Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
150 Asymptotic analysis of blood flow in stented arteries
On the other hand, for any test function φ ∈ H 1 0 (Ωε):
Ωε
Lk ˜ R ε 0φdx = Ĉk
Ωε\Ω0
φdx+Mk
Γ0
φdx.
Then, using Cauchy-Schwarz and Poincaré like inequalities, we obtain the upper bound:
≤ c | Ĉk|ε+|Mk| √
ε ˜ R ε 0H1 (Ωε), (5.3.9)
ik ˜ R ε 0 2 L 2 (Ωε) +∇˜ R ε 0 2 L 2 (Ωε)
where c is a non negative constant depending on the Poincaré inequality. And |Mk| is
controlled as follows:
|Mk| =
Ĉkr (2−e r −e−r
)
|ik(er −e−r )|
≤ |Ĉk|
√k
≤ 2| Ĉk|.
2
|er −e−r | +1
(5.3.10)
Finally, combining (5.3.8)-(5.3.10), we get the H 1 -error estimate.
For the L 2 error, we use the concept of a very weak solution. Namely, one solves the
dual problem: for a given φ ∈ L 2 (Ω0), φ being x1 periodic on Γin ∪Γout, find ˆv ∈ H 2 (Ω0)
such that ⎧ ⎨
⎩
Lk ˆv = φ in Ω0,
ˆv = 0 on Γ1 ∪Γ0,
ˆv is x1-periodic on Γin ∪Γout.
Then, considering the L 2 (Ω0) scalar product . , . , and using the Green formula:
(R ε 0
, φ) =
Ω0
R ε 0 Lkˆv = −ik
Ω0
R ε 0
=
ˆv +
Ω0
ˆv , ∂Rε 0
∂n
∇R ε
0∇ˆv −
Γin∪Γout
−
∂Ω0
R ε 0
∂ˆv
∂n
R ε 0 , ∂ˆv
∂n
Γ0∪Γ1
(5.3.11)
, (5.3.12)
where the brackets refer to the dual product in H−1 , H1 (∂Ω0) and the rest of the
products are in L2 either on Γ0 or in Ω0. Then one computes:
|(R ε 0 , φ)| ≤ R ε
0L2
∂ˆv
(Γ0) ∂n
L 2 (Γ0)
≤ √ ε∇R ε 0 L 2 (Ωε\Ω0) φ L 2 (Ω0)
≤ √ ε∇R ε 0 L 2 (Ωε) φ L 2 (Ω0)
≤ ε 3
2 φ L 2 (Ω0) .