CQ H ©

YD

5. EndDo

Considering only the iterates, the above iteration would resemble a form of Gauss-Seidel

procedure on the Schur complement matrix (13.14). In fact, it is mathematically equivalent,

provided a consistent initial guess is taken. This is stated in the next result established by

Chan and Goovaerts [48]:

¥

™ # m…¨ …v¤

¥¦ ¥¡ #% ©§

¨ ¦d

Let the guess for the Schwarz procedure in each subdomain

¤¢

¨ ¦d ¦

be chosen such that

Pn H¦ j

r o r o

˜ ’

˜

c

5 C ¥ – vC C C

C| D ml

h

Then the iterates produced by the Algorithm 13.3 are identical to those of a Gauss-Seidel

sweep applied to the Schur complement system (13.12).

™

˜ kw ‘ ¡ w— — p7w ¡w £ ˜

} p {7 | ¢ &

£¤§ §¤ X

“

! ¨

§

¡

£ %

¤¢¡

£

We start by showing that with the choice (13.21), the components of the initial

residuals produced by the algorithm are identical to those of the Schur complement system

(13.12). Refer to Section 13.2.3 and the relation (13.10) which de¬nes the ™s from )C 5

£ )C

˜

the block structure (13.8) of the global matrix. Observe that and note Dx) ) )C 5

from (13.11) that for the global system the components of the initial residual vectors are

r o r o r o r o

‚

a

D CG Y £

¤ ) )C 5

C C C

C |C

$) $G

d

r o r o r o

‚

a

5 C ¥– c v C £ C D ) )C 5

C C C C

l

G$) d

r o r o

‚

¥c £

£ ) )C 5

C C C

C C

D h

v

G$) d

This is precisely the expression of the residual vector associated with the Schur comple-

r o

ment system (13.12) with the initial guess . C

r G o

Now observe that the initial guess has been selected so that for all . Because ¢£D C Y w

only the components of the residual vector are modi¬ed, according to line 4 of Algorithm

13.3, this property remains valid throughout the iterative process. By the updating equation

(13.20) and the relation (13.7), we have

¤Yc eC G

£

D C t ’

C t

vC

which is precisely a Gauss-Seidel step associated with the system (13.14). Note that the

update of the residual vector in the algorithm results in the same update for the compo-

nents as in the Gauss-Seidel iteration for (13.14).

It is interesting to interpret Algorithm 13.2, or rather its discrete version, in terms of

projectors. For this we follow the model of the overlapping block-Jacobi technique seen in

the previous chapter. Let be an index set CB

t I H ii¢it

¨ H uIF D C B

tcH ad thhh

where the indices are those associated with the mesh points of the interior of the

p H Ci

discrete subdomain . Note that as before, the ™s form a collection of index sets such

C CB

that

¡

$CB

D t "e2i¢i’F

a ithhhte

G ¦¦¦G c sC

¥¥¥

W

and the ™s are not necessarily disjoint. Let be a restriction operator from to . C'#

CEB C

By de¬nition, belongs to and keeps only those components of an arbitrary vector

d)#

|C C

that are in . It is represented by an matrix of zeros and ones. The matrices ¢

| C i

C

i

associated with the partitioning of Figure 13.4 are represented in the three diagrams of

# C

Figure 13.7, where each square represents a nonzero element (equal to one) and every other

element is a zero. These matrices depend on the ordering chosen for the local problem.

Here, boundary nodes are labeled last, for simplicity. Observe that each row of each has # C

exactly one nonzero element (equal to one). Boundary points such as the nodes 36 and 37

are represented several times in the matrices , and because of the overlapping ¨ # c…# ¤#

t