Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 263

Fig. 56. Schwarz alternating iteration with one-dimensional subspaces V and
W in Euclidean 2-space. The iterates u 1 ,u 3 ,u 5 ,... lie in V ⊥ and u 2 ,u 4 ,... in
W ⊥ . The angle between V ⊥ and W ⊥ is the same as between V and W .

5.2 Convergence Theorem. Assume that there is a constant γ< 1 such that for
the inner product in H

| a(v, w) |≤ γ v w for v ∈ V, w ∈ W. ( 5 . 3 )

Then we have for the iteration with the Schwarz alternating method the error
reduction

− ≤

u k −

≥

u k + 1 u γ u for k 1 . ( 5 . 4 )

Proof. Because of the symmetry of the problem we may confine ourselves to even
k . Since u k is constructed by a minimization in the subspace W ,wehave

a(u k − u, w) = 0 for w ∈ W ( 5 . 5 )

We decompose u k − u = v + w with v ∈ V, w ∈ W . From (5.5) it follows with

w = w that

2 .

a(v, w) =− w ( 5 . 6 )

By the strengthened Cauchy inequality (5.3) we have a(v, w) =− α k v w
with some α k ≤ γ . Without loss of generality let α k = 0. It follows from (5.6)

v = α − 1

2 2 2 2 2

that w and u k − u = v + w = v − 2 w + w =

k

(α − 2

2 .
Since u k + 1 is the result of an optimization in V , we obtain an upper estimate
from the simple test function u k + (α k −

− 1 ) w

k

1 )v .

2

≤ u k + (α k −

2

u k + 1 − u 1 )v

= α k v + w

2

− α k ) w

2

= α k u k − u

2 .

= ( 1

Noting that α k ≤ γ , the proof is complete.
The bound in (5.4) is sharp. This becomes obvious from an example with
one-dimensional spaces V and W depicted in Fig. 56 and also from Problem 5.8.