requires fewer iterations for solving the linear systems arising at each time step and the computational cost for the vector ODE seems neglectable. Several numerical examples in both two- and three-dimensional spaces are investigated.