${\bf B}=\nabla\times{\bf A}$ and ${\bf E}=-\partial_t{\bf A}-\nabla\phi$ denote the induced magnetic and electric fields, respectively.

Besides -, an additional gauge condition is needed for the uniqueness of the solution $(\psi,\phi,{\bf A})$. Under the gauge $\phi=-\nabla\cdot{\bf A}$, the TDGL reduces to $$\begin{aligned}
&\eta\frac{\partial \psi}{\partial t} -i\eta \kappa \psi \nabla\cdot{\bf A}
+ \bigg(\frac{i}{\kappa} \nabla + \mathbf{A}\bigg)^{2} \psi
 + (|\psi|^{2}-1) \psi = 0,
\label{PDE1}\\[5pt]
&\frac{\partial \mathbf{A}}{\partial t} 
+ \nabla\times(\nabla\times{\bf A})
-\nabla(\nabla\cdot{\bf A}) 
+  {\rm Re}\bigg[\overline\psi\bigg(\frac{i}{\kappa} \nabla 
+ \mathbf{A}\bigg) \psi\bigg] =  \nabla\times {\bf H} ,
\label{PDE2}\end{aligned}$$ and the boundary conditions can be written as [^3] $$\begin{aligned}
&\nabla\psi\cdot{\bf n} = 0
&&\mbox{on}\,\,\,\,\partial\Omega ,  \label{PDEBC-00}\\
&{\bf n}\times(\nabla\times{\bf A})= {\bf n}\times{\bf H}  
&&\mbox{on}\,\,\,\,\partial\Omega , \label{PDEBC-0}\\
&{\bf A}\cdot{\bf n}=0 
&&\mbox{on}\,\,\,\,\partial\Omega .
\label{PDEBC}\end{aligned}$$ Given the initial conditions $$\begin{aligned}
\label{PDEini}
\psi(x,0)=\psi_0(x)\quad\mbox{and}\quad
{\bf A}(x,0)={\bf A}_0(x),\quad\mbox{for}\,\, x\in\Omega , \end{aligned}$$ the solution $(\psi,{\bf A})$ can be solved from -. Other gauges can also be used, and the solutions under different gauges are equivalent in the sense that they produce the same quantities of physical intereset [@CHL; @Tinkham], such as the superconducting density $|\psi|^2$ and the magnetic field ${\bf B}$.

In a smooth domain, well-posedness of - has been proved in [@CHL] and convergence of the Galerkin finite element method (FEM) was proved in 