Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{const}$ on each electrode). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance.
With FEM we can obtain the the potential, then the electric field and then $E/|E|$.
I was wondering, are there any other methods to obtain $E/|E|$?
This corresponds to the current streamlines. Is there some equations the streamlines follow?
In 2-D the situation is different from 3-D, as is exemplified in this answer to the question:
At the bottom, we have an example with Electric Fields. There replace the exact differential equation by the following (orthogonal trajectories one) to find the streamlines of the currents (giving straight lines through the origin): $$ E_x\, dy - E_y\, dx = \frac{x\,dy - y\,dx}{r^2} = 0 \quad \Longrightarrow \quad \ln(y) = \ln(x)+\ln(c) \quad \Longrightarrow \quad y = c.x $$ Another reference that might be interesting: