G

a §¨“ ` d f

f

3.

˜ §` ¡ w

a ¨¨ f

4.

f

5. EndDo

The algorithm requires the solution of the linear system

uiµ ‘ x¤£

¦¥

U

‚¢

¨ §

f

at each iteration. By substituting the result of line 3 into line 4, the iterates can be

¶ 8˜ ’ ”p20("'&}%#”8³ª%¡"p!¡©’ ”W

8¥1 )© ¥ ’$ ©˜ ’8 © 8¥ ˜ 8©

¡¡

¢ ¡¨ ¨¦ ¤

© §¥ &

$

eliminated to obtain the following relation for the ™s,

ªa § ¨‚ ` d ˜ § 5

w

¨

¡

¨ 7

f

f

which is nothing but a Richardson iteration for solving the linear system

˜§

˜ GGµ ‘ x¤£

¦

R¨ ¨‚ 'd

¬f

§ § fd

Apart from a sign, this system is the reduced system resulting from eliminating the vari-

able from (8.30). Convergence results can be derived from the analysis of the Richardson

iteration.

˜ 5§ w¢‚¢ ¢ ¦ ¦ ¦

¦

¡

Let be a Symmetric Positive De¬nite matrix and a matrix of §

§ d ©

full rank. Then is also Symmetric Positive De¬nite and Uzawa™s algorithm

§

f ¢

converges, if and only if

H

GGµ ‘ x¤£

¦µ

E

¬ ¨ ` e&h 8 ²‘¡ ²

a pi ¢£

In addition, the optimal convergence parameter is given by

¡ H

¬ ¨ ` e&h 8

¡ s8

§¥¢

¦¤ w

"¢£` S h a pi

a ¤

¢

©¨¦£

§ §¥

The proof of this result is straightforward and is based on the results seen in

Example 4.1.

E

P¨

It is interesting to observe that when and is Symmetric Positive De¬nite, then

the system (8.32) can be regarded as the normal equations for minimizing the -norm

d

f

@

¨

of . Indeed, the optimality conditions are equivalent to the orthogonality conditions

§ E

%

@ a ˜ § % “`

§ ¨

%

fd ˜ §

which translate into the linear system . As a consequence, the prob- § § fd

lem will tend to be easier to solve if the columns of are almost orthogonal with respect §

to the inner product. This is true when solving the Stokes problem where represents §

d

f ˜§

the discretization of the gradient operator while discretizes the divergence operator,

and is the discretization of a Laplacian. In this case, if it were not for the boundary

˜

conditions, the matrix would be the identity. This feature can be exploited in de-

§ § 'd

f

veloping preconditioners for solving problems of the form (8.30). Another particular case

E

¨

is when is the identity matrix and . Then, the linear system (8.32) becomes the sys-

™

¨

tem of the normal equations for minimizing the 2-norm of . These relations provide §

insight in understanding that the block form (8.30) is actually a form of normal equations

3

for solving in the least-squares sense. However, a different inner product is used.

§

In Uzawa™s method, a linear system at each step must be solved, namely, the system

(8.31). Solving this system is equivalent to ¬nding the minimum of the quadratic function

6 µ ‘ x¤£

¦

o ¦ “f % x` ”a % `

¨ ¨ qa

¬

minimize §

GH

¦`

a

x ¦

Apart from constants, is the Lagrangian evaluated at the previous iterate. The

a`

solution of (8.31), or the equivalent optimization problem (8.34), is expensive. A common