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Houška’s simulations.
Velocity Microstructure
0.0 0.2 0.4 0.6 0.8 1.0
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
0.80
0.90
1.00 2D-FEM
Discrete
Figure [...] V′, and Nu : V× V −→ V′ as follows:
〈Lu(u, λ)u,v〉 =
∫ Ω
2µ(DII,r, λ)D(u) : D(v) dx ∀u,v ∈ 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 [...] norm. With the definitions of the operators Lu, Nu, Mλ, and Nλ, and the embedding property of H1 in L4, we show the boundedness of all operators, that is there are positive constants ||Lu||, ||Nu||, ||Mλ|| …