2D quad mapping for finite difference to solve Poisson's equation

Question

I've been reading about the method of finite differences to solve Poisson's equation and I think I got the main idea of how it works on a 2D regular grid. However, I'm not sure how I could apply it to other domains. It seems a common approach is to create a mapping and reformulate the Poisson equation with new coordinates. Let's say I want to solve Poisson's equation for a domain like the following trapezoid. How could I make a suitable mapping to/from the regular grid to that domain so that I can apply the finite difference method?

Answer 1

You can create a mapping from the unit square to a trapezoid given by points A, B, C, D with the following function:

$$f(t_1, t_2) = A(1 - t_1)(1 - t_2) + B \ t_1 (1 - t_2) + C\ t_1\ t_2 + D (1 - t_1) t_2$$

One thing to keep in mind is that the Laplacian at a given point is independent of the coordinate system you choose, but you might still have to transform some equations from one domain to another, depending on your problem, like boundary conditions and such.