2. Solve the reduced system via an iterative method. © Q
8D
3. Backsubstitute, i.e., compute via (13.3). 
The different methods relate to the way in which step 2 is performed. First observe
£
that the matrix need not be formed explicitly in order to solve the reduced system by
an iterative method. For example, if a Krylov subspace method without preconditioning £
is used, then the only operations that are required with the matrix are matrixbyvector
£
operations . Such operations can be performed as follows.
£ D
5 8
1. Compute , D
2. Solve
a8 D
3. Compute .
£
£ D
The above procedure involves only matrixbyvector multiplications and one lin
ear system solution with . Recall that a linear system involving translates into 
independent linear systems. Also note that the linear systems with must be solved ex
actly, either by a direct solution technique or by an iterative technique with a high level of
accuracy. £
While matrixbyvector multiplications with cause little dif¬culty, it is much harder £
to precondition the matrix , since this full matrix is often not available explicitly. There
have been a number of methods, derived mostly using arguments from Partial Differential
Equations to precondition the Schur complement. Here, we consider only those precondi
tioners that are derived from a linear algebra viewpoint.
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§ ¤ ©
'D¡$"
# # % (F 10 ) 7U 9 ( 9 ¥ F0
F 0(
) 2
I
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One of the easiest ways to derive an approximation to is to exploit Proposition 13.1
and the intimate relation between the Schur complement and Gaussian elimination. This £
proposition tells us that a preconditioning operator to can be de¬ned from the (ap
˜
proximate) solution obtained with . To precondition a given vector , i.e., to compute
c £
, where is the desired preconditioner to , ¬rst solve the system
Dp£ v
¥ ¥
2n1 1 ¦ j
¢ ˜

˜ D t
¦ ¦
then take . Use any approximate solution technique to solve the above system. Let
£
D
˜
be any preconditioner for . Using the notation de¬ned earlier, let represent the ¤
' #
restriction operator on the interface variables, as de¬ned in Proposition 13.1. Then the
—t
7 u pU¢ 7 p z7 yw— y w u
{ {˜p p { p
˜
¡
“£§
¢
! hC §¤ !
¤ …S !
§
¥
¤ ¤
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preconditioning operation for which is induced from is de¬ned by
'
¥
¢
c c c
¤ ¤¦# ¤
#D # v'
D h
v v' w