equations in nonsmooth domains with incompatible initial data [^2]** '
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Introduction
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The time-dependent Ginzburg–Landau equation (TDGL) is a macroscopic phenomenological model for the superconductivity phenomena in both low and high temperatures [@GL; @GE; @Gennes; @Tinkham], and has been widely accepted in the numerical simulation of transition and vortex dynamics of both type-I and type-II superconductors [@FUD91; @LMG91]. In a non-dimensionalization form, the TDGL is given by $$\begin{aligned}
&\eta\frac{\partial \psi}{\partial t} + i\eta\kappa\psi\phi
+ \bigg(\frac{i}{\kappa} \nabla + \mathbf{A}\bigg)^{2} \psi
 + (|\psi|^{2}-1) \psi  = 0,
\label{GLLPDEq1}\\[5pt]
&\frac{\partial \mathbf{A}}{\partial t}  +\nabla \phi
+ \nabla\times(\nabla\times{\bf A})
+  {\rm Re}\bigg[\overline\psi\bigg(\frac{i}{\kappa} \nabla 
+ \mathbf{A}\bigg) \psi\bigg] =  \nabla\times {\bf H} ,
\label{GLLPDEq2}\end{aligned}$$ where the order parameter $\psi$ is complex scalar-valued, the electric potential $\phi$ is real scalar-valued and magnetic potential ${\bf A}$ is real vector-valued; $\eta>0$ and $\kappa>0$ are physical parameters, and ${\bf H}$ is a time-independent external magnetic field. In a domain $\Omega\subset\R^3$ occupied by a superconductor, the following physical boundary conditions are often imposed: $$\begin{aligned}
&\Big(\frac{i}{\kappa}\nabla\psi +
\mathbf{A}\psi\Big) \cdot{\bf n} = 0
&\mbox{on}\,\,\,\,\partial\Omega , \\
&{\bf n}\times{\bf B}={\bf n}\times {\bf H}
&\mbox{on}\,\,\,\,\partial\Omega , \\
&{\bf E}\cdot{\bf n}=0  &\mbox{on}\,\,\,\,\partial\Omega , 
\label{OrBDC3}\end{aligned}$$ where $\mathbf{n}$ denotes the unit normal vector on the boundary of the domain, 