Biot-Stokes coupling for interfacial flow in the eye Ruiz-Baier, Taffetani, Westermeyer, Yotov
the Bochner space-time norms, for a separable Banach space V and f : (0, T ) → V, kfk
2
L
2
(0,T ;V)
:=
R
T
0
kf(t)k
2
V
dt and kfk
L
∞
(0,T ;V)
:= ess sup
t∈(0,T )
kf(t)k
V
. By C we will denote generic constants that are
independent of the mesh size.
3.1. Cartesian case
The initial step in deriving the finite element scheme consists in stating a weak form for (2.1a)-
(2.3c). We proceed to test these equations against suitable smooth functions and to integrate over the
corresponding subdomain. After applying integration by parts wherever adequate, we formally end up
with the following remainder on the interface
I
Σ
= −h(2µ
f
(u) − p
F
I)n, vi
Σ
+ h(2µ
s
(d) − ϕI)n, wi
Σ
+ h
κ
µ
f
∇p
P
· n, q
P
i
Σ
,
where h·, ·i
Σ
denotes the duality pairing between the trace functional space space H
1/2
(Σ) and its dual
H
−1/2
(Σ). Then, as in, e.g., [36], we proceed to use each of the transmission conditions (2.8a)-(2.8d),
yielding the expression
I
Σ
= hp
P
, (v − w) · ni
Σ
+
d−1
X
j=1
h
γµ
f
√
κ
(u − ∂
t
d) · t
j
, (v − w) · t
j
i
Σ
− h(u − ∂
t
d) · n, q
P
i
Σ
.
This interfacial term is well-defined because of the regularity of the entities involved, and this implies
that we do not require additional Lagrange multipliers to realise the coupling conditions. Also, in view
of the boundary conditions we define the Hilbert spaces
H
1
?
(Ω
F
) = {v ∈ H
1
(Ω
F
) : v|
Γ
u
F
= 0}, H
1
?
(Ω
P
) = {w ∈ H
1
(Ω
P
) : w|
Γ
d
P
= 0},
H
1
?
(Ω
P
) = {q
P
∈ H
1
(Ω
P
) : q
P
|
Γ
p
P
P
= 0},
associated with the classical norms in H
1
(Ω
F
), H
1
(Ω
P
), and H
1
(Ω
P
), respectively. Consequently we
have the following mixed weak form: For t ∈ [0, T ], find u ∈ H
1
?
(Ω
F
), p
F
∈ L
2
(Ω
F
), d ∈ H
1
?
(Ω
P
),
p
P
∈ H
1
?
(Ω
P
), ϕ ∈ L
2
(Ω
P
), such that
a
F
1
(∂
t
u, v) + a
F
2
(u, v) + c
F
(u, u; v) + b
F
1
(v, p
F
)
+ b
Σ
2
(v, p
P
) + b
Σ
3
(v, ∂
t
d) = F
F
(v) ∀v ∈ H
1
?
(Ω
F
), (3.1a)
− b
F
1
(u, q
F
) = 0 ∀q
F
∈ L
2
(Ω
F
), (3.1b)
b
Σ
3
(u, w) + b
Σ
4
(w, p
P
) + a
P
1
(d, w)
+ a
Σ
2
(∂
t
d, w) + b
P
1
(w, ϕ) = F
P
(w) ∀w ∈ H
1
?
(Ω
P
), (3.1c)
− b
Σ
2
(u, q
P
) − b
Σ
4
(∂
t
d, q
P
) + a
P
3
(∂
t
p
P
, q
P
)
+ a
P
4
(p
P
, q
P
) − b
P
2
(∂
t
ϕ, q
P
) = G(q
P
) ∀q
P
∈ H
1
?
(Ω
P
), (3.1d)
− b
P
1
(d, ψ) − b
P
2
(ψ, p
P
) + a
P
5
(ϕ, ψ) = 0 ∀ψ ∈ L
2
(Ω
P
), (3.1e)
where the bilinear and trilinear forms and linear functionals are defined as
a
F
1
(u, v) = ρ
f
Z
Ω
F
u · v, a
F
2
(u, v) = 2µ
f
Z
Ω
F
(u) : (v) +
d−1
X
j=1
h
γµ
f
√
κ
u · t
j
, v · t
j
i
Σ
,
7
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