I've read Boxerman's thesis and I feel that there is possibly a mistake.
We have to resolve $$Ax=b$$ $A$ is a positive-definite symmetric matrix and is very sparse so the conjugate gradient method is chosen.
He wrote (p70 to p73) that to improve the convergence, we can use the following preconditionner : $$P=SC+(I-S)$$ where $C$ is the 3x3 block-diagonal sub-matrix of $A$, $I$ is the identity matrix and $S$ is a 3x3 block-diagonal matrix where each block can be
$S_i=I_3$ or $S_i=0$ or $S_i=I_3-nn^T$ ($n$ is a 3-dimension unit vector)
So, the preconditionner $P$ is a 3x3 Block-Diagonal matrix and can be easily inverted BUT I think it is not necessary symmetric. As far as I know, a preconditionner for the conjugate gradient method must be symmetric.
Am I missing something or there is a mistake in the thesis ?