The finite differences algorithm

Suppouse that we are given the values of $V(x,y)$ in the boundaries of a rectangular area. If we divide the rectangle in a grid, we can the apply (78) to calculate the values of $V$ inside. However, we notice right away that if we apply this equation in a point in the middle, we get zero as a result. So the first obvious observation is that the initial estimate for the method can not be a constant value. It is evident now that the procedure has to be iterative. We start from the bottom-left corner, and we sweep across the grid. As we obtain the first estimate for $V$, we repeat the sweep until the values of $V$ do not change considerably, and we can say that we have converged with certain accuracy. Here we have started with the initial values $V(x,y)$ inside the grid that in principle can be any initial guess (exept a constant, as we have seen). The rate of convergence will be determined by the proximity of this guess to the actual solution.

The algorithm can be decribed as follows:

1. Divide the rgion of interest into a grid. The region must be enclosed by boundaries with known values of $V$.

2. Assingn to all points of the boundary the given values, and all interior cells an arbitrary potential (preferably a reasonable guess)

3. sweep the internal cells by rows or columns and calculate the new values of $V$ as the average in the four neighboring points.

4. Repeat the sweep using the values obtained in the previous iteration, until the potential does not change within certain level of accuracy.