3. Compute 85£

D

£

4. Solve 8© 9 D

5. Compute .

¥

8A D | 8 5

In a practical implementation, all the matrices are factored and then the systems C

and are solved. In general, many columns in will be zero. These

¥ ¥

5 D 8C 5 C D 8C C 5

C C C

zero columns correspond to interfaces that are not adjacent to subdomain . Therefore, w

any ef¬cient code based on the above algorithm should start by identifying the nonzero

columns.

© ¤© ¨¤ ¨ ¡I

§ §© ¤

! #$! $"

#% 7 9 7 ) )¥

F I 9 )0 4

1 9 3 7 A ) 3 ) (

Now the connections between the Schur complement and standard Gaussian elimination

will be explored and a few simple properties will be established. Start with the block-LU

˜

factorization of , )

¥ ¥ ¥

An ¦ j

)¢

a5 5

˜

D c £

¢

£ £

¦ ¦ ¦

v

which is readily veri¬ed. The Schur complement can therefore be regarded as the (2,2)

˜˜

block in the part of the block-LU factorization of . From the above relation, note that

˜ £

if is nonsingular, then so is . Taking the inverse of with the help of the above equality

yields )

¥ ¥ ¥

c

)¢

c

5c v c

£

v

a5 v v

D c c

£

¢

£ £

¦ ¦ ¦

v v

¥

An! ¦ j

c c c c c c

£ £

£

5v 5v

˜

v v v v

t

D h

c c c

£ £

£

¦

v v v

™

p w}g3y w

{7 | p˜ 7 { f p uq£ … ¢ 7

}¢ ˜

7| &§

¥£¤§

¡ !

§ #

"

! §¤ "

…¤"’ hC §

¤

˜˜

c

£

Observe that is the (2,2) block in the block-inverse of . In particular, if the original

v

˜ c £