= ({\bf H} \, , \nabla\times {\bf a}_{h}) 
-{\rm Re}\bigg(\overline\psi_h^{n}\bigg(\frac{i}{\kappa} \nabla 
+{\bf A}_h^{n}\bigg) \psi_h^{n} , {\bf a}_h\bigg) \, , 
  &&\forall\, {\bf a}_h\in  {\mathbb N}_{h}^{k} .
\label{FEM2} \end{aligned}$$ After solving $\psi_h^{n+1}$, $\phi_h^{n+1}$ and ${\bf A}_h^{n+1}$ from the equations above, the magnetic and electric fields can be computed by ${\bf B}_h^{n+1}=\nabla\times{\bf A}_h^{n+1} $ and ${\bf E}_h^{n+1}=-D_\tau{\bf A}_h^{n+1}
-\nabla\phi_h^{n+1} $.

For simplicity, we have chosen $(\psi_h^0,{\bf A}_h^0)=(\psi_0,{\bf A}_0)$ at the initial step, which are not finite element functions. Due to the nonlinearities and the choice of the initial data, some integrals in and may need to be evaluated numerically in practical computations. In this paper, we focus on the analysis of the discretization errors of the finite element method and assume that all the integrals are evaluated accurately.

Since we have not assumed any extra regularity of the PDE’s solution, we need the condition to be compatible with the nonlinear structure of the equations in order to control a nonlinear term arising from (see for the details). If the PDE’s solution is smooth enough, (e.g. consider the problem in a smooth domain), then the condition can be relaxed.

We define the discrete energy $$\begin{aligned}
{\cal G}_{h}^{n} 
&=\int_\Omega 
\bigg(\frac{1}{2}\bigg|\frac{i}{\kappa} 
\nabla\psi_h^{n}  + {\bf A}_h^{n}\psi_h^{n} \bigg|^2
