preconditioning can be combined with a subsidiary relaxation-type preconditioning such as SSOR

[2, 153]. Finally, polynomial preconditionings can be useful in some special situations such as that

of complex linear systems arising from the Helmholtz equation [93].

Multicoloring has been known for a long time in the numerical analysis literature and was used

in particular for understanding the theory of relaxation techniques [232, 213] as well as for deriving

ef¬cient alternative formulations of some relaxation algorithms [213, 110]. More recently, it became

an essential ingredient in parallelizing iterative algorithms, see for example [4, 2, 82, 155, 154, 164].

It is also commonly used in a slightly different form ” coloring elements as opposed to nodes ”

in ¬nite elements techniques [23, 217]. In [182] and [69], it was observed that -step SOR pre-

3

conditioning was very competitive relative to the standard ILU preconditioners. Combined with

multicolor ordering, multiple-step SOR can perform quite well on supercomputers. Multicoloring

is especially useful in Element-By-Element techniques when forming the residual, i.e., when multi-

plying an unassembled matrix by a vector [123, 88, 194]. The contributions of the elements of the

same color can all be evaluated and applied simultaneously to the resulting vector. In addition to the

parallelization aspects, reduced systems can sometimes be much better conditioned than the original

system, see [83].

Independent set orderings have been used mainly in the context of parallel direct solution tech-

niques for sparse matrices [66, 144, 145] and multifrontal techniques [77] can be viewed as a par-

ticular case. The gist of all these techniques is that it is possible to reorder the system in groups of

equations which can be solved simultaneously. A parallel direct solution sparse solver based on per-

forming several successive levels of independent set orderings and reduction was suggested in [144]

and in a more general form in [65].

—§™

Equation.

sic concepts, as well as the terminology, are introduced from a model Partial Differential

this chapter considers these techniques from a purely linear algebra view-point, the ba-

neering, a discipline which is not dominated by Partial Differential Equations. Although

the earliest practical uses for domain decomposition approaches was in structural engi-

principles have been exploited in other contexts of science and engineering. In fact, one of

ing Partial Differential Equations over regions in two or three dimensions. However, similar

the principle of divide-and-conquer. Such methods have been primarily developed for solv-

Domain decomposition methods refer to a collection of techniques which revolve around

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An L-shaped domain subdivided into three sub-

domains.

Consider the problem of solving the Laplace Equation on an L-shaped domain parti-

tioned as shown in Figure 13.1. Domain decomposition or substructuring methods attempt

to solve the problem on the entire domain

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from problem solutions on the subdomains . There are several reasons why such tech- C

niques can be advantageous. In the case of the above picture, one obvious reason is that the

subproblems are much simpler because of their rectangular geometry. For example, fast

solvers can be used on each subdomain in this case. A second reason is that the physical

problem can sometimes be split naturally into a small number of subregions where the

modeling equations are different (e.g., Euler™s equations on one region and Navier-Stokes

in another). Substructuring can also be used to develop “out-of-core” solution techniques.

As already mentioned, such techniques were often used in the past to analyze very large

mechanical structures. The original structure is partitioned into pieces, each of which

is small enough to ¬t into memory. Then a form of block-Gaussian elimination is used

to solve the global linear system from a sequence of solutions using subsystems. More

recent interest in domain decomposition techniques has been motivated by parallel pro-

cessing.

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