§ © ©(
§
$&$"
# %# % 9 ( C
) 4
) )
˜
Throughout this section, it is assumed that is Symmetric Positive De¬nite. The projectors
de¬ned by (13.23) play an important role in the convergence theory of both additive and
2 C
multiplicative Schwarz. A crucial observation here is that these projectors are orthogonal
˜˜
with respect to the inner product. Indeed, it is suf¬cient to show that is selfadjoint 2 C
with respect to the inner product,
˜ 0˜ ˜ ˜ ˜ ˜
c
' } C  x D } C # c v C
' }  C 2 x # x D }  C # v C
# #t  2t
wC wC
t D t h
x
Consider the operator,
un ¦ ¦ j
‚
˜
˜ W
¡
h$2
X
D C
c sC
˜ ˜
Since each is selfadjoint with respect to the inner product, i.e., selfadjoint, their
)2
˜ ˜
sum is also selfadjoint. Therefore, it will have real eigenvalues. An immediate con
sequence of the fact that the ™s are projectors is stated in the following theorem.
2 C
Y% ™ # 9r…¨ …¡v¤
¥¦ ¥
˜
The largest eigenvalue of is such that
˜ }
¥£)
¤¢ U t
x
where is the number of subdomains.
£ %
¤¢¡
£
˜ s˜ ˜
¡ %U } ¡ x ¥¢
For any matrix norm, . In particular, if the norm is used,
¤
)
we have
‚
˜ W
}
¤¦£)
¢ 'C 2
U h
x
c sC
˜ ˜
Each of the norms of is equal to one since is an orthogonal projector. This
"2 C"2
C
proves the desired result.
This result can be improved substantially by observing that the projectors can be grouped
in sets that have disjoint ranges. Graph coloring techniques seen in Chapter 3 can be used §
3 3
to obtain such colorings of the subdomains. Assume that sets of indices iii¢’fVet C
thhhte D w
§
are such that all the subdomains for have no intersection with one another. ) H C
Then,
un! ¦ ¦ j
‚
˜
©2
¨ )2
V
Dd
$) ¨ d
is again an orthogonal projector.
This shows that the result of the previous theorem can be improved trivially into the
following.
™Y% ™ # 9r…¨ …¡v¤
¥¦ ¥
Suppose that the subdomains can be colored in such a way that two
subdomains with the same color have no common nodes. Then, the largest eigenvalue of
˜ is such that
¡
˜ } 3
¡ x ¥¢
¤
) U t
—t
7 u pU¢ 7 p z7 yw— y w u
{ {˜p p { p
˜
¡
“£§
¢
! hC §¤ !
¤ …S !
§
¥
¤ ¤
3
where is the number of colors.
In order to estimate the lowest eigenvalue of the preconditioned matrix, an assumption
must be made regarding the decomposition of an arbitrary vector into components of . C

„
Assumption 1. There exists a constant such that the inequality
£tC £ £t £
‚
„
˜ ˜
W } }
C U t
x x
c sC