2 DGM for elliptic problems

lem:AP
16 2 Elliptic problems
By ( 2.31), Ltwoproj π K,p is the operator of L 2 (K)-projection. This implies that Π hp
is the L 2 (Ω)-projection on S hp : if ϕ ∈ L 2 (Ω), then
∫
∫
Π hp ϕ ∈ S hp , (Π hp ϕ)v dx = ϕv dx ∀v ∈ S hp . (2.33) LtwoprOm
Lemma 2.7. Let assumption (A1) be valid. Then
Ω
Ω
|Π hp v − v| Hq (Ω,T h ) ≤ C Ah µ−q |v| Hµ (Ω,T h ), v ∈ H s (Ω, T h ), (2.34) eq:AP
where µ = min(p + 1,s), 0 ≤ q ≤ s and C A is the constant from ( 2.30).
eq:ap
Proof. Using ( 2.32), eq:Pi_h definition of a seminorm in a broken Sobolev space and
the approximation properties ( 2.30), eq:ap we obtain ( 2.34). eq:AP
⊓⊔
sec:2.2
2.3 DGM based on a primal formulation
In this section we shall describe and analyze the DGM for the solution of
problem ( 2.1) A0.1 – ( 2.3) A0.2b based on a primal formulation. The approximate solution
will be sought in the space S hp ⊂ H 1 (Ω, T ). However, the weak formulation
( 2.5) A0.3 given in Section 2.1 sec:2.0 is not suitable for the derivation of the DGM, because
( 2.5) A0.3 does not make sense for u ∈ H 1 (Ω, T ) ⊄ H 1 (Ω). Therefore, we shall
introduce a “weak form of ( 2.1) A0.1 – ( 2.3) A0.2b in the sence of broken Sobolev spaces”.
Let as assume that u is a sufficiently regular solution of ( 2.1) A0.1 – ( 2.3). A0.2b We
shall consider the so-called strong solution, i.e. a solution u ∈ H 2 (Ω). We
proceed in the following way: We multiply ( 2.1) A0.1 by a function v ∈ H 2 (Ω, T h )
integrate over K ∈ T and use Green’s theorem. Summing over all K ∈ T h , we
obtain the identity
∑
∇u · ∇v dx − ∑
∫
(n K · ∇u)v dS = f v dx, (2.35) A2.4
K∈T h
∫K
K∈T h
∫∂K
where n K denotes the unit outer normal to ∂K. The surface integrals over
∂K make sense due to the regularity of u. We split them according to the
type of faces Γ that form the boundary of the element K ∈ T h :
∑
K∈T h
∫∂K
= ∑
Γ ∈F D h
+ ∑
∫
Γ ∈F I h
Γ
∫
(n K · ∇u)v dS (2.36) A2.5
(n Γ · ∇u)v dS + ∑
Γ
(
n Γ ·
Γ ∈F N h
(∇u| (L)
Γ
)v|(L) Γ
∫
Γ
Ω
(n Γ · ∇u)v dS
− (∇u|(R) Γ
)v|(R) Γ
)
dS.
(There is a sign “−” in the last integral, since n Γ is the unit outer normal to
but the unit inner normal to ∂K (R) sec:2.1.1
Γ
, see Section 2.2.1 or Figure 2.2.
fig:normals
∂K (L)
Γ