¨#
9#
c
represented in all three matrices.
™ —t
7 u pU¢ 7 p z7 yw— y w u
{ {˜p p { p
˜
§&
“£§
¢
! hC §¤ !
¤ …S !
§
¥
¤ ¤
# Dc
¨# D
¤# D
™
& $"
% #!
Patterns of the three matrices associated
# C
with the partitioning of Figure 13.4.
From the linear algebra point of view, the restriction operator is an matrix formed ¢
# C #Ci
i
by the transposes of columns of the identity matrix, where belongs to the index ¢
)˜ i i H
set . The transpose of this matrix is a prolongation operator which takes a variable
#
B
C wC
from and extends it to the equivalent variable in . The matrix
C
˜ # ˜ )#
DEC 8
C wC
˜ ˜
of dimension de¬nes a restriction of to . Now a problem associated with
¢
Ci Ci C C
can be solved which would update the unknowns in the domain . With this notation, the C
multiplicative Schwarz procedure can be described as follows:
1. For Do
2hi¢hit GD$w
e
th
˜ ˜
c }
2. v' wC # t pD  # mx C 
z
C
3. EndDo
We change notation and rewrite step 2 as
An H¦ j
˜ S
˜
˜
# c vC }
!¢  )# mx C
t pD
 wC  h
z
£
If the errors are considered where is the exact solution, then notice that
£  D £

¢!˜ £
˜ }
and, at each iteration the following equation relates the new error
x £  x
z `#
D
£
and the previous error ,
!¢ £ £ £
˜ ˜
# c vC
)#
D wC C h
™
˜ kw ‘ ¡ w— — p7w ¡w £ ˜
} p {7  ¢ &&
£¤§ §¤ X
“
! ¨
§
¡
£

Starting from a given whose error vector is , each subiteration produces
£ fD
 
an error vector which satis¬es the relation
£ £ £
˜ ˜
c
)# # vC
DC C C C
c wC c
t
v v
for . As a result, )
"w
D 2hi¢hit e
th
£ £}
$2
EC
D C C c
x
v
in which
un¦ ¦ j
˜
˜ ˜