3 ¢ ¢

4

—t

7 u pU¢ 7 p z7 yw— y w u

{ {˜p p { p

˜

# “£§

¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

¤

Then, it is easy to recover by applying it to three well-chosen vectors. Consider the three

vectors

}

£ t w i¢it¢ t ¢ ’¢t ¢ t¢ t it ¢ t ¢ ’e x

#c

D t

e

te

thhh

}

¨£ t w i¢it¢ ’¢t ¢ t ¢ t it¢ t ¢ ’¢t ¢ x

D te

e

te

thhh

}

¤£ h w ti¢it it ¢ t ¢ ’it¢ t¢ ’¢t ¢ t ¢ x

D te

te

hhh e

Then we have

¤ }

3 ¦ 3 ’ t z ’¨ ¨ sC ¤ 3 t c sC ¤ 4 t c sC ¤ z ¢ii¢t

t c 4x t4

£ Dc t thhh

r

r

r t w i¢it

hhh

¤ }

¦ t ¦ z t ¤ 3 ’¨ t w ih¢it ¤ sC ¤ 3 ¨ sC ¤ 4 ’£sC ¤ z iii¢ 3 t

¨

4t 4

¨£ t¨r

D t thhht

tr

r

hh

{x

z

¤ }

3 ’¤ 3 t C ¤ 4 t C ¤ z iii¢§ 3 t

4¤ t z t t

4

¤£ Aw hii¢c sC ¤

hht r

D t thhht

h

{x

z

¤

This shows that all the coef¬cients of the matrix are indeed all represented in the above

three vectors. The ¬rst vector contains the nonzero elements of the columns , , , , ¨

e ¢ih

hh

¤

, , in succession written as a long vector. Similarly, contains the columns ¨£

¢ih e t w

hh

¤

£

, and contains the columns . We can easily compute ¦ t¤ t

¤£ £ t’fD"et C

hi¢it 4¡ t

t

hh ¢i¢t

hhh w

e

d ¤ £

and obtain a resulting approximation which can be used as a preconditioner to . The £

idea can be extended to compute any banded approximation to . For details and analysis

see [49].

©DR)

¨§I

¡$©

§) ¤ ¤© H§¢

¦ ©

&D $"

#%

%# 7 9 ( E¡C 0F

9F

( (F C

) C

) 5 I 14

§ ¤

7GE9

3 A ) 3 ) ( 4

I

We now discuss some issues related to the preconditioning of a linear system with the

matrix coef¬cient of (13.14) associated with a vertex-based partitioning. As was mentioned

before, this structure is helpful in the direct solution context because it allows the Schur

complement to be formed by local pieces. Since incomplete LU factorizations will utilize

the same structure, this can be exploited as well.

Note that multicolor SOR or SSOR can also be exploited and that graph coloring

can be used to color the interface values in such a way that no two adjacent interface C

variables will have the same color. In fact, this can be achieved by coloring the domains.

In the course of a multicolor block-SOR iteration, a linear system must be solved with the

£

diagonal blocks . For this purpose, it is helpful to interpret the Schur complement. Call

C

the canonical injection matrix from the local interface points to the local nodes. If

2 Ci

points are local and if is the number of the local interface points, then is an ¢"i

§ §