Uppsala University
Department of Information Technology

### Technical Report 2007-008

# Finite Element Block-Factorized Preconditioners

### Erik Bängtsson and Maya Neytcheva

March 2007

**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 *fine* and *coarse*, namely,

* A = begin{bmatrix} A*_{11}&A_{12}\ A_{21}&A_{22}end{bmatrix} \beginmatrixfine,\ coarse.end{matrix} .

The matrix *A* admits the exact factorization * A = \beginbmatrixA*_{11}&0\ A_{21}&S_{A}end{bmatrix} \beginbmatrixI_{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} = \beginbmatrixB_{11}&0\ A_{21}&Send{bmatrix} \beginbmatrixI_{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.

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