¦¤ Y ™ Q$@
63
% #
£
1. Choose an initial guess to the solution
2. Until convergence Do:
3. For Do:
£ ¢ ’f$w
te D t
£ £
VTVT
T
B
4. Solve in with in ¥ )C )C
BC
D D
£
5. Update values on ) tC H
6. EndDo
7. EndDo
The algorithm sweeps through the subdomains and solves the original equation in each
£
of them by using boundary conditions that are updated from the most recent values of .
Since each of the subproblems is likely to be solved by some iterative method, we can take
advantage of a good initial guess. It is natural to take as initial guess for a given subproblem
the most recent approximation. Going back to the expression (13.11) of the local problems,
observe that each of the solutions in line 4 of the algorithm will be translated into an update
of the form
£ £
© tC
C tC
D
™ —t
7 u pU¢ 7 p z7 yw— y w u
{ {˜p p { p
˜
U&
“£§
¢
! hC §¤ !
¤ …S !
§
¥
¤ ¤
where the correction solves the system
© C
˜ h C D C © C
Y
˜
Here, is the local part of the most recent global residual vector , and the above
CY z #

system represents the system associated with the problem in line 4 of the algorithm when
˜
a nonzero initial guess is used in some iterative procedure. The matrix has the block Ch
structure (13.9). Writing
¥ ¥ ¥
C G ¡© C G ¢Y
£ C
©
DC DC D CY
t t t
GC © GC Y
¤ ¤
C ¦ ¦ ¦
the correction to the current solution step in the algorithm leads to
¥ ¥ ¥ ¥c
An" H¦ j
CG Y
v
a5
˜

C C C C
D h
t
CG Y ¤
£
C C C ¦C
¦ ¦ ¦
After this step is taken, normally a residual vector would have to be computed again to Y
get the components associated with domain and to proceed with a similar step for tw e
the next subdomain. However, only those residual components that have been affected by
the change of the solution need to be updated. Speci¬cally, employing the same notation
)G ¤ Y
used in equation (13.11), we can simply update the residual for each subdomain for H
which as
w )
Q
5 ) G ¤ YpD ) G ¤ Y h CG
¤ ©C)
This amounts implicitly to performing Step 5 of the above algorithm. Note that since the
matrix pattern is assumed to be symmetric, then the set of all indices such that , 0w
)
H Q
£
i.e., , is identical to . Now the loop starting in line 3 of Algorithm
w P IF
D H bC
a C
C
Q Q Q
13.2 and called domain sweep can be restated as follows.
™Y ™ # vx¤…¨©§¥¢
¡ ¦ ¦¤ §&%¥y¨ (0¥¤ (9y§# 5§&74 ' ¨5 '2 £¨ % ) ¡ ¥ D) 30
$ @ 4 2 0 $ ¦4 ¢
¦ ¤@ ) 42
6
¨
¤
1. For Do:
t VT§T t GD$w
e
T
©˜
2. Solve ¢YpDE¢ C8
C
C
D C CG CG
3. Compute , , and set
¤©
© ¢ D SY
t ’
t pD C 
C C C
)G Y C G ¤ C ) 5 ) G