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Houška’s simulations.
Velocity Microstructure
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y
0.00
0.02
0.04
0.06
0.08
0.10
x -v
e lo
c it y
2D-FEM
Discrete
0.0 0.2 0.4 0.6 0.8 1.0
y
0.70
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1.00 2D-FEM
Discrete
Figure 1: [...] L2-norm ||·||0, respectively, V′, Q′, and T′ are their corresponding dual spaces, respectively [2, 3, 6]. We introduce the strong operators Au,Lu : V× T× V −→ V′, and Nu : V× V −→ V′ as follows:
〈Lu(u, λ)u [...] ∈ V, λ ∈ T, (4)
〈Nu(w)u,v〉 =
∫ Ω
w · ∇uv dx ∀u,v,w ∈ V, (5)
and set Au(u, λ) := Nu(u) + Lu(u, λ). (6)
The operators Aλ,Mλ : V× T −→ T′, and Nλ : V× T −→ T′ are defined as follows:
〈Mλ(u, λ), ξ〉 =
∫ Ω
( …