Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 238

For most of our discussion we can get by with weaker hypotheses, but in that
case the proofs become more technical. For example, H 1 + α -regularity with α> 0
would suffice for this section; see Problem 2.12.
There is a different theory due to Bramble, Pasciak, Wang, and Xu [1991]
and Xu [1992]. Multigrid algorithms are connected with a decomposition of finite
element spaces. The space decomposition method has the advantage that it does
not require regularity assumptions. On the other hand, it does not model the fact
that the convergence rate is improved by increasing the number of smoothing steps
and it applies only to the energy norm. Since the arguments of the theory are far
away from the finite element theory in this topic, we restrict ourselves in §5 to the
motivation and the proof of the central tool.
We will also not discuss quantitative results on the convergence rate obtained
by Fourier methods [Brandt 1977, Ries, Trottenberg, and Winter 1983].

Discrete Norms

So far, the quality of the approximation of a function by finite elements has been
expressed in terms of higher Sobolev norms. This no longer works in dealing with
the approximation of a function v h ∈ S h by a function in the space S 2 h on the coarse
grid. In particular, if S h consists of C 0 elements, then in general S h ⊂ H 2 () , and
we cannot employ estimates in the H 2 norm.
This leads us to assign another Hilbert scale to the N h -dimensional space S h .
The new scale should be connected with the scale of the Sobolev spaces as closely
as possible. - It is easiest to discuss how to do this in the following abstract form.
For a symmetric positive definite matrix A , the powers A s are well defined
for all values of s , and not just for integers. Thus, we go back to the spectral
decomposition. The matrix A has a complete system of orthonormal eigenvectors
{ z i }

N
i = 1 :

Az i = λ i z i , i = 1 , 2 ,...,N,

(z i ,z j ) = δ ij , i, j = 1 , 2 ,...,N.

N

Every vector x ∈ R can be written in the form

N

x = c i z i , ( 2 . 3 )

i = 1

and

A s x = c i λ i z i ( 2 . 4 )

i

is well defined.