Complex Step Laplace

We can use the results from the previous post about complex step finite differences to solve the 2D Laplace's equation. Instead of solving the equation on a single rectangular grid as we would with normal finite differences, we now solve on three parallel grids (shown below).

One of the grids lies on the real x-y plane, one of the grids is offset slightly in the positive imaginary direction (blue arrow in the figure above), and one of the grids is offset slightly in the negative imaginary direction. We care about the real part of those two imaginary solutions for our second derivative approximation:


The central difference takes care of our solution on the real grid, but what about the two imaginary grids? For those we need asymmetric differences, and I'll put those up in another post.

You might recognize the look of the example mesh if you've ever played around with the 3D modeling/animation program Blender.