£¤ c

¢ 85

¤ §

A

c c

vc

¤ § ¨ 5c v ¢

¨

¤ § ¨

.

.. .

.

' .

D h

¥ ©

5c v ¢

W W

W

Each pair is an incomplete LU factorization of the local matrix. These ILU

bA¢

tC C C

c c

factorizations can be computed independently. Similarly, the matrices and $£

)¢ #5

vC C C vC

can also be computed independently once the LU factors are obtained. Then each of the

matrices

a

5 c vC ¢ c v C C £ C

£

D C tC

which are the approximate local Schur complements, is obtained. Note that since an incom-

plete LU factorization is being performed, some drop strategy is applied to the elements in

¤

£

. Let be the matrix obtained after this is done,

C C

¤ £

# C

DC hC

Then a ¬nal stage would be to compute the ILU factorization of the matrix (13.14) where

¤

£

each is replaced by . C C

©

! D $"

#%

# 7 9@ &D

(F

To derive preconditioners for the Schur complement, another general purpose technique

exploits ideas used in approximating sparse Jacobians when solving nonlinear equations.

£

In general, is a dense matrix. However, it can be observed, and there are physical justi-

¬cations for model problems, that its entries decay away from the main diagonal. Assume

£

that is nearly tridiagonal, i.e., neglect all diagonals apart from the main diagonal and the £

two codiagonals, and write the corresponding tridiagonal approximation to as

£¤ §¦

4¤ §

¨z

c

¤ §

¨3 ¤ §

¨4 ¤z

.. .. ..

¤

. . .

D h

¥ ©

¢3 ¢ ¢z

4

c c