Apply second order finite difference discretization for mixed boundary condition

Question

I want to solve the problem below

\begin{equation} \begin{aligned} \eta u-\Delta u &=f, &\text{in $\Omega$}\\ \end{aligned} \end{equation}

where $\Omega=(0,1)\times (0,1)$, My boundary are like this $\frac{\partial u}{\partial n} = 0$ if $y=0$ or $1$, and $u=0$ if $x=0$ or $1$. Using ghost nodes can handle the low order discretization of the neumann boundary, but the problem that i'm facing is that I want to include the four corner points in the neumann boundary not in the dirichlet one, and this causes me problem since I cannot use the interior equation on the corners.

I hope my question was clear.

Answer 1

Mathematically speaking, you can only pose boundary conditions on parts of the boundary that has nonzero $d-1$ dimensional measure. In other words, it doesn't make sense to say that you want a particular boundary condition to hold at individual points. Whether you do or do not do anything at a corner point, you will nevertheless converge to the same solution. In other words, just don't bother with figuring out whether or not you have Dirichlet or Neumann conditions at these corners.