I've a problem regarding the Poisson's equation:
The version of the Poisson's equation being solved here is
$$\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} u(x, y) = f(x, y)$$
$$u(x, y) = g(x, y) \text{ on $\partial D$}$$
over the rectangle $$0 \leq x \leq 1\\0 \leq y \leq 1\\$$
What's the meaning of "a Poisson's equation being solved over a rectangle"?
The rectangle is the domain of your equations, so where you have the boundary of D, D is your rectangle.