2 DGM for elliptic problems

2.5 Error estimates 29
lem:2.7
where C J = 2C A
√
CW C M .
Proof. i) Using the multiplicative trace inequality ( 2.19) eq:MTI and ( 2.30), eq:ap we obtain
∑
h −1
K ‖η‖2 L 2 (∂K) (2.118) A3.23
K∈T h
∑ (
)
≤ C M K
h −1
K ‖η‖2 L 2 (K) + ‖η‖ L 2 (K)|η| H1 (K)
≤ C M C 2 A
K∈T h
h −1
∑
h −1
K
K∈T h
≤ 2C M CAh ∑ 2 2µ−2 |u| 2 H µ (K) ,
K∈T h
which proves ( 2.114).
A3.29b
ii) Similarly, by ( 2.19) eq:MTI and ( 2.30), eq:ap we have
∑
(
)
h −1
K h2µ K + hµ K hµ−1 K
|u| 2 H µ (K)
h K ‖∇η‖ 2 L 2 (∂K) (2.119) A3.23+
K∈T h
∑ (
)
≤ C M h K h −1
K ‖∇η‖2 L 2 (K) + ‖∇η‖ L 2 (K)|∇η| H 1 (K)
K∈T h
∑ ( )
= C M |η|H1 (K) + h K |η| H1 (K)|η| H2 (K)
K∈T h
≤ 2C M CAh 2 2µ−2 |u| 2 H µ (K) ,
iii) By ( 2.77) A3.4 and ( 2.80),
A3.6
Jh σ (η,η) = ∑ ∫
σ[η] 2 dS (2.120) A3.30
≤
∑
Γ ∈F ID
h
Γ ∈F ID
h
Γ
C W
h Γ
‖[η]‖ 2 L 2 (Γ) ≤ 2C W
∑
K∈T h
1
h K
‖η‖ 2 L 2 (∂K) ,
which together with ( 2.114) A3.29b gives ( 2.116).
A3.29c
iv) Inequality ( 2.117) A3.29d immediately follows from ( 2.68), en ( 2.34) eq:AP and ( 2.116).
A3.29c
⊓⊔
Lemma 2.28. Let us assume that s ≥ 1,p ≥ 1 are integers, u ∈ H s (Ω),
µ = min(p + 1,s) and a h (·, ·) is a bilinear form defined by ( 2.44) ah_S or ( 2.45) ah_N or
( 2.46). ah_I Then there exists a constant C a > 0 such that
and,
|a h (Π hp u − u,v h )| ≤ C a h µ−1 |u| Hµ (Ω)|||v h ||| ∀v h ∈ S hp , (2.121) A4.a
|Jh σ (Π hp u − u,v h )| ≤ C J h µ−1 |u| Hµ (Ω)|||v h ||| ∀v h ∈ S hp ,(2.122) A4.j
where the constant C J was introduced in Lemma 2.27.
lem:2.6