Im just working on a preconditioner for the linear equation system $Ax = b$ arising in FEM for elliptic PDE. $A$ is a s.p.d Matrix with real valued entries. I read something about the element by element preconditioner and now I have a question about the following:
Defining a preconditioner $P$ such that $P \approx A$. One can solve the equation system (left preconditioning) instead:
\begin{equation} P^{-1}Ax = P^{-1}b \end{equation}
I read that some people choosing the preconditioner as the Assembly of Element Preconditioners.
\begin{equation} P =\prod\limits_{e =1}^n P_e \end{equation}
where $P_e$ is defined for every element $e$ as \begin{equation} P_e = I + diag(A_e)^{-1/2}\cdot(A_e - diag(A_e))\cdot diag(A_e)^{-1/2} \end{equation}
Now to calculate $P^{-1}$ you just need to assembly all $P_e^{-1}$.
My question is now: If I use a Cholesky decomposition would for $P$ such that $P = C\cdot C^T$ would it be the same as if I use the decomposition for every element and assemble afterwards:?? \begin{equation} P_e = C_e C_e^T \rightarrow C\cdot C^T = \prod\limits_{e = 1}^n C_e C_e^T \end{equation}
I dont have the feeling that this holds because of overlapping Elements. But Im not sure.
By the way my goal is to use a decomposition to solve
\begin{align} C^{-1}AC^{-T}y &= C^{-1}b \\ x &= C^{-T}y \end{align}
