$E2 h )# h 8#
DC wC
vC hC
¤C
˜ ˜
c
Observe that the operator is a projector since
2 # # vC C
wC
¨}
˜ ˜ ˜ ˜˜ ˜ ˜ ˜
c c
c
c
# h w C # C # v' } w C # C # x ' w C # D
# ' wC # x C hC
vC vC
D
vC h
C
Thus, one sweep produces an error which satis¬es the relation ) ) )
2n1 ¦ j
£ } "2 x i¢h } c 2 x } 2 x D £ ˜
hh
c h
vW W
W
In the following, we use the notation )
)
)
un4 ¦ j
¤ ˜
2 x} 2 x
}
}
'
h cV2 x iih c
hh
v
W W W
¦ ¨¡
¤ § ¤ §
© ¡$©
§) ¤
! $&$"
# %# % 3 1 &¥ A
7F A 5@ F CCF
) I 5 7R 9 ( 9¥0F
F &4(
(F
Because of the equivalence of the multiplicative Schwarz procedure and a block Gauss
Seidel iteration, it is possible to recast one Multiplicative Schwarz sweep in the form of a
¡
global ¬xedpoint iteration of the form . Recall that this is a ¬xedpoint ¥
!¢  t) D
)c
˜ c
iteration for solving the preconditioned system where the precondition v D # v
 z
˜
c
ing matrix and the matrix are related by . To interpret the operation G
D
v
c
associated with , it is helpful to identify the result of the error vector produced by this
v
}£
iteration with that of (13.24), which is . This comparison yields, ' D £ ) W ! ¢ 
 ‚ x

W
}
' D !¢  t£
' xt
W W
and therefore, )
}
¥ h£
'f ' )x D
D
W W
˜
c
Hence, the preconditioned matrix is . This result is restated as follows.
'
D
v
W
x¤ …¨ r£
£¦ Y% ™ #
£¨ ¡¨
¢ The multiplicative Schwarz procedure is equivalent to a ¬xed
point iteration for the “preconditioned” problem
˜
c c
D tz
v v
in which )
un ¦ j
˜
˜
c
' ) )
D
v
un! ¦ j
W ˜
˜
c } } c
D£
' '
Dz hz
v v
x x
W W
—t
7 u pU¢ 7 p z7 yw— y w u
{ {˜p p { p
˜
“£§
¢
! hC §¤ !