To solve these two differential equations, we can first discretize the time derivative by finite differences. If u is a functions. Let us note \(u^n\) the quantity designating \(u\) at time \(n\).

Let’s go back to our equations :

\[\begin{equation} \int_{\Omega} \frac{1}{\mu} \, (\nabla \times \phi) \cdot (\nabla \times A) = - \int_{\Omega_C} \sigma \phi \cdot (\nabla V + \frac{\partial A}{\partial t}) \end{equation}\]

(or the following one if the edges are curved :)

\[\begin{equation} \int_{\Omega} \frac{1}{\mu} \, (\nabla \times \phi) \cdot (\nabla \times A) - \int_{\Gamma_D} \frac{1}{\mu} A_D \cdot (\nabla \times A) = - \int_{\Omega_C} \sigma \phi \cdot (\nabla V + \frac{\partial A}{\partial t}) \end{equation}\]

and :

\[\begin{equation} - \int_{\Omega_C} \sigma ( -\nabla V -\frac{\partial A}{\partial t}) \cdot \nabla \psi = 0 \end{equation}\]

Let us note \(\Delta t > 0\) the step time, such that \(t_n = n\Delta t\). Let us note \(A^n(x) := A(t_n,x)\). We have, using an implicit euler’s schema : \(\frac{\partial A}{\partial t} = \frac{A^n-A^{n-1}}{\Delta t}\).

So, in the case where the edges are not curved, we have :

\[\begin{equation} \int_{\Omega} \frac{1}{\mu} \, (\nabla \times \phi) \cdot (\nabla \times A^n) = - \int_{\Omega_C} \sigma \phi \cdot (\nabla V + \frac{A^n-A^{n-1}}{\Delta t}) \end{equation}\]

In other words :

\[\begin{equation} \int_{\Omega} \frac{\Delta t}{\mu} \, (\nabla \times \phi) \cdot (\nabla \times A^n) + \int_{\Omega_C} \sigma \phi \cdot (A^n + \Delta t\nabla V) = \int_{\Omega_C} \sigma \phi \cdot A^{n-1} \end{equation}\]

and the second equations :

\[\begin{equation} \int_{\Omega_C} \sigma (A^n + \Delta t\nabla V) \cdot \nabla \psi = \int_{\Omega_C} \sigma A^{n-1} \cdot \nabla \psi \end{equation}\]

If the edges are curved, our first equation becomes :

\[\begin{equation} \int_{\Omega} \frac{\Delta t}{\mu} \, (\nabla \times \phi) \cdot (\nabla \times A^n) - \int_{\Gamma_D} \frac{1}{\mu} A_D \cdot (\nabla \times A^n) + \int_{\Omega_C} \sigma \phi \cdot (A^n + \Delta t \nabla V) = \int_{\Omega_C} \sigma \phi \cdot A^{n-1} \end{equation}\]