C C C

matrix whose columns are the last columns of the identity matrix. Then it is ¢

5§

C y i

C

Ci

easy to see that

An 1 H¦ j

˜

‚

˜ c } c

£

2 C G £¡h w'2 x EC

D

¢ v™ h

v

˜ ˜

¢ D C G ¢£¤™ C G £¡™

If is the LU factorization of then it can be veri¬ed that

¢

nA! 1 H¦ j

˜

c

2c v ¢c v 2 c v ¢ w'2 2 c v

£

w'2 '2

vC D D w t

2c v ¢ w 2

which indicates that in order to operate with , the last principal submatrix ¢C

§ § C

2c v w 2

of must be used. The same is true for which requires only a back-solve with

¢

˜ C G £¤™

the last principal submatrix of . Therefore, only the LU factorization of is

¢

§ §

C C ¢

‚ U¢ 7 fz— ||k}

˜ p {7 ¡#

#

£¤§ #¤

…¤

#

!

¡

£

needed to solve a system with the matrix . Interestingly, approximate solution methods C

˜ CG

associated with incomplete factorizations of can be exploited. ™

¢¡

9

’ hv k –z

‘

¢

&E& 3 7 h

@

¨U ¨ ¤

£

—˜ •

C

We call any technique that iterates on the original system (13.2) a full matrix method. In the

˜

same way that preconditioners were derived from the LU factorization of for the Schur

˜

complement, preconditioners for can be derived from approximating interface values.

Before starting with preconditioning techniques, we establish a few simple relations

˜ £

between iterations involving and .

x¤ …¨ r£

£¦ ™#

£¨ ¡¨

¢ %

Let )

¥ ¥

un# 1 ¦ j

)¢ )5 ˜

'¢ '

D t D

c

¢

£ ¦ ¦

v

and assume that a Krylov subspace method is applied to the original system (13.1) with left

preconditioning and right preconditioning , and with an initial guess of the form

'¢ '

¥ ¥