3 a S £` £

™„D¬BD% G

¨ %¬ ¬

3. For each and for Do:

£G S

£ S £

4. Compute

¨ £ 5S £

5. Compute . £ S £

5S 5

S

6. Update the levels of ¬ll of the nonzero ™s using (10.18)

C£

7. EndDo

a ¡£ ` £

8. Replace any element in row with by zero

S

9. EndDo

There are a number of drawbacks to the above algorithm. First, the amount of ¬ll-in and

E

computational work for obtaining the ILU( ) factorization is not predictable for .

Second, the cost of updating the levels can be quite high. Most importantly, the level of

¬ll-in for inde¬nite matrices may not be a good indicator of the size of the elements that

are being dropped. Thus, the algorithm may drop large elements and result in an inaccurate §

¦

¨

incomplete factorization, in the sense that is not small. Experience reveals ¤

that on the average this will lead to a larger number of iterations to achieve convergence,

although there are certainly instances where this is not the case. The techniques which will

be described in Section 10.4 have been developed to remedy these three dif¬culties, by

producing incomplete factorizations with small error and a controlled number of ¬ll-ins. ¤

« y¨ f y h

f

£¨

« «k

£y

£

h

f S ¨ y h S

S SS

f

s

¡¨

s

s sy s

)0d ¥ ¤¥£ ¢

¢§ ¡

Matrix resulting from the discretization of an el-

liptic problem on a rectangle.

¢D!¥ ¤e

¡ BH P ¥

3§ 3¢6

2 2¡ ¨gf

Y9 CbDB e &!@ p3e a Y P

9T 8H

@eY CeA@ Y

H

Often, the original matrix has a regular structure which can be exploited to formulate the

ILU preconditioners in a simpler way. Historically, incomplete factorization precondition-

ers were developed ¬rst for such matrices, rather than for general sparse matrices. Here, we

call a regularly structured matrix a matrix consisting of a small number of diagonals. As an

¶

’ ”8 ” &’ ¤ ¦§ ’ ”p¡ &ª8 ¤ ¥ c

$ © $ 8¥ ’ ¨¥

§

1 ¡¡© &

$

¢

¡

example, consider the diffusion-convection equation, with Dirichlet boundary conditions W

´´ ¥¤¬G£wV´ ¡¨ ¦

Wc& U ¦

¢

¦§

E

¦

where is simply a rectangle. As seen in Chapter 2, if the above problem is discretized

using centered differences, a linear system is obtained whose coef¬cient matrix has the

structure shown in Figure 10.4. In terms of the stencils seen in Chapter 4, the representation

of this matrix is rather simple. Each row expresses the coupling between unknown and

w w

k

¨ ±

unknowns , which are in the horizontal, or direction, and the unknowns

G G

±w

¨

and which are in the vertical, or direction. This stencil is represented in Figure 10.5.

h S

S ¨

y

S S f

c

S

"0# ¥ ¤¥£ ¡¥

¢§ Stencil associated with the 5-point matrix shown

in Figure 10.4.

¦

The desired and matrices in the ILU(0) factorization are shown in Figure 10.6.

¦

h¦

«¨

f¢ f

qG«

h

f

s¦