2 DGM for elliptic problems

lem:A11a
2.4 Properties of the bilinear forms 23
Lemma 2.13. Any form a h defined by ( 2.44), ah_S ( 2.45) ah_N or ( 2.46) ah_I satisfies the
estimate
|a h (u,v)| ≤ ‖u‖ 1,σ ‖v‖ 1,σ ∀u,v ∈ H 2 (Ω, T h ), (2.83) Aa2.31
where
‖v‖ 2 1,σ = |v| 2 H 1 (Ω,T h ) + ∑
Γ ∈F ID
h
Proof. It follows from ( ah_S 2.44) – ( ah_N 2.45) that
∫
Γ
(
σ −1 (n · 〈∇v〉) 2 + σ[v] 2) dS (2.84) Aa2.31b
|a h (u,v)| ≤ ∑
|∇u · ∇v| dx
(2.85) Aa2.32
K∈T
∫K h
} {{ }
χ 1
+ ∑ ∫
|n · 〈∇u〉 [v]| dS + ∑ ∫
|n · 〈∇u〉 [v]| dS
Γ ∈F ID
h
Γ
} {{ }
χ 2
Γ ∈F ID
h
Γ
} {{ }
χ 3
.
(For the form a i h term χ 3 vanishes of course.) Obviously,
Moreover, the Cauchy inequality implies that
χ 2 ≤
∑
Γ ∈F ID
h
⎛
≤ ⎝ ∑
Γ ∈F ID
h
(∫
∫
Γ
Γ
χ 1 ≤ |u| H 1 (Ω,T h )|v| H 1 (Ω,T h ). (2.86) Aa2.33
) 1/2 (∫ 1/2
σ −1 (n · 〈∇u〉) 2 dS σ[v] dS) 2 (2.87) Aa2.34
Γ
⎞
σ −1 (n · 〈∇u〉) 2 dS⎠
1/2 ⎛
⎝ ∑
Γ ∈F ID
h
∫
Γ
⎞
σ[v] 2 dS⎠
where the penalty weight σ is given by ( A3.1 2.69). Similarly we find that
⎛
χ 3 ≤ ⎝ ∑
Γ ∈F ID
h
∫
Γ
⎞
σ −1 (n · 〈∇v〉) 2 dS⎠
1/2 ⎛
⎝ ∑
Γ ∈F ID
h
∫
Γ
⎞
σ[u] 2 dS⎠
Using the Cauchy inequality, from ( Aa2.33 2.86) – ( Aa2.35 2.88) we derive the bound
1/2
1/2
,
. (2.88) Aa2.35
|a h (u,v)| ≤ |u| H1 (Ω,T h )|v| H1 (Ω,T h ) (2.89) Aa2.36
⎛
⎞
+ ⎝ ∑
1/2 ⎛
⎞
∫
σ −1 (n · 〈∇u〉) 2 dS⎠
⎝ ∑
1/2
∫
σ[v] 2 dS⎠
Γ ∈F ID
h
Γ
Γ ∈F ID
h
Γ