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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}

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  • $\begingroup$ This seems pretty sketchy to me. Is the "sum of inverses" (what you're doing) and the "inverse of a sum" (the ideal preconditioner, $\mathbf A^{-1}$) related in any meaningful way? And won't the end result just be a Cholesky triangle with the same stencil as incomplete Cholesky0 (ie no fill-ins)? I think that you'd be on theoretically firmer ground just doing incomplete/no-fill Cholesky, especially using a modification/compensation strategy to "correct" for the missing fill-in. $\endgroup$ – rchilton1980 Nov 5 '18 at 16:16
  • $\begingroup$ Thank you for the fast respond to my question. I should have mentioned that "Now to calculate $P^{-1}$ you just need to assembly all $P_e^{-1}$" ist just meant in a symbolically way. At the end I want to calculate the inverse by using a decomposition. Even if I use incomplete Cholesky my question remains the same. Can I calculate the decomposition of my global preconditioner by assembly the decomposition of every element? $\endgroup$ – Kerem Nov 5 '18 at 17:01

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