An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg–Landau equations of superconductivity

The paper focuses on numerical study of the time-dependent Ginzburg–Landau (TDGL) equations under the Lorentz gauge. The proposed method is based on a fully linearized backward Euler scheme in temporal direction, and a mixed finite element method (FEM) in spatial direction, where the magnetic field σ=curlA is introduced as a new variable. The linearized Galerkin-mixed FEM enjoys many advantages over existing methods. In particular, at each time step the scheme only requires the solution of two linear systems for ψ   and (σ,A)(σ,A), respectively, with constant coefficient matrices. These two matrices can be pre-assembled at the initial time step and these two linear systems can be solved simultaneously. Moreover, the method provides the same order of optimal accuracy for the density function ψ, the magnetic potential A, the magnetic field σ=curlA, the electric potential divA and the current curlσ. Extensive numerical experiments in both two- and three-dimensional spaces, including complex geometries with defects, are presented to illustrate the accuracy and stability of the scheme. Our numerical results also show that the proposed method provides more realistic predictions for the vortex dynamics of the TDGL equations in nonsmooth domains, while the vortex motion influenced by a defect of the domain is of high interest in the study of superconductors.