where a star denotes a matrix whose actual expression is unimportant. Recall that by de¬-

nition, )

}

¢ x D'¤ # t

where this partitioning conforms to the above ones. This means that

c c c

¤ ¤

# # v' #¢

w D 9

v v

and, therefore, according to (13.45), . This result is stated in the following

¢

D

proposition.

r¤ …¨ r£

£¦ Y™% ™ #

£¨ ¥ ¡¨

¢ ˜

Let be an ILU preconditioner for . Then the

' '¢ '

D

£

preconditioner for induced by , as de¬ned by (13.45), is given by

'

with ¤ ¤

¤ ¤

'¢ '

¢D ¢ #D t )# #D h )#

t w w

7f p uq£ ˜

| }¢ ˜ £ p‘ ¡

£¤ ¢

§¤ “ W …"’ h¤C §

¤

" $

§

§

}

In words, the proposition states that the L and U factors for are the blocks

dt dx

˜

of the L and U factors of the ILU factorization of . An important consequence of the

above idea is that the parallel Gaussian elimination can be exploited for deriving an ILU

£

preconditioner for by using a general purpose ILU factorization. In fact, the and ¢

factors of have the following structure: '

˜ witht

'¢ #'

D

£¤ §¦

¤ §

¢ c

¤¤ §

§

¨¢

..

.

D '¢

¥ ©

¢

c c c

W

£ 1£ £ ¢

¨

c vc v¨ v

VVT

TT

W