Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 267

Next we derive an equality that is useful also in other contexts

(B T w, T w) = (T w, B B − 1 A P w)
= (T w, A P w) = a(T w, P w) = a(T w, w).

( 5 . 23 )

The first factor on the right-hand side of (5.22) can be estimated by A1 . Since
T = B − 1

A P , we can insert (5.23) into the summands of the second factor, and
the proof of the lemma is complete.
It is more than a coincidence that a(T w, w) is a multiple of the discrete
norm ||| P w ||| 2 that we encountered in §2 if B is a multiple of the identity on the

subspace S .

Convergence of Multiplicative Methods

First we estimate the reduction of the error by the multigrid algorithm on the level
from below.

5.4 Lemma. Let ≥ 1 . Then

2 2

v − (I − T )v ≥ a(T v, v).

( 5 . 24 )

Proof. From the binomial formula we obtain that the left-hand side of (5.24) equals

2 a(T v, v) − a(T v, T v). ( 5 . 25 )

Next, we consider the second term using A ≤ B and (5.23)

a(T v, T v) ≤ (B T v, T v) = a(T v, v).

Therefore the negative term in (5.25) can be absorbed by the term a(T v, v) by
subtracting 1 from the factor 2. and the proof is complete.
Now we turn to the central result of this §. It yields the convergence rate of
the multigrid iteration in terms of the constants in the assumptions A1 and A2 .

5.5 Theorem. Assume that A1 and A2 hold. Then the energy norm of the error
propagation operator E of the multigrid iteration satisfies

1

2

E ≤ 1 −

+ K 2 ) 2 . K 1 ( 1

Proof. By applying Lemma 5.4 to E − 1 v and noting that E = (I − T )E − 1 we
obtain

2 2

E − 1 v − E v ≥ a(T E − 1 v, E − 1 v).