== The Finite Difference Technique ==
The general form of Poisson's equation for any field component potential ψ can be expressed as:
<math> \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} + \frac{\partial^2\psi}{\partial z^2} = -f(\mathbf{r}) </math>
The second derivative of ψ with respect to the coordinate x coordinate can be approximated by the second-order difference:
<math> \frac{\partial^2\psi(\mathbf{r})}{\partial x^2} \approx \frac{\psi(x+\Delta x,y,z)-2\psi(x,y,z)+\psi(x-\Delta x,y,z)}{(\Delta x)^2} </math> Similar expressions can be written for the second derivative with respect to the y and z coordinates. In the finite difference method, the computational domain is discretized using a 3D rectangular grid. The grid spacing along the three principal coordinate axes is denoted by Δx, Δy and Δz, respectively. The potential at any point (x,y,z) in the space can be expressed in terms of the potentials at six neighboring grid points along the principal axes. The creates a 7-point computational molecule shown in the figure below:
== 2D Quasi-Static Solution of TEM Transmission Line Structures ==