£

is satis¬ed by the representation of as the sum

£ £ £

‚ W

C

D bC

t eC

h

c sC

The following theorem has been proved by several authors in slightly different forms and

contexts.

™ # m…¨ …v¤

¥¦ ¥¡ %

If Assumption 1 holds, then

e

˜ }

¢ x C

) „ h

£ %

£

Unless otherwise stated, all summations in this proof are from to . Start with

e

£ £ £

an arbitrary decomposed as and write D C

$£

££ £x £tC £C £x £C

‚ ‚ ‚

'} t x } } }

' ' '

2x 2tC

tC

D D D h

˜

The last equality is due to the fact that is an -orthogonal projector onto and it is

V2

C C

therefore self-adjoint. Now, using Cauchy-Schwarz inequality, we get

¨ ¤c ¨ ¤c

££ ££ ££ ££

‚ ‚ ‚

£ £

'} t x '} C 2tC x e '}C tC x e '} C 2t C 2x

b b

D U h

By Assumption 1, this leads to

¨ ¤c

¨£ £ e '} £C 2t £C 2x

‚ £

¨ ¤c „

£

b '

U ' t

which, after squaring, yields

„ U ¨' £ £

}£

‚

h ' C 2t C 2x

˜

Finally, observe that since each is an -orthogonal projector, we have

V2

C

£C £C £t £C ££

‚ ‚ ‚

} } e

' ' b

2x 2t 2x ' t C2

D D h

£

Therefore, for any , the inequality

£ £ ££ e

˜ '} t '} t x „

x

holds, which yields the desired upper bound by the min-max theorem.

—

˜ kw ‘ ¡ w— — p7w ¡w £ ˜

} p {7 | ¢

£¤§ §¤ X

“

! ¨

§

¡

Note that the proof uses the following form of the Cauchy-Schwarz inequality:

¨ ¤c ¨ ¤c

£ £

g g g

‚ ‚ ‚

} } }

C tC |x C tC x

C et C | x

U | h

¡ ¡

c sC

c sC

c sC

See Exercise 1 for a proof of this variation.

We now turn to the analysis of the Multiplicative Schwarz procedure. We start by

recalling that the error after each outer iteration (sweep) is given by

£ £

'D h

W

We wish to ¬nd an upper bound for . First note that (13.31) in Lemma 13.1 results

'

'

in

W

' C 2 ¨c C ' D C ' C

uc

t

v v

˜

from which we get, using the -orthogonality of ,

2 C

¨

¨ ¨

' c vC) 'C 2 ' c vC ' D ' C ' h

¤

The above equality is valid for , provided . Summing these equalities from

' e 8w

D

to gives the result,

eG$w

D

un# ¦ ¦ j

‚