2 DGM for elliptic problems

24 2 Elliptic problems
⎛
+ ⎝ ∑
⎛
Γ ∈F ID
h
∫
≤ ⎝|u| 2 H 1 (Ω,T h ) + ∑
⎛
Γ
⎞
σ −1 (n · 〈∇v〉) 2 dS⎠
Γ ∈F ID
h
× ⎝|v| 2 H 1 (Ω,T h ) + ∑
= ‖u‖ 1,σ ‖v‖ 1,σ .
∫
Γ ∈F ID
h
Γ
1/2 ⎛
⎝ ∑
Γ ∈F ID
h
∫
Γ
⎞
σ[u] 2 dS⎠
⎞
(
σ −1 (n · 〈∇u〉) 2 + σ[u] 2) dS⎠
∫
Γ
1/2
⎞
(
σ −1 (n · 〈∇v〉) 2 + σ[v] 2) dS⎠
1/2
1/2
exer3
exer4
lem:est_sigma
⊓⊔
Exercise 2.14. Prove that ‖ · ‖ 1,σ intoduced by ( 2.83) Aa2.31 defines a norm on the
broken Sobolev space H 2 (Ω, T h ).
Corollary 2.15. By virtue of ( 2.50) A2.20a – ( 2.51), A2.20b Lemma 2.13 lem:A11a and Exercise 2.14,
exer3
the bilinear forms Bh s and Bn h are bounded with respect to the norm ‖ · ‖ 1,σ on
the broken Sobolev space H 2 (Ω, T h ).
Exercise 2.16. Prove that the bilinear forms B s,σ
h
respect to the norm ‖ · ‖ 1,σ on H 2 (Ω, T h ).
and B n,σ
h
are bounded with
Lemma 2.17. Under assumptions (A1) and (A2), for any v ∈ H 2 (Ω, T h ) the
following estimate holds:
∑
∫
σ −1 (n · 〈∇v〉) 2 dS (2.90) A2.30a
Γ ∈F ID
h
Γ
≤ C HC M
C W
∑ (
)
h K ‖∇v‖ L2 (K) |∇v| H1 (K) + h −1
K ‖∇v‖2 L 2 (K)
K∈T h
Proof. Using ( 2.81) A3.7 and ( 2.19), eq:MTI we find that
∑
∫
σ −1 (n · 〈∇v〉) 2 dS (2.91) A2.35ab
Γ ∈F ID
h
≤ C H
C W
≤ C HC M
C W
Γ
∑
h K ‖∇v‖ 2 L 2 (∂K)
K∈T h
∑ (
h K ‖∇v‖ L 2 (K) |∇v| H 1 (K) + h −1
K ‖∇v‖2 L 2 (K)
K∈T h
which we wanted to prove.
⊓⊔
Now, we prove the continuity of the bilinear forms B defined by ( 2.50) A2.20a –
( 2.54) A2.20e with respect to the norm ||| · ||| given by ( 2.68) en on the space S hp .
)
,