@TechReport{ it:2007-008,
author = {Erik B{\"a}ngtsson and Maya Neytcheva},
title = {Finite Element Block-Factorized Preconditioners},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2007},
number = {2007-008},
month = mar,
abstract = {In this work we consider block-factorized preconditioners
for the iterative solution of systems of linear algebraic
equations arising from finite element discretizations of
scalar and vector partial differential equations of
elliptic type.
For the construction of the preconditioners we utilize a
general two-level standard finite element framework and the
corresponding block two-by-two form of the system matrix,
induced by a splitting of the finite element spaces,
referred to as {\em fine} and {\em coarse}, namely, $$ A =
\begin{bmatrix} A_{11}&A_{12}\\ A_{21}&A_{22}\end{bmatrix}
\begin{matrix}fine,\\ coarse.\end{matrix}. $$ The matrix
$A$ admits the exact factorization $$ A =
\begin{bmatrix}A_{11}&0\\ {A}_{21}&{S_A}\end{bmatrix}
\begin{bmatrix}I_{1}&A_{11}^{-1}A_{12}\\ 0&
I_2\end{bmatrix}, $$ where
$S_A=A_{22}-A_{21}A_{11}^{-1}A_{12}$ and $I_1$, $I_2$ are
identity matrices of corresponding size. The particular
form of preconditioners we analyze here is $$ M_{B} =
\begin{bmatrix}B_{11}&0\\ {A}_{21}&{S}\end{bmatrix}
\begin{bmatrix}I_{1}&Z_{12}\\ 0& I_2\end{bmatrix}, $$ where
$S$ is assumed to be some available good quality
approximation of the Schur complement matrix $S_A$.
We propose two methods to construct an efficient, sparse
and computationally cheap approximation $B_{11}^{-1}$ of
the inverse of the pivot block $A_{11}^{-1}$, required when
solving systems with the block factorized preconditioner
$M_B$. Furthermore, we propose an approximation $Z_{12}$ of
the off-diagonal matrix block product $A_{11}^{-1}A_{12}$,
which further reduces the computational complexity of the
preconditioning step. All three approximations are based on
element-by-element manipulations of local finite element
matrices.
The approach is applicable for both selfadjoint and
non-selfadjoint problems, in two as well as in three
dimensions. We analyze in detail the 2D case and provide
extensive numerical evidence for the efficiency of the
proposed matrix approximations.}
}