also leads to low regularity of the solution.

Numerical analysis of the TDGL under the zero electric potential gauge $\phi=0$ has also been done in many works [@ASPM11; @GKL02; @GKLLP96; @MH98; @RPCA04; @VMB03; @WA02; @Yang]; also see the review paper [@Du05]. Since $\|\nabla\times{\bf A}\|_{L^2 }$ is not equivalent to $\|\nabla{\bf A}\|_{L^2 }$, both theoretical and numerical analysis are difficult under this gauge without extra assumptions on the regularity of the PDE’s solution. Again, convergence of these numerical methods have been proved in the case that the PDE’s solution is smooth enough.

Under either gauge, convergence of the numerical solutions has not been proved in nonsmooth domains such as general curved polyhedra, possibly nonconvex and multi-connected. Meanwhile, correct numerical approximations of the TDGL in domains with edges and corners are important for physicists and engineers [@ASPM11; @BKP05; @VMB03]. The difficulty of the problem is to control the nonlinear terms in the equations only based on the a priori estimates of the finite element solution. In this paper, we introduce a decoupled mixed FEM for solving - which guarantees that the discrete solution has bounded discrete energy, and prove convergence of the fully discrete finite element solution in general curved polyhedra without 