Gauss-Seidel and red-black ordering

Hi,

For 1D elliptic equation discretized with finite differences : Does the red/black ordering method for Gauss-Seidel algorithm imposes to have an odd number of points if one wants periodic boundary conditions?

If $i = 0,...,n$, $n$ is even, then the update of $i=0$ (which is black, say) needs $i=1$ and $i=n-1$. But because $n$ is even, $n-1$ is black too, there seems to be a problem because black should only use red points and vice versa.

So does red-black ordering and periodic boundary imposes an odd number of points ?

Your reasoning seems correct to me, apart from the fact that the interval $0,\dots,n$ contains an odd number of points for even $n$ and viceversa. So o/e would need an even number of points, which makes sense if you consider it as a colored grid on $\mathbb{Z}\times\mathbb{Z}$ and then quotient. –  Federico Poloni Oct 2 '11 at 7:50