  In this paper, we propose a fully discrete mixed finite element method for
solving the time-dependent Ginzburg--Landau equations, and prove the
convergence of the finite element solutions in general curved polyhedra,
possibly nonconvex and multi-connected, without assumptions on the regularity
of the solution. Global existence and uniqueness of weak solutions for the PDE
problem are also obtained in the meantime. A decoupled time-stepping scheme is
introduced, which guarantees that the discrete solution has bounded discrete
energy, and the finite element spaces are chosen to be compatible with the
nonlinear structure of the equations. Based on the boundedness of the discrete
energy, we prove the convergence of the finite element solutions by utilizing a
uniform $L^{3+\delta}$ regularity of the discrete harmonic vector fields,
establishing a discrete Sobolev embedding inequality for the N\'ed\'elec finite
element space, and introducing a $\ell^2(W^{1,3+\delta})$ estimate for fully
discrete solutions of parabolic equations. The numerical example shows that the
constructed mixed finite element solution converges to the true solution of the
PDE problem in a nonsmooth and multi-connected domain, while the standard
Galerkin finite element solution does not converge.
