H

£

¢

An# H¦ j

£ £ £ £

˜ ¥

˜

¢ ¢

¥ ¡¢ ¥¤ E

)

&¤ h

) &¤

)

D| t| h

E E i

¡ ¡

This gives rise to a system of the form (13.17) in which the part of the right-hand side p

incorporates the Neumann data related to the second integral on the right-hand side of

(13.18).

It is interesting to note that if a problem were to be solved with all-Dirichlet conditions,

i.e., if the Neumann conditions at the interfaces were replaced by Dirichlet conditions, the

resulting matrix problem would be of the form,

¥ ¥ ¥

)5 An¢ H¦ j

p p p| pz

˜ ¥

˜

D

¢ p ¦p

¦ ¦

where represents precisely the Dirichlet data. Indeed, according to what was seen in

p

Section 2.3, Dirichlet conditions are handled simply by replacing equations associated with

boundary points by identity equations.

i “w xim “ m– ¥ i w “D u

’‘

‚f¥ 3p£¥ @

¢£

&3¤ 7 8

3 6 7 "7

@

•

—

C

The original alternating procedure described by Schwarz in 1870 consisted of three parts:

alternating between two overlapping domains, solving the Dirichlet problem on one do-

main at each iteration, and taking boundary conditions based on the most recent solution

obtained from the other domain. This procedure is called the Multiplicative Schwarz pro-

cedure. In matrix terms, this is very reminiscent of the block Gauss-Seidel iteration with

overlap de¬ned with the help of projectors, as seen in Chapter 5. The analogue of the

block-Jacobi procedure is known as the Additive Schwarz procedure.

¦§ ¨ §¡

¤ § ¤ © £9@©

§ ©

'$&$"

# %# % G3

1 7C A

F A 5@ F C¥F

) I G

5 S

7 ) 0

1 $

)

In the following, assume that each pair of neighboring subdomains has a nonvoid overlap-

ping region. The boundary of subdomain that is included in subdomain is denoted by C H

B

)G C

.

t&™

˜ kw ‘ ¡ w— — p7w ¡w £ ˜

} p {7 | ¢

£¤§ §¤ X

“

! ¨

§

¡

B

¤G c

¤

B

¤G c

B

cG

¤

G c B B B G ¨ B

cG ¨G c

¨ ¨

c

™

P% # 3¡

!

An L-shaped domain subdivided into three over-

lapping subdomains.

This is illustrated in Figure 13.6 for the L-shaped domain example. Each subdomain ex-

B

tends beyond its initial boundary into neighboring subdomains. Call the boundary of C C

G C B B

)G C

consisting of its original boundary (which is denoted by ) and the ™s, and denote

£ £ B

by the restriction of the solution to the boundary . Then the Schwarz Alternating

) )

C C

Procedure can be described as follows.