Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 85

performed in two steps. Given a nodal point x j , let Q j : L 2 (ω j ) → P 0 be the L 2 -
projection onto the constant functions. It follows from the Bramble-Hilbert lemma
that

v − Q j v ≤ ch j | v | 1 ,ω j , ( 6 . 16 ) 0 ,ω j

where h j is the diameter of ω j . In order to cope with homogeneous Dirichlet bound-
ary conditions on D ⊂ ∂ we modify the operator and set

0

if x j ∈ D ,

Q j v =

( 6 . 17 )

Q j v otherwise.

Here we get an analogous estimate to (6.16) by adapting the technique of the proof
of Friedrich's inequality

v − Q j v 0 ,ω j = v 0 ,ω j ≤ ch j | v | 1 ,ω j if x j ∈ D . ( 6 . 18 )

Next we define

( Q j v)v j ∈ M

1

I h v : =

0 .

( 6 . 19 )

j

The shape functions v j constitute a partition of unity. Specifically, for each x , I h v
contains at most three nonzero terms. For each relevant term, v − Q j v can be
estimated by (6.16) or (6.18). resp.

(v − Q j v)v j 0 ,T ≤ (v − Q j v) 0 ,ω j ≤ v − I h v 0 ,T ≤ 3 ch T v 1 , ω T .

j j

0. For the H 1 -stability we refer to Corollary 7.8.
The construction is easily modified to get an analogous mapping from H 0 ()

This proves (6.15a) for m =

1
0

H 0 () .If x j ∈

to M ∩ ∂ , then P j v may be set to zero and (6.16) follows from

Friedrichs' inequality.

Appendix: On the Optimality of the Estimates

6.10 Remark. The inverse estimates show that the above approximation theorems
are optimal (up to a constant). The assertions have the following structure:
Suppose the complete normed linear space X is compactly imbedded in Y .
Then there exists a family (S h ) of subspaces of X satisfying the approximation
property

S h u − v h Y ≤ ch α

inf
v h ∈

u X for all u ∈ X, ( 6 . 20 )

and (with β = α ) the inverse estimate

v h X ≤ ch − β

v h Y for all v h ∈ S h . ( 6 . 21 )