@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.}
}