Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
154 Asymptotic analysis of blood flow in stented arteries
5.3.4 First order estimates
The gain obtained when introducing the microscopic corrector is of order √ ε. Indeed,
the following error estimates hold.
Proposition 2. There exist two positive constants c3 and c4, depending only on the mode
Ĉk and not on the frequency k, such that:
ûǫ,k − Ûǫ,k H 1 (Ωε) ≤ c3ε , ûǫ,k − Ûǫ,k L 2 (Ω0) ≤ c4ε 3/2 . (5.3.18)
Proof. Denote Rε := ûǫ,k − Ûǫ,k the error to estimate. It is solution of the problem:
⎧
⎪⎨
⎪⎩
Lk Rε = ĈkχΩε\Ω0 −ikMkx2χ Ωε\Ω0 −ikMkε β( x
ε )−β +βχ
Ωε\Ω0 −εMkβδΓ0 , in Ωε
x1 1
Rε = −εMk β( ε , ε )−β
on Γ1,
Rε = 0 on Γε,
Rε is x1-periodic on Γin ∪Γout.
(5.3.19)
The existence and uniqueness of Rε are standard. We focus again on the a priori estimates:
test the system above by Rε and estimate the lhs from below as in (5.3.8), then estimate
from above the rhs. The last step includes new terms wrt the zeroth order approximation,
listed below :
⎧
⎪⎨
⎪⎩
A1 = Ĉk
Ωε\Ω0
A2 = −ikMk
A3 = −ikMkε
Rε dx,
Ωε\Ω0
Ωε
A4 = −ikMkεβ
A5 = −εβMk
Then, estimating these terms, one gets
|
5
j=1
Γ0
x2Rε dx,
(β( x
ε )−β)Rε dx,
Ωε\Ω0
Rε dx1.
Rε dx,
Aj| ≤ ε 3
2 c∇Rε L 2 (Ωε)
(5.3.20)