Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 262

methods in this context; see below, In contrast to §3 the multilevel iteration is not
treated as a perturbation of the 2-grid procedure.
We want to present the basic ideas of the theory of Bramble, Pasciak, Wang,
and Xu. A complete theory without regularity is beyond the scope of this topic;
rather we demonstrate its advantage for another application which is not covered
by the standard theory. The extension to locally refined meshes will be illustrated.
For convenience, we restrict ourselves to symmetric smoothing operators
and to nested spaces. For more general results see Xu [1992], Wang [1994], and
Yserentant [1993]. All these theories do not reflect the improvement of the con-
vergence rate when the number of smoothing steps is increased. This was only
achieved via more involved considerations by Brenner [2000].
The norm without subscript refers to the energy norm )) 1 / 2 · : = (a( · , ·

since the theory applies only to this norm.

Schwarz' Alternating Method

For a better understanding of space decomposition methods we first consider the
alternating method which goes back to H.A. Schwarz [1869]. There is a simple
geometrical interpretation, cf. Fig. 56, when the abstract formulation for the case
of two subspaces is considered.
We are interested in the variational problem

a(u, v) = f, v for v ∈ H.

Here a(., .) is the inner product of the Hilbert space H and · is the corresponding

norm. Let H be the direct sum of two subspaces

H = V ⊕ W,

and the determination of a solution in the subspaces V or W is assumed to be easy.
Then an alternating iteration in the two subspaces is natural.

5.1 Schwarz Alternating Method. Let u 0 ∈ H .
When u 2 i is already determined, find v 2 i ∈ V such that

a(u 2 i + v 2 i ,v) = f, v for v ∈ V.

Set u 2 i + 1 = u 2 i + v 2 i .

When u 2 i + 1 is already determined, find w 2 i + 1 ∈ W such that

a(u 2 i + 1 + w 2 i + 1 ,w) = f, w for w ∈ W.

Set u 2 i + 2 = u 2 i + 1 + w 2 i + 1 .
Obviously, projections onto the two subspaces alternate during the iteration.
The strengthened Cauchy inequality (5.3) is crucial in the analysis.