The interface values can be obtained by employing a form of block-Gaussian elimination

which may be too expensive for large problems. In some simple cases, using FFT™s, it is

possible to explicitly obtain the solution of the problem on the interfaces inexpensively.

Other methods alternate between the subdomains, solving a new problem each time,

with boundary conditions updated from the most recent subdomain solutions. These meth-

ods are called Schwarz Alternating Procedures, after the Swiss mathematician who used

the idea to prove the existence for a solution of the Dirichlet problem on irregular regions.

The subdomains may be allowed to overlap. This means that the ™s are such that C

¡

¡C

E¥£)

h ¤ ¢D

D eC

t

G c sC

W

For a discretized problem, it is typical to quantify the extent of overlapping by the number

of mesh-lines that are common to the two subdomains. In the particular case of Figure

13.3, the overlap is of order one.

™ —t

7 u pU¢ 7 p z7 yw— y w u

{ {˜p p { p

˜

U§

“£§

¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

30 31 32 33

26 27 28 29

22 23 24 25

40 39 38 37 19 20 21

7 8 9 36 16 17 18

4 5 6 35 13 14 15

3 34 10

1 2 11 12

™Y&™ $"!

%#

Discretization of problem shown in Figure 13.1.

™

& $"

%#!

Matrix associated with the ¬nite difference mesh

of Figure 13.3.

The various domain decomposition techniques are distinguished by four features:

¢

¡

Type of Partitioning. For example, should partitioning occur along edges, or along

™

p w}g3y w

{7 | p˜ 7 { f p uq£ … ¢ 7

}¢ ˜

7| §

¥£¤§

¡ !

§ #

"

! §¤ "

…¤"’ hC §

¤

vertices, or by elements? Is the union of the subdomains equal to the original do-

main or a superset of it (¬ctitious domain methods)?

¡

Overlap. Should sub-domains overlap or not, and by how much?

¡

Processing of interface values. For example, is the Schur complement approach

¡

used? Should there be successive updates to the interface values?

¡

Subdomain solution. Should the subdomain problems be solved exactly or approx-

¢

imately by an iterative method?

The methods to be discussed in this chapter will be classi¬ed in four distinct groups. First,

direct methods and the substructuring approach are useful for introducing some de¬nitions

and for providing practical insight. Second, among the simplest and oldest techniques are

the Schwarz Alternating Procedures. Then, there are methods based on preconditioning

the Schur complement system. The last category groups all the methods based on solving

˜

the linear system with the matrix , by using a preconditioning derived from Domain

Decomposition concepts.

…w hx gu 9 mz ˜‘

‘ ’ “ `“ ’’ w

’“ –m

7 5@

& 3 p9¤

@7£ E&86

7 7

¥

£ X

¨ 4

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