c
1. Input: , Output: .
D v
2.
V2 D
c
3. For Do "w
D 2¢hiit
th h
d
}
4.
xC 2 t
D
5. EndDo
In summary, the Multiplicative Schwarz procedure is equivalent to solving the “pre
conditioned system” )
un4 ¦ ¦ j
˜
}
'
) D
x
W
} c
£
where the operation can be computed from Algorithm 13.5 and
'xD 9
D v z
can be computed from Algorithm 13.4. Now the above procedures can be used within an
W
accelerator such as GMRES. First, to obtain the righthand side of the preconditioned
)
system (13.35), Algorithm 13.4 must be applied to the original righthand side . Then z
GMRES can be applied to (13.35) in which the preconditioned operations are '
W
performed by Algorithm 13.5.
Another important aspect of the Multiplicative Schwarz procedure is that multicolor
ing can be exploited in the same way as it is done traditionally for block SOR. Finally, note
that symmetry is lost in the preconditioned system but it can be recovered by following the
sweep 1, 2, by a sweep in the other direction, namely, . This yields
¢i¢h
thh d " ’"
te ¢¢iit
ethhh
a form of the block SSOR algorithm.
¦§ ©§¡
¤ ¨
© © § ©
% $&$"
# %# % 5 C ¥ F0
F
) I G
5 R
7 9 ) 10 )
The additive Schwarz procedure is similar to a blockJacobi iteration and consists of up
dating all the new (block) components from the same residual. Thus, it differs from the
multiplicative procedure only because the components in each subdomain are not updated
until a whole cycle of updates through all domains are completed. The basic Additive
Schwarz iteration would therefore be as follows:
1. For Do 2¢hiit eG"w
D
th h
˜
˜
c }
2. Compute D C© # v' w C # mx C 
z
C
3. EndDo
4. pD ! ¢ 
© c sWC C

$t
The new approximation (obtained after a cycle of the substeps in the above algorithm
‘
—t
7 u pU¢ 7 p z7 yw— y w u
{ {˜p p { p
˜
“£§
¢
! hC §¤ !
¤ …S !
§
¥
¤ ¤
are applied) is
‚
˜ ˜
c
W }
!¢  # h )# z{x C
t pD
 wC
vC #
 h
c sC
Each instance of the loop rede¬nes different components of the new approximation and
there is no data dependency between the subproblems involved in the loop.
The preconditioning matrix is rather simple to obtain for the additive Schwarz proce
dure. Using the matrix notation de¬ned in the previous section, notice that the new iterate
satis¬es the relation
)