16 15 13 32 26 27 28

14

7 9

8 12 31 23 24 25

5 6 30 20

4 11 21 22

29 19

1 2 3 10 17 18

&™ $"!

%#

Discretization of problem shown in Figure 13.1

and associated matrix.

Consider the Schur complement system obtained with this new labeling. It can be

written similar to the edge-based case using a reordering in which all interface variables

are listed last. The matrix associated with the domain partitioning of the variables will have

™

p w}g3y w

{7 | p˜ 7 { f p uq£ … ¢ 7

}¢ ˜

7| #&

¥£¤§

¡ !

§ #

"

! §¤ "

…¤"’ hC §

¤

a natural -block structure where is the number of subdomains. For example, when D8

(as is the case in the above illustration), the matrix has the block structure de¬ned by the

solid lines in the ¬gure, i.e.,

£ ¦

˜˜ ¨ ©˜ ˜ ©˜ ˜ ¤c un# ¦ j

c c

˜

˜ ¥ © ¤© ¨ ¨

¨

D h

c

¤˜

¤ ˜ ¨¤ ˜

c

In each subdomain, the variables are of the form

¥

C|

DC t

C ¦

where denotes interior nodes while denotes the interface nodes associated with sub- C

C|

˜ ˜

domain . Each matrix will be called the local matrix. The structure of is as follows:

C C

w

¥

un¢ ¦ j

#5

˜

C C

˜ a

$C

D

$£

C C ¦

in which, as before, represents the matrix associated with the internal nodes of subdo-

C

a

main and and represent the couplings to/from external nodes. The matrix is the

$£

5

w C C C

a

local part of the interface matrix de¬ned before, and represents the coupling between

local interface points. A careful look at the matrix in Figure 13.5 reveals an additional

˜

structure for the blocks . Each of these blocks contains a zero sub-block in the

¢D H ) C w

part that acts on the variable . This is expected since and are not coupled. There- ¡|

) &|

)

C|

fore,

¥

un ¦ j

¢ ˜ ¥

˜

˜

)8

C D h

¦ )C 5

In addition, most of the matrices are zero since only those indices of the subdomains

)C 5 H

that have couplings with subdomain will yield a nonzero . )C 5

w

Now write the part of the linear system that is local to subdomain , as w

sn ¦ j

¥

a5 C C

C |C C

D ˜ ¢¥

˜˜

t

h

) ) C 5 EG $ )

£ C C C

C |C D

t t d

$

The term is the contribution to the equation from the neighboring subdomain number

) )C 5

, and is the set of subdomains that are adjacent to subdomain . Assuming that is

H C w C

Q

nonsingular, the variable can be eliminated from this system by extracting from the ¬rst

“|

C