Existence and Uniqueness of Eddy Current Problems in Bounded ...

4 MICHAEL KOLMBAUER
I n
n 2
n 1
2
Σ⩵0
air
1
Σ0
conductor
Figure 1. Unbounded exterior domain
• semi-coercive, i.e. 〈A(u), u〉 ≥ c‖ curl u‖ 2 L 2(Ω)
• bounded, i.e. 〈A(u), v〉 ≤ c‖u‖ H(curl,Ω) ‖v‖ H(curl,Ω)
• hemicontinuous.
Proof. see [3, Lemma 2.6].
For linear operators the whole analysis simplifies as the following remark states.
Remark 1. Let M be any linear operator. If M is semi-coercive, i.e. 〈M(w), w〉 ≥
0, then M is also monotone.
Hence the operator A resulting from the conducting part of our computational
domain naturally fulfills the requirements of Theorem 1 and Theorem 2. Therefore
the idea of reducing the computational domain to the conducting parts only, arises
quite natural in this context.
3. The eddy current problem in R 3
Let Ω be R 3 and consist of two subdomains, Ω 1 and Ω 2 , with the following properties.
Ω 1 is Lipschitz polyhedron that is simply connected. Ω 2 is the complement
of Ω 1 in R 3 , i.e R 3 \Ω 1 , and hence also simply connected. Furthermore we denote
by Γ I the interface of the two subdomains, i.e. Γ I = Ω 1 ∩ Ω 2 . By n we denote the
exterior unit normal vector field of Ω 1 on Γ I , pointing from Ω 1 to Ω 2 (see Figure 1).
Additionally to the partial differential equations in Ω 1 and Ω 2 , the solution has to
be sinusoidal in Ω 2 . The system is completed by appropriate decay and interface
conditions and an initial condition. Hence we deal with the following problem:
(1)
⎧
⎪⎨
⎪⎩
∂u
σ 1 ∂t + curl (ν 1(|curl u|) curl u) = f 1 , in Ω 1 × (0, T )
curl (curl u) = 0 in Ω 2 × (0, T )
div u = 0 in Ω 2 × (0, T )
u = O(|x| −1 ) for |x| → ∞
curl u = O(|x| −1 ) for |x| → ∞
u = u 0 on Ω 1 × {0}
u 1 × n = u 2 × n on Γ I × (0, T )
ν 1 (|curl u 1 |)curl u 1 × n = curl u 2 × n on Γ I × (0, T )
Here u 1 and u 2 are the restrictions of u to Ω 1 and Ω 2 , i.e. u 1 = u| Ω1 and u 2 = u| Ω2 .
Remark 2. Due to scaling arguments, it can always be achieved that ν 2 = 1.
(Otherwise ν 1 = ν 1 /ν 2 and σ 1 = σ 1 /ν 1 .)
We show, that the degenerated parabolic problem on the whole domain (1) can
be reduced to an initial value problem in the conducting region by using the tools
of boundary integral operators. For the resulting parabolic equation, standard
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