2 DGM for elliptic problems

lem:II
lem:III
def:Pi_h
2.2 Spaces of discontinuous functions 15
‖v‖ 2 L 2 (∂K) ≤ 1 [
]
2h K ‖v‖ L
ρ 2 (K)|v| H 1 (K) + d‖v‖ 2 L 2 (K) (2.25) A1.57
K
[
≤ C R 2‖v‖ L2 (K)|v| H1 (K) + d ]
‖v‖ 2 L
h 2 (K) ,
K
which proves ( eq:MTI 2.19) with C M = C R max{2,d}. ⊓⊔
Lemma 2.4. (Inverse inequality) Let (A1) be satisfied. Then there exists
a constant C I > 0 independent of v, h and K such that
|v| H1 (K) ≤ C I h −1
K ‖v‖ L 2 (K), ∀v ∈ P p (K), ∀K ∈ T h , forallh ∈ (0,h 0 ).
(2.26) eq:InvI
Proof. Let ˆK be a reference triangle and F K : ˆK → K, K ∈ Th be an
affine mapping such that F K ( ˆK) = K. By [Cia79], Ciarlet Theorem 3.1.2 for any v ∈
H m (K), where m ≥ 0 is an integer,the function ˆv(ˆx) = v(F K (ˆx)) ∈ H m ( ˆK)
and
|v| Hm (K) ≤ c c h d 2 −m
K |ˆv| H m ( ˆK)
, (2.27) eq:ci1
|ˆv| Hm ( ˆK) ≤ c ch m− d 2
K |v| H m (K), (2.28) eq:ci2
where c c > 0 depends on C R but not on K and v. From [Sch98], Schwab-book Theorem
4.76 we have
|ˆv| H 1 ( ˆK) ≤ c sp 2 ‖ˆv‖ L 2 ( ˆK) , ˆv ∈ P p( ˆK), (2.29) eq:sch
where c s > 0 depends on d but not on ˆv and p. A simple combination of ( 2.27)
eq:ci1
– ( 2.29) eq:sch proves ( 2.26) eq:InvI with C I = c s c 2 c p 2 . ⊓⊔
Lemma 2.5. (Approximation properties) Let assumption (A1) be valid.
Then there exist a mapping π K,p : H 1 (K) → P p (K) and a constant C A > 0
such that
|π K,p v − v| H q (K) ≤ C A h µ−q
K |v| H µ (K) (2.30) eq:ap
∀v ∈ H s (K) ∀K ∈ T h ∀h ∈ (0,h 0 ),
where µ = min(p + 1,s), 0 ≤ q ≤ s and p,s are integers.
Proof. We define π K,p as the L 2 (K) projection on P p (K), i.e, for ϕ ∈ L 2 (K),
∫
∫
π K,p ϕ ∈ P p (K), (π K,p ϕ)v dx = ϕv dx ∀v ∈ P p (K). (2.31) Ltwoproj
K
The approximation properties ( 2.30) eq:ap follow immediately from [Cia79, Ciarlet Theorem
3.1.4].
⊓⊔
Definition 2.6. Let T h be a triangulation and p ≥ 0. We define the mapping
Π hp : H 1 (Ω, T h ) → S hp by
K
(Π hp u) | K = π K,p (u| K ) ∀K ∈ T h , (2.32) eq:Pi_h
where π K,p : H 1 (K) → P p (K) is the mapping introduced in Lemma 2.5.
lem:III