Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 346

This requires one degree of freedom per edge, which can be the value at the
midpoint of the edge. In analogy with h ,weset

h v : = π h v + π h (v − π h v),

so that e ( h v − v)ds = 0 for every edge e of the triangulation. Using Green's

formula, we have

grad ( h v − v) · γ h dx = (π h v − v)γ h · nds − (π h v − v) div γ h dx = 0 .

T ∂T T

The contour integral vanishes by construction, and the second integral also vanishes
since γ h is constant on T . The boundedness of h follows as for π h , and so
the hypotheses of Fortin's criterion are satisfied since ( grad h w − h θ,γ h ) 0 =
( grad w − θ,γ h ) 0 for γ h ∈ M

0 .
More recently, Chapelle and Stenberg [1998] analyzed the finite element
discretization with h :

⊕ B 3 ) 2 keeping W h and h as in Fig. 63. They
avoided the H − 1 ( div ) -norm by using mesh-dependent norms and showed stability
with respect to the norm whose square is

1
0 , 0

= ( M

1

2
1

2
1

2
0

+ (h 2 + t 2 ) γ

2

w + θ +

+ t 2 ∇ w − θ

0 .

h 2

In particular, the duality argument of Aubin-Nitsche could be more easily per-
formed with these norms.
Another mesh-dependent norm which less conceals the connection with
H − 1 ( div ) was introduced by Carstensen and Sch oberl [2000].

Problems

6.9 Let η ∈ L 2 () 2 . Show that the spaces for the components of the decom-
position (6.5) can be exchanged, i.e., that we can choose ψ ∈ H 1 ()/ R and

H 0 () . To this end, decompose η ⊥ according to (6.5), and write the result
for η ⊥ as a decomposition of η .

p ∈

Does ψ ∈ H 1 (), q ∈ H 1 ()/ R

6.10 suffice to establish the orthogonality

relation

( ∇ ψ, curl q) 0 = 0 ,

or is a zero boundary condition required?