Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 270

This proves A1 with a constant K 1 = c that is independent of the number of levels.

In the cases with less regularity we perform the decomposition by applying
the L 2 -orthogonal projectors Q instead of P

L

v = v ,

( 5 . 36 )

= 0

v 0 = Q 0 v, v = Q v − Q − 1 v for = 1 , 2 ,...,L.

From Lemma II.7.9 we have Q 0 v ≤ c v . Next from (II.7.15) it follows that
v 0 ≤ v − Q v 0 + v − Q − 1 v 0 ≤ ch v . Recalling the approximate
solvers from (5.34) we proceed as in the derivation of (5.35)

L L

h − 2 v

2
1

2
0

(B v ,v ) ≤ v 0

+ c

( 5 . 37 )

= 0 = 1

2 .

≤ c(L + 1 ) v

1 / 2 . Although this
result is only suboptimal, it has the advantage that no regularity assumptions are
required. As mentioned above, the logarithmic factor arises since we stay in the
framework of Sobolev spaces. An analysis with the theory of Besov spaces shows
that the factor can be dropped, see Oswald [1994].

1 ) 1 / 2

This proves A1 with a constant K 1 ≤ c(L + ≤ c | log h L |

Local Mesh Refinements

An inspection of the proof of Lemma II.7.9 shows that the estimate (5.37) remains
true if the orthogonal projector Q is replaced by an operator of Clement type,
e.g., we may choose I = I h

from (II.6.19). That interpolation operator is nearly

local.
This has a big advantage when we consider finite element spaces which arise
from local mesh refinements. Assume that the refinement of the triangulation on
the level is restricted to a subdomain ⊂ and that

L ⊂ L − 1 ⊂ ... ⊂ 0 = . ( 5 . 38 )

Given v ∈ S L , its restriction to \ coincides there with some finite element
function in S . Now we modify I v at the nodes outside and set

= v(x j )

if x j ∈ . (I v)(x j ) :

Specifically, when defining I , the construction of the operator Q j in (II.6.17) is
augmented by the rule (II.6.23). We have

(I v)(x) = v(x) ( 5 . 39 )