c }

equation which yields, upon substitution in the second equation,

¥

C C#5 x v C

DEC | C

un ¦ j

‚ ˜ I¥

˜

¥c

£

E£

) C#5 )

t ’ C ’

C D C C t

C t2i¢i’f"w

hhhte D

vC

d$G $ )

£

in which is the “local” Schur complement

C

unH¦ ¦ j

a ˜ ¥

˜

c

£

$£ h#5

EC

D C C C

vC

When written for all subdomains , the equations (13.12) yield a system of equations which w

involves only the interface points , and which has a natural block structure d ’fD H )

te ii¢t

thhh

—&™

—t

7 u pU¢ 7 p z7 yw— y w u

{ {˜p p { p

˜

“£§

¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

associated with these vector variables

£¤ §¦

£

¤ §

¨ 85 ¤ 85 85

c c c c

V§T

TT

¤ §

£

W

¨ 5 ¤¤ §

¨ ¤¨ 5 ¨5

c V§T

TT

§ unH1 H¦ j

. .

.. W

. .

˜ ¥

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.

£

. .

D h

. .

..

. .

.

¥ ©

. .

£

5 5 5

¨ ¤

c V§T

TT

W W W W

£

The diagonal blocks in this system, namely, the matrices , are dense in general, but the C

offdiagonal blocks are sparse and most of them are zero. Speci¬cally, only if

)C 5 ¢ ¢D ) C 5

subdomains and have at least one equation that couples them.

w H

£

A structure of the global Schur complement has been unraveled which has the fol-

lowing important implication: For vertex-based partitionings, the Schur complement ma- £

trix can be assembled from local Schur complement matrices (the ™s) and interface-to- C

interface information (the ™s). The term “assembled” was used on purpose because a

) 85

C

similar idea will be exploited for ¬nite element partitionings.

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