[@CD01; @GLS] with different time discretizations by assuming that the PDE’s solution is smooth enough, e.g. ${\bf A}\in L^\infty(0,T;{\bf H}^1)\cap L^2(0,T;{\bf H}^2)$. In a nonsmooth domain such as a curved polyhedron, the magnetic potential ${\bf A}$ may be only in $L^\infty(0,T;{\bf H}({\rm curl,div}))\cap 
L^2(0,T;{\bf H}^{1/2+\delta})$, where $\delta>0$ can be arbitrarily small (depending on the angle of the edges or corners of the domain), and so the Galerkin finite element solution may not converge to the solution of -. Some mixed FEMs were proposed in [@Chen97; @GS15], and the numerical simulations in [@GS15] show better results in nonsmooth domains, compared with the Galerkin FEM. Some discrete gauge invariant numerical methods [@Du98; @DJ05] are also promising to approximate the solution correctly. Convergence of these numerical methods have been proved in the case that the PDE’s solution is smooth enough. However, whether the numerical solutions converge to the PDE’s solution in nonsmooth domains where the magnetic potential is only in $L^\infty(0,T;{\bf H}({\rm curl,div}))\cap 
L^2(0,T;{\bf H}^{1/2+\delta})$ is still unknown. Another problem is that the initial data ${\bf A}_0$ are often incompatible with the boundary condition (see the numerical examples in [@ASPM11; @CD01; @Mu97], where ${\bf n}\times(\nabla\times{\bf A}_0)=0$ but ${\bf n}\times{\bf H}\neq 0$), and this 