Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 347

6.11 Show that

(γ, η) 0
γ H( rot ,) ,

div η − 1 ≤ const sup

γ

and thus that div η ∈ H − 1 () for η ∈ (H 0 ( rot , )) . Since H 0 ( rot ,) ⊃ H 0 ()
implies (H 0 ( rot , )) ⊂ H − 1 () , this completes the proof of (6.9).

6.12 In what sense do the solutions of (5.9) and (6.2) satisfy

div γ = f ?

6.13 Let t> 0, and suppose H 1 () is endowed with the norm

2
0

+ t 2

2

1 ) 1 / 2 .

||| v ||| = ( v v

:

(u,v) 0
|||

The norm of the dual space ||| u ||| − 1 : = sup v can be estimated easily from

v |||

above by

t u − 1 .
Give an example to show that there is no corresponding estimate from below with
a constant independent of t by computing the size of

min

1

||| u ||| − 1 ≤ u 0 ,

n sin n 2 x H 0 [0 ,π ]

u(x) : = sin x + ∈ ( 1 /t ≤ n ≤ 2 /t)

sufficiently exactly in each of these norms.

The finite element space W h contains H 1

6.14 conforming elements, and thus

lies in C() . Show that ∇ W h ⊂ H 0 ( rot ,) . What property of the rotation is

responsible for this?