Partial Differential Equations - Modelling and ... - ResearchGate

Let
A Lagrange Multiplier Based Domain Decomposition Method 133
E(t) = 1 2
∫
Ω
2
ε(x)
∂u
∣ ∂t ∣ dx + 1 ∫
2
be the energy of the system. We take w = ∂u
∂t
∫
∫
dE(t)
dt
√
+
ε 1 µ −1
1
Γ ext
( ∂u
∂t )2 dΓ =
since E(0) = 0, the following stability inequality holds:
Ω
Ω
µ −1 (x)|∇u| 2 dx
in (3) and obtain:
f ∂u
∂t dx ≤‖f‖ L 2 (Ω)‖ ∂u
∂t ‖ L 2 (Ω),
E(t) ≤ const T ‖f‖ L 2 (Ω×(0,T )),
∀t ∈ (0,T).
In order to use a structured grid in a part of the domain Ω, we introduce
a rectangular domain R with sides parallel to the coordinate axes, such that
Ω 2 ⊂ R ⊂ Ω with γ the boundary of R (Fig. 1).
Define ˜Ω = Ω \ ¯R and let the subscript 1 of a function v 1 mean that
this function is defined over ˜Ω × (0,T), while v 2 is a function defined over
R × (0,T).
Now we formulate the problem (3) variationally as follows: Let
{
W 1 = v ∈ L ∞ (0,T; H 1 ( ˜Ω)),
{
W 2 = v ∈ L ∞ (0,T; H 1 (R)),
∂v
∂t ∈L∞ (0,T; L 2 ( ˜Ω)),
∂v
∂t ∈ L∞ (0,T; L 2 (R)))
}
∂v
∂t ∈L2 (0,T; L 2 (Γ ext )) ,
}
,
Find a pair (u 1 ,u 2 ) ∈ W 1 × W 2 , such that u 1 = u 2 on γ × (0,T) and for a.a.
t ∈ (0,T)
⎧ ∫
˜Ω
∫
⎪⎨
+
R
∂ 2 ∫
u 1
ε 1
∂t 2 w 1dx + µ −1
˜Ω
∫
1 ∇u 1 ·∇w 1 dx +
√ ∫
µ −1 (x)∇u 2 ·∇w 2 dx+ ε 1 µ −1
1
R
Γ ext
∂u 1
∂t w 1dΓ =
ε(x) ∂2 u 2
∂t 2 w 2dx
∫ ∫
f 1 w 1 dx+
˜Ω
R
f 2 w 2 dx,
for all (w 1 ,w 2 ) ∈ H 1 ( ˜Ω) × H 1 (R) such that w 1 = w 2 on γ,
⎪⎩ u(x, 0) = ∂u
∂t (x, 0) = 0. (4)
Now, introducing the interface supported Lagrange multiplier λ (a function
defined over γ × (0,T) ), the problem (4) can be written in the following way:
Find a triple (u 1 ,u 2 ,λ) ∈ W 1 × W 2 × L ∞ (0,T; H −1/2 (γ)), which for a.a.
t ∈ (0,T) satisfies