# Element Preconditioner

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}

• 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. Nov 5, 2018 at 16:16
• 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?
– user29088
Nov 5, 2018 at 17:01