 
Summary: An Iterated Tangential Filtering Decomposition
Yves Achdou ,
Frederic Nataf y
May 14, 2002
Abstract
Large linear systems arising from the discretization of partial dierential equations with
nite dierences or nite elements on structured grids in dimension d (d 3) require eĘcient
preconditioners. For a symmetric and positive denite (SPD) matrix, we propose a SPD
block LDL T preconditioner whose factorized form requires a smaller amount of memory
than the original matrix. Moreover, the computing time for the preconditioner solves is
linear with respect to the number of unknowns. The preconditioner is built in d stages: in
a rst stage, we use the tangential ltering decomposition of Wittum et al [12, 13, 9, 10],
and obtain a preconditioner which remains rather diĘcult to factorize. Then, in a second
stage, we apply tangential ltering decomposition recursively to the diagonal blocks of this
rst preconditioner. The nal stage consists of factorizing exactly the blocks corresponding
to one dimensional problems. Such preconditioners can also be computed adaptively and
combined in a multiplicative way. A generic programming implementation is discussed and
numerical tests are presented, in particular for problems with highly heterogeneous media.
1 Introduction
For applications involving non linear partial dierential equations in heterogeneous media, the
