v v

) c sC )

c c sC

c

Adding these two inequalities, squaring the result, and using (13.41) leads to the inequality

(13.40).

From (13.38), it can be deduced that if Assumption 2 holds, then,

An 1 H¦ j

‚

e

¨ ¨ ¨

˜ u

W

' ' U ' ' C 2

¨}c „

h

t ex

W

c sC

¨ $2

Assumption 1 can now be exploited to derive a lower bound on . This will

'C c sWC

$

yield the following theorem.

% ™ # m…¨ …v¤

¥¦ ¥¡ Assume that Assumptions 1 and 2 hold. Then,

¨ ¤c

£

An¦ 1 H¦ j

e

˜

'

' U „ e

¨}c „ h

t ex

W

£ %

£

' }¤ t C 2 x D ¨' C 2

Using the notation of Section 13.3.3, the relation yields

‚ ‚

¨

˜

W W ¤

}

¡x

h' t

' $2 $2

C D C t D

¡

c sC

c sC

'

„ ' 4 t x

˜ ˜

} ' 4 t

} }

c

According to Theorem 13.4, , which implies .

¡x C ¢

) x

¤

¨

Thus,

' 4 t x

}

‚ ¨

W

' C 2 „

t

c sC

which upon substitution into (13.42) gives the inequality

¨

' e

'

„ e U

¨}c „ h

¨ W

t ex

'

The result follows by taking the maximum over all vectors .

This result provides information on the speed of convergence of the multiplicative

Schwarz procedure by making two key assumptions. These assumptions are not veri¬able

from linear algebra arguments alone. In other words, given a linear system, it is unlikely

that one can establish that these assumptions are satis¬ed. However, they are satis¬ed for

equations originating from ¬nite element discretization of elliptic Partial Differential Equa-

tions. For details, refer to Drya and Widlund [72, 73, 74] and Xu [230].

˜ £ p‘ 7f p uq£ ˜

| }¢ ¡

¢

£¤§ §¤ “ W …"’ h¤C §

¤

" $

§

`x’ z

“ ’ “

“b m– “m

@ E86

7& 7 6 Q3

6 3 "7

@

¥

£ X

¨ 4

¨ ¥

£

—˜ •

C

Schur complement methods are based on solving the reduced system (13.4) by some pre-

conditioned Krylov subspace method. Procedures of this type involve three steps.

¥c

1. Get the right-hand side . £

… ©

D8

v

£