s

¨

s s

s

¢

G

""0# ¥ ¥£ ¥

¢ § ¦

¤¡

and factors of the ILU(0) factorization for

the 5-point matrix shown in Figure 10.4.

¦

Now the respective stencils of these and matrices can be represented at a mesh

point as shown in Figure 10.7.

p¥¶

¶ 8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢

¡

h S ¦

¨ S

S S¢ f

G

S

¤ )0d ¥ ¤¥£ ¢

¢§ ¦

¡

Stencils associated with the and factors

shown in Figure 10.6.

§

¦

The stencil of the product can be obtained easily by manipulating stencils directly ¦

rather than working with the matrices they represent. Indeed, the -th row of is obtained

by performing the following operation:

¡C ¡C ¡C ¡C

§

¦ ¦ ¦ ¦

w w

`S `S ` d S ` h dS

¯ G ”a

¯ S ¯ qa

¬

S

a a

f

This translates into a combination of the stencils associated with the rows:

§

¦ ¦ ¦ ¦h

w w

®¢¢ ® ¢ ¢ ¯ G ”a S ® ¢ ¢ ¯ S S ® ¢ ¢ ¯

S` S`

¨ ¨ ` d „¨ ¨

S

a a ` a

f

d

¨ ®¢¢ 4$`

in which represents the stencil of the matrix based at the mesh point labeled

a §

¦

. This gives the stencil for the matrix represented in Figure 10.8.

¥

h S ¦S h S ¦

d

f

¦

w w

S ¨S

S¢ £ S S

S ¨

f

SS

fd ¢

h S ¨S

h S ¢S

d

f

d

)0d ¥

¢§

¥£ ¢

¤¡

Stencil associated with the product of the and

¦

factors shown in Figure 10.6.

In the ¬gure, the ¬ll-in elements are represented by squares and all other nonzero elements

¦§

of the stencil are ¬lled circles. The ILU(0) process consists of identifying with in

S

locations where the original ™s are nonzero. In the Gaussian eliminations process, this

£

”

®”

is done from to . This provides the following equations obtained directly from

G

comparing the stencils of LU and (going from lowest to highest indices)

h S ¢S c

S

d

¶

’ ”8 ” &’ ¤ ¦§ ’ ”p¡ &ª8 ¤ ¥ c

$ © $ 8¥ ’ ¡¥

1 ¡¡© &

$

¢

¡

SS S

¦ d ¢ w ¨

f w y

S¢

S S S S S

S ¨

S ¨

f¦ f

h S

¬