2 DGM for elliptic problems

34 2 Elliptic problems
∫
〈∇(v − I hp v)〉 · n dS = 0 ∀Γ ∈ F h , v ∈ H 2 (Ω, T h ), (2.146) A7.3
Γ
|I hp v − v| H q (Ω,T h ) ≤ ¯C A h µ−q |v| H µ (Ω,T h ), v ∈ H s (Ω, T h ), h ∈ (0,h (2.147) 0 ), A7.3a
where µ = min(p + 1,s), q = 0,1 and ¯C A is a constant.
Let u ∈ H s (Ω) (s ≥ 1 is an integer) be an exact solution of ptoblem ( 2.1)
A0.1
– ( 2.2). A0.2a Then, in view of ( 2.145) A7.2 and ( 2.146),
A7.3
B h (u − I hp u,v h ) = 0 ∀v h ∈ S h0 , (2.148) A7.4
where S h0 denotes the space of piecewise constant fucntions on T h . Hence, if
Π 0 is the orthogonal projection of L 2 (Ω) onto S h0 , then ( 2.96) A2.41 and ( 2.148)
A7.4
imply that
|B h (u − I hp u,v h )| ≤ ∣ ∣ Bh (u − I hp u,v h − Π 0 v h ) ∣ ∣ +
∣ ∣Bh (u − I hp u,Π 0 v h ) ∣ ∣
≤ C B |||u − I hp u||| |||v h − Π 0 v h ||| ∀v h ∈ S hp . (2.149) A7.5
Obviously,
|u − Π 0 u| H 1 (K) = |u| H 1 (K), K ∈ T h . (2.150) A7.6a
Moreover, it follows from the approximation properties ( 2.34) eq:AP that
‖v − Π 0 v‖ L 2 (K) ≤ C A h K |v| H 1 (K), v ∈ H 1 (K), K ∈ T h . (2.151) A7.6
Let ψ ∈ H 1 (Ω, T h ). Then, using ( eq:MTI 2.19) and ( A3.30 2.120), we find that
|||ψ||| 2 = |ψ| 2 H 1 (Ω,T h ) + Jσ h(ψ,ψ)
∑
(2.152) A7.7
≤ |v| 2 H 1 (Ω,T h ) + 2C W h −1
K ‖ψ‖2 L 2 (∂K)
K∈T h
≤ |ψ| 2 H 1 (Ω,T h ) +2C WC M
∑
K∈T h
(
h −2
K ‖ψ‖2 L 2 (K) + h−1 K ‖ψ‖ L 2 (K)|ψ| H1 (K)
For v ∈ H 1 (Ω, T h ) let us set ψ := v − Π 0 v in ( 2.152). A7.7 Then from ( 2.150) A7.6a –
( 2.151) A7.6 we get
|||v − Π 0 v||| 2 ≤ (1 + 4CAC 2 W C M ) ∑
|v| 2 H 1 (K) (2.153) A7.8
K∈T h
= (1 + 4C 2 AC W C M )|v| 2 H 1 (Ω,T h ) .
On the other hand, if we set ψ := u − I hp u, then by ( 2.152) A7.7 and ( 2.147)
A7.3a
we obtain
)
.
|||u − I hp u||| 2 ≤ ¯C 2 Ah 2µ−2 |u| 2 H µ (Ω + 4 ¯C 2 AC W C M h 2µ−2 |u| 2 H µ (Ω)
= ¯C 2 A(1 + 4C W C M )h 2µ−2 |u| 2 H µ (Ω) .
(2.154) A7.9