  We prove well-posedness of time-dependent Ginzburg--Landau system in a
nonconvex polygonal domain, and decompose the solution as a regular part plus a
singular part. We see that the magnetic potential is not in $H^1$ in general,
and the finite element method (FEM) may give incorrect solutions. To remedy
this situation, we reformulate the equations into an equivalent system of
elliptic and parabolic equations based on the Hodge decomposition, which avoids
direct calculation of the magnetic potential. The essential unknowns of the
reformulated system admit $H^1$ solutions and can be solved correctly by the
FEMs. We then propose a decoupled and linearized FEM to solve the reformulated
equations and present error estimates based on proved regularity of the
solution. Numerical examples are provided to support our theoretical analysis
and show the efficiency of the method.
