2 DGM for elliptic problems

36 2 Elliptic problems
‖ψ‖ H 2 (Ω) ≤ ¯c‖z‖ L 2 (Ω). (2.162) A8.4
This is true, provided the domain Ω is convex and Γ N = ∅, as follows from
grisvard
[Gri92]. Let us note that H 2 (Ω) ⊂ C(Ω).
Let B h (·, ·) be the symmetric bilinear form given by ( 2.52), A2.20c i.e.
lem:2.13
B h (u,v) = a s h(u,v) + Jh σ (u,v), u,v ∈ H 2 (Ω, T h ), (2.163) A8.5
where a s h and Jσ h
are defined by (ah_S 2.44) and ( 2.77), A3.4 respectively. Let us assume
that the constant C W satisfies condition ( 2.102).
A3.40
Theorem 2.33. (L 2 (Ω)-optimal error estimate) Let us assume that u ∈
H s (Ω),s ≥ 1, is the solution of problem ( 2.1) A0.1 – ( 2.3), A0.2b the system {T h } h∈(0,h0)
of triangulation of Ω satisfies assumptions (A1) and (A2), S hp is the space of
discontinuous piecewise polynomial functions ( 2.16)and A1.23 u h ∈ S hp the approximate
solution of problem ( 2.1) A0.1 – ( 2.3) A0.2b obtained by the SIPG method ( 2.62).
A2.22c
Let the weight σ from the penaly terms be defined by ( 2.69) A3.1 – ( 2.70) A3.2 with a
constant C W satisfying ( 2.102). A3.40 Let there exist the weak solution of problem
( 2.159) A8.1 from H 2 (Ω) satisfying ( 2.162). A8.4 Then
‖e h ‖ L 2 (Ω) ≤ C 3 h µ |u| H µ (Ω) (2.164) A8.20
where e h = u h − u, C 3 is a constant independent of h and µ = min{p + 1,s}.
Proof. Let ψ ∈ H 2 (Ω) be the solution of the dual problem ( 2.161) A8.3 with z :=
e h = u h − u ∈ L 2 (Ω) and let Π h1 ψ ∈ S h1 be the approximation of ψ defined
by ( 2.32) eq:Pi_h with p = 1. Taking into account the regularity of the solution ψ of
problem ( 2.159), A8.1 we can show that
B h (ψ,v) = (u h − u,v) L2 (Ω) ∀v ∈ H 2 (Ω, T h ). (2.165) AB.1
The symmetry of B h (·, ·), the Galerkin orthogonality of the error ( 2.67) A2.23a and
( 2.165) AB.1 with v := e h yield
‖e h ‖ 2 L 2 (Ω) = B h(ψ,u h − u) = B h (u h − u,ψ) (2.166) AB.2
= B h (u h − u,ψ − Π h1 ψ) = B h (e h ,ψ − Π h1 ψ).
Moreover, from ( 2.52), A2.20c ( 2.83) Aa2.31 and ( 2.99) A2.J it follows that
B h (e h ,ψ − Π h1 ψ) (2.167) AB.3
= a s h(e h ,ψ − Π h1 ψ) + J σ h(e h ,ψ − Π h1 ψ)
≤ ‖e h ‖ 1,σ ‖ψ − Π h1 ψ‖ 1,σ + J σ h(e h ,e h ) 1/2 J σ h(ψ − Π h1 ,ψ − Π h1 ψ) 1/2 ,
where
‖v‖ 2 1,σ = |v| 2 H 1 (Ω,T h ) + Jσ h (v,v) + ∑
= |||v||| + ∑
Γ ∈F ID
h
∫
Γ
Γ ∈F ID
h
∫
Γ
σ −1 (n · 〈∇v〉) 2 dS.
σ −1 (n · 〈∇v〉) 2 dS (2.168)
AB.4