We have the inhomogeneous poisson equation $-\Delta u = f$ on the unit interval and a non-uniform mesh $0=x_0 \ldots x_N=1$. When we write down the stiffness matrix we get a linear system of dimension $N+1$.
Enforcing Dirichlet boundary conditions, the dimension reduces by 1 for each side of the boundary.
What about von Neumann boundary conditions? For example, if I use a Dirichlet boundary condition $u(1)=b$ and a von neumann boundary condition $u'(0)=a$ how do the dimensions change?
First of all the Dirichlet side will cut the dimension by 1 again. It seems to me that the dimension should stay at $N$ even with von neumann at the other side but I don't know how to show it.