2 DGM for elliptic problems

22 2 Elliptic problems
estsigma
Lemma 2.11. Let (A2) be valid. Then for each v ∈ H 1 (Ω, T h ) we have
∑
∫
[v] 2 dS ≤ ∑ ∫
|v| 2 dS, (2.78) A3.6a
Γ ∈F ID
h
∑
Γ ∈F ID
h
h −1
Γ
h Γ
∫
Γ
Γ
2h −1
K
K∈T h
〈v〉 2 dS ≤ C H
∑
∂K
K∈T h
h K
∫∂K
|v| 2 dS. (2.79) A3.7a
Hence,
∑
Γ ∈F ID
h
∑
Γ ∈F ID
h
σ| Γ ‖[v]‖ 2 L 2 (Γ) ≤ 2C W
1
σ| Γ
‖〈v〉‖ 2 L 2 (Γ) ≤ C H
C W
Proof. a) By ( A1.15 2.14), the inequality
∑ 1
‖v‖ 2 L
h 2 (∂K), K
K∈T h
(2.80) A3.6
∑
. (2.81) A3.7
K∈T h
h K ‖v‖ 2 L 2 (∂K)
exer2
and ( 2.73) A3.5 we have
∑
∫
h −1
Γ [v]2 dS = ∑
Γ ∈F ID
h
≤ 2 ∑
Γ
Γ ∈F I h
≤ 2 ∑
Γ ∈F ID
h
≤ 2 ∑
∫
h −1
Γ
Γ
h −1
K
K∈T h
h −1
K (L)
Γ
∫
(γ + δ) 2 ≤ 2(γ 2 + δ 2 ), γ,δ ∈ IR, (2.82) AC3.6
Γ ∈F I h
( ∣∣∣v| (L)
∣ 2 ∣
+
∫
∂K
∣
Γ
∣v| (L)
Γ
Γ
|v| 2 dS.
∫
h −1
Γ
Γ
∣v| (R)
Γ
∣ 2 dS + ∑
∣
∣v| (L)
Γ
− v|(R) Γ
∣ 2) dS + ∑
Γ ∈F I h
h K
(R)
Γ
Γ ∈F D h
∫
∣
∣ 2 dS + ∑
∫
h −1 ∣
Γ
Γ
∣v| (R)
Γ
Γ
∣ 2 dS
Γ ∈F D h
∣v| (L)
Γ
∫
h −1 ∣
Γ
Γ
∣ 2 dS
∣v| (L)
Γ
This and ( 2.72) A3.3 immediately imply ( 2.80).
A3.6
b) In the proof of ( 2.79) A3.7a we proceed similarly, using ( 2.14), A1.15 ( 2.73) A3.5 and
( 2.82). AC3.6 Inequality ( 2.81) A3.7 is a consequence of ( 2.79) A3.7a and ( 2.72). A3.3
⊓⊔
Exercise 2.12. Formulate and prove Lemma 2.11 estsigma for definitions ( 2.74) A3.1a and
( 2.76).
A3.1c
∣ 2 dS
2.4.1 Continuity of the bilinear forms B
First, we shall prove an auxiliary assertion.