¬

& s s £¤( £ s £¤( s £¡ % R

¢¡ ¢$

¢¡

«

eDG

PW

PG

PG

and

¥

¬R (8& s s £¡ s s §¨( £ s £¡ £ s ¢©¨( s £¡ s § $¦

¢ ¦( ¢

¢

¦

¢

¢

« «

PWG

PeDG

W

PG

PG

PG

PG

(¤

¤¥

u¡

where each is the -th column of the identity matrix, and represents a

u s ¢¦

PG

weight factor chosen so that

¥

¬R R

r¤

R¦

When there is no overlap, i.e., when the ™s form a partition of the whole set ,

Q¡˜™(

© ((

§

y

¬ P uG s

then de¬ne . ©¦

¢

yR

§ ¤¥ ¤

vR

u u

Let be the matrix

u ¥

R

¬ uR § §

and de¬ne similarly the partitioned vectors

¥

R ¥

¬ RF ¬ R S (© ¨

–R T

X

R ¥ R T

«fª R R!

Note that is a projector from to the subspace spanned by the columns ,

X y

¤R R

..., . In addition, we have the relation

!

f

¬© R FR

£R

W

¥ R ¥ R

R

¤ © ©©

R

The -dimensional vector represents the projection of with respect to

R R

the basis spanned by the columns of . The action of performs the reverse operation.

bR b R

That means is an extension operation from a vector in (represented in the basis

R ¥

bR «ª

R

consisting of the columns of ) into a vector in . The operator is termed a

R

restriction operator and is an prolongation operator.

Each component of the Jacobi iteration can be obtained by imposing the condition that

R¦

the projection of the residual in the span of be zero, i.e.,

f

¥ ¥ R ‚

R ¥ u

u

y¨ R

§² © © ¬

!„

w " ¡Y

"UQH

W H

£u

R

¥

©u ¬

uF

Remember that , which can be rewritten as T

a±!w°i ¯®

X

¥

R

¬ § V§ A¨

©²

w QIGp` rW"VIGpF

P UHR

PHRF W I“ R

R

H

This leads to the following algorithm:

• ’A

‘ A

“

˜h¤ v ¢

¡¦ PD £7V95UT RPH77 9FD B CA9745( 4CDB9

B @8 5 6

GE

6 Q S @8 S Q I D E @ 3A

0 (%&$

'#

) 2

1

¬

1. For until convergence Do:

V( ¡Y (

( W

T

y

v

¬

2. For Do:

E

¡ (Q˜ (