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Cell Adhesion and Detachment in Shear Flow 215
Fig. 3. Subdivision of a triangle of T 2h .
W g0,h(t) ={v h | v h ∈ (C 0 (Ω)) 2 , v h | T ∈ P 1 × P 1 , ∀T ∈T h , v h = g 0 (t)
on the top and bottom of Ω and v is periodic at Γ
in the x 1 -direction}, (8)
W 0,h = {v h | v h ∈ (C 0 (Ω)) 2 , v h | T ∈ P 1 × P 1 , ∀T ∈T h , v h = 0
on the top and bottom of Ω and v is periodic at Γ
in the x 1 -direction}, (9)
L 2 h = {q h | q h ∈ C 0 (Ω), q h | T ∈ P 1 , ∀T ∈T 2h ,q h is periodic at Γ
in the x 1 -direction}, (10)
∫
L 2 0,h = {q h | q h ∈ L 2 h, q h dx =0}. (11)
Ω
In (8)–(11), P 1 is the space of polynomials in two variables of degree ≤ 1.
Remark 4. A different choice of finite element, the Taylor–Hood finite element,
for the velocity field has been considered in [JGP02] for simulating the
fluid/particle interaction via distributed Lagrange multiplier based fictitious
domain method for non-neutrally buoyant particles. ⊓⊔
A finite dimensional space approximating Λ 0 (t) isdefinedasfollows:let
{x i } N i=1 be a set of points covering B(t) (see Figure 4, for example); we define
then
{
}
N∑
Λ h (t) = µ h | µ h = µ i δ(x − x i ), µ i ∈ R 2 , ∀i =1, ..., N , (12)
i=1
where δ(·) is the Dirac measure at x = 0. Then, instead of the scalar product
of (H 1 (B(t))) 2 we shall use 〈·, ·〉 Bh (t) defined by
〈µ h , v h 〉 Bh (t) =
N∑
µ i · v h (x i ), ∀µ h ∈ Λ h (t), v h ∈ W 0,h . (13)
i=1
Then we approximate Λ 0 (t) by
{
Λ 0,h (t) = µ h | µ h ∈ Λ h (t), 〈µ h , e i 〉 Bh (t) =0, i =1, 2, 〈µ h , −→ }
Gx ⊥ 〉 Bh (t) =0 .
(14)