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218 5. Iterative Methods for Systems of Linear Equations

and from (5.50) we can conclude immediately that
Ae (k) = g (k) , e (k+1) = e (k) + α k d (k) , (5.52)
g (k+1) = g (k) + α k Ad (k) . (5.53)
We consider the energy norm
T 1/2
x A := x Ax (5.54)
induced by the energy scalar product. For a finite element stiffness matrix
A with a bilinear form a we have the correspondence
x A = a(u, u) 1/2 = u a
m
for u = x i ϕ i if the ϕ i are the underlying basis functions. Comparing
i=1
the solution x = A −1 b with an arbitrary y ∈ R m leads to
1 2
f(y)= f(x)+ y − x , (5.55)
A
2
so that condition (5.44) also minimizes the distance to x in · A .The
energy norm will therefore have a special importance. Measured in the
energy norm we have, due to (5.52),
2 (k) T (k) (k) T −1 (k)
e
(k) = e g = g A g ,
A
and therefore due to (5.52) and (5.51),
2 (k+1) T (k)
e = g e .
(k+1)
A
(k) (k)
The vector −∇f x in x points in the direction of the locally steepest
descent, which motivates the gradient method, i.e.,
d (k) := −g (k) , (5.56)
and thus
T
d (k) d (k)
α k = T . (5.57)
d (k) Ad (k)
The above identities imply for the gradient method

T

2 (k) (k) T (k) (k) 2 d (k) d (k)
e
(k+1) = g + α k Ad e = e
A 1 − α k T
d (k) A −1 (k)
d
and thus by means of the definition of α k from (5.57)
 
T 2
 (k) (k) 
 d d 
2 2
(k)
(k+1) − x = x − x 1 − .
x
A A (k) T (k) (k) T
d
 d Ad d A −1 (k) 
 
With the inequality of Kantorovich (see, for example, [28, p. 132]),
T
T
x Ax x A −1 x 1 1/2 1 −1/2 2
≤ κ + κ ,
2
T
(x x) 2 2