Diploma thesis as pdf file - Johannes Kepler University, Linz

2.2 A predictor-corrector approach for the following the central path 21
And for the next corrector step it holds
We have
J ′′
κ(ũ)(δũ,v) = −J ′ κ(ũ)v ∀v ∈ V. (2.7)
∫
J ′ (ũ)v = J ′ (u + )v + ρ ∇p∇vdω ∀v ∈ V,
Ω
then by use of (2.6) we obtain
(
∫
∀v ∈ V J ′ (ũ)v = J ′ (u + )v + ρ −J ′ (u + )v − κ
∫
= (1 − ρ)J ′ (u + )v − ρκ
Ω
(
∫
= (1 − ρ) J κ(u ′ + )v + κ
=
and consequently, denoting by ξ = δu
u−ψ ,:
∀v ∈ V − J ′ κ + (ũ)v = −J ′ (ũ)v + κ + ∫
∫
Ω
=
=
=
Ω
pv
(u + − ψ) 2
Ω
)
pv
(u + − ψ) 2 dω
)
v
u + − ψ dω − ρκ
∫
Ω
pv
(u + − ψ) 2
−κ + (δu) 2 v
(u + − ψ)(u − ψ) 2 + v
κ+
u + − ψ − ρκ pv
(u + − ψ) 2 dω
∫
Ω
∫
κ +
Ω
∫
κ +
Ω
Ω
( ξ
κ + 2 v
u + − ψ −
v
ũ − ψ dω
v
u + − ψ +
v )
ũ − ψ
pv
+ ρκ
(u + − ψ) 2 dω
ξ 2 v
u + − ψ − pv
ρκ+ (u + − ψ)(ũ − ψ) + ρκ pv
(u + − ψ) 2 dω
ξ 2 v
u + − ψ + ρ2 κ p(u+ + p − ψ)v
(u + − ψ) 2 (ũ − ψ) dω.
Then using (2.7) with v = δũ and application of the Cauchy-Schwartz inequality
yield
∫
Ω
|∇δũ| 2 dω + κ + ∫
Ω
(δũ) 2 ( ∫
(ũ − ψ) 2 dω ≤ (δũ) 2 ) 1
2 κ+ Ξ
Ω (ũ − ψ) 2 dω ,
where
∫
Ξ =
Ω
(
ξ 2 p
(1 + ρ
u + − ψ ) +
ρ2
1 − ρ
(
p
u + 1 + p )) 2
− ψ u + dω.
− ψ