Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
5.5 Numerical results 159
• ˆ Vε,k,h, the implicit discrete wall-law.
In order to compute the cell problem and β, the constant at infinity related to the specific
roughness f, we discretize a cell problem defined on a truncated domain : find βL solving
⎧
⎪⎨ −∆βL = 0 in Z
⎪⎩
+ ∪Γ∪P ∩{y2 < L}
∂nβ = 0 on {y2 = L}
β = −y2
on P0
It is shown in [118] that the solutionβL converges exponentially fast, whenL → ∞, towards
the solution of (5.3.15). So solving the problem above provides a good approximation of
β L. And we use this numerical value in the boundary condition on Γ0 in (5.4.6) in order
to compute ˆ Vε,k,h. The code is written in freefem++ language [111]: it is very well suited
for solving complex valued variational problems with finite elements. Our code is available
through Internet (1) .
5.5.2 Error estimates
We compute numerical equivalent norms for a priori and very weak estimates. We plot
this results in the log-log scale for various sizes ε (in abscissa) in fig. 5.4. We recover better
· H 1 (Ω0 )
meshsize
1
û ε,k,h − û 0,k,h
ǫ 0.75
ûε,k,h − û1,k,h ǫ
0.1
0.1 1
ǫ
0.525
0.1
100000
10000
0.01
0.1 1 1000
# triangles
ǫ
h
# vertices
# vertices & triangles
· L 2 (Ω0 )
1
0.1
0.01
ûε,k,h − û0,k,h 3
ǫ2
ûε,k,h − û1,k,h ǫ
0.001
0.1 1
ǫ
2.25
Figure 5.4: Numerical error estimates in H 1 (Ω0) (left) and L 2 (Ω0) (middle) norms, and
mesh parameters (right)
orders of convergence than expected: the very weak estimates provide ε 3
2 convergence for
the Poiseuille profile while they give ε 9
4 for the wall-law. The H1 (Ω0) norm (∼ ε 3
4) is
better than expected for the Poiseuille profile while surprisingly the error is worse for the
1 http://ljk.imag.fr/membres/Vuk.Milisic/Software/complexWallLaw.edp