can all be computed in parallel and do not depend on one another. The operations

¢¢ 2

t

D

can be processed in parallel for any group of elements that do not share any vertices. This

grouping can be found by performing a multicoloring of the elements. Any two elements

which have a node in common receive a different color. Using this idea, good performance

can be achieved on vector computers.

™

˜ ` q £ — ¢ wp

}{ ¢ 7 7

£¤¢

¡ "

"

§ $

¡

¢

EBE preconditioners are based on similar principles and many different variants have

been developed. They are de¬ned by ¬rst normalizing each of the element matrices. In the

˜

sequel, assume that is a Symmetric Positive De¬nite matrix. Typically, a diagonal, or

˜ ˜

block diagonal, scaling is ¬rst applied to to obtain a scaled matrix ,

2n1 j

˜

˜¨ ¨ ¤c

˜ c

¤

£ £

D h

v„ v„

W ¢R

˜ W ¢R ˜

This results in each matrix and element matrix being transformed similarly:

©¤

¦

W ¢R

˜ ¨¨ ¨ ¤c

˜ c £ £

D v„ v„

¨£c

P˜

£c

v „ ©¨ ¢ 2 v„D

¦¤

W ¢ R } ¨ ¤c

¨ £¤c

˜

} £

¢ 2 v „ '2 x ¢ 2 D

v „ ¢ 2x

w'2 w¢

¢

¤

˜ h w'2 ¦ ¤ ¢ 2

¢

The second step in de¬ning an EBE preconditioner is to regularize each of these trans- W ¢R

˜

formed matrices. Indeed, each of the matrices is of rank at most, where is the ¢ ¢

y y

˜

size of the element matrix , i.e., the number of nodes which constitute the -th ele- ˜

¤ ¦

W ¢R

˜

ment. In the so-called Winget regularization, the diagonal of each is forced to be the

identity matrix. In other words, the regularized matrix is de¬ned as )

un4 j

W ¢R ¡ W ¢R } W ¢R

¢

˜

˜ ˜ ˜

x ¤ A

£

D h

t

These matrices are positive de¬nite; see Exercise 8.

The third and ¬nal step in de¬ning an EBE preconditioner is to choose the factorization

itself. In the EBE Cholesky factorization, the Cholesky (or Crout) factorization of each

W ¢R ¡

˜

regularized matrix is performed,

un j

W ¢R ¡

˜

˜ ¢ „¢ ¢ h ¢A¢

D w

The preconditioner from it is de¬ned as

¢ ™¢

™ c

un! j

˜

¢ ¢

¢¢ ¢ ¢ h ¢w

D „

c b¢ c ¢ ¢ ¢

™

Note that to ensure symmetry, the last product is in reverse order of the ¬rst one. The fac- ¡

˜

torization (12.26) consists of a factorization of the small matrix . Performing ¢

¢ ¢

y Q

y ¤ ¦

the preconditioning operations will therefore consist of a sequence of small back- ¢

¢ ¢

y 9

y

ward or forward solves. The gather and scatter matrices de¬ned in Chapter 2 must also ¢2

be applied for each element. These solves are applied to the right-hand side in sequence. In

addition, the same multicoloring idea as for the matrix-by-vector product can be exploited

to perform these sweeps in parallel.

One of the drawbacks of the EBE Cholesky preconditioner is that an additional set of

element matrices must be stored. That is because the factorizations (12.26) must be stored W ¢R ¡

˜

for each element. In EBE/SSOR, this is avoided. Instead of factoring each , the usual

W ¢R ¡

˜

splitting of each is exploited. Assuming the Winget regularization, we have )

un# j

W ¢R ¡

˜

˜ ¢5 w…5

¢

D

W ¢R ¡

˜

in which is the strict-lower part of . By analogy with the SSOR preconditioner,

¢ 5£

™ —t

7 —— p w7 z p — |w z

˜ { { | |

§

“£§

¢ ¢

¡