Given the 2D Poisson equation
$$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\ \partial_n u (x, 1-x) =0, 0<x<1$$
defined on the domain $\Omega := \{(x,y) \in \mathbb{R}^2_+: x+y<1\}$, discretize the equation using Shortley-Weller approximation with 5-point stencils.
On this domain, however, it appears that all near-boundary points use standard 5-point stencils. Is my understanding correct? I.e., with a uniform mesh non-boundary points' descritization uses the same step $h$ in all directions.
I'm, however, puzzled about the Neumann condition. Is this where Shortley-Weller approximation actually applied? I would appreciate some help with pointing me in the right direction towards discretizing the Neumann condition, and this equation as a whole.
Update: I figure the Neumann condition for this domain can be discretized as follows: since $\partial_n u = \nabla u \cdot n$, we have $\partial_{n} u_{i,j} = \frac{u_{i, j+1}-u_{i,j-1}+u_{i+1,j}-u_{i-1,j}}{h}=0$ (because $n=(1, 1)^T$). Now it looks like we encounter two ghost shells in each case. Am I on the right track?
