I have following Problem,
Consider poisson equation $-\Delta y =f$ on triangle with boundary conditions $$y=g^{(1)}$$ $$y=g^{(2)}$$ $$\frac{\partial y}{\partial n}=g^{(3)}$$
(https://i.stack.imgur.com/7gUAY.jpg) with $Q$ being the the sides
for smooth functions $f,g^{(1)},g^{(2)},g^{(3)}$ and outer normal $n$. Lets take a equidistant discreditization of triangle $\Omega$ with grid width $h=1/N$.
i)I wanna devise a difference scheme (using a 5-point stencil) and approximate the derivatives in the Neuman boundary condtion up to first order.
ii)this should lead to following finite volume method (same grid with node-centered Voronoi polygons)
$4u_{i,j}-u_{i,j-1}-u_{i,j+1}-u_{i-1,j}-u_{i+1,j}=h^2 f_{i,j}$ $\quad$ for $x_{i,j}\in \Omega_h$
$u_{i,j}=g_{i,j}^{(l)}$ $\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ for $x_{i,j}$ $\in$ $Q_l$ $\quad$for $l=1,2 $
$u_{i,j}-u_{i,j}-u_{i,j}=\sqrt{2}hg_{i,j}+\frac{1}{2}h^2f_{i,j}$ $\quad\quad\quad\quad\quad$ for $x_{i,j} \in Q_3$
I have no clue how to solve that. Anyone has an idea? The triangle somehow just confuses me.
Thanks in advance