Partial Differential Equations - Modelling and ... - ResearchGate

134 S. Lapin et al.
∫
∂ 2 ∫
∫
u 1
ε 1
˜Ω ∂t 2 w 1dx + µ −1
1 ∇u 1 ·∇w 1 dx + ε(x) ∂2 u 2
˜Ω
R ∂t 2 w 2dx
∫
√ ∫
∫
+ µ −1 (x)∇u 2 ·∇w 2 dx + ε 1 µ −1 ∂u 1
1
R
Γ ext
∂t w 1dΓ + λ(w 2 − w 1 )dγ
γ
∫
∫
= f 1 w 1 dx + f 2 w 2 dx for all w 1 ∈ H 1 ( ˜Ω), w 2 ∈ H 1 (R), (5)
˜Ω
R
∫
ζ(u 2 − u 1 )dγ = 0 for all ζ ∈ H −1/2 (γ), (6)
γ
and the initial conditions from (1).
Remark 1. We selected the time dependent approach to capture harmonic
solutions since it substantially simplifies the linear algebra of the solution
process. Furthermore, there exist various techniques to speed up the convergence
of transient solutions to periodic ones (see, e.g., [BDG + 97]).
2 Time Discretization
In order to construct a finite difference approximation in time of the problem
(5), (6), we partition the segment [0,T]intoN intervals using a uniform
discretization step ∆t = T/N.Letu n i ≈ u i (n∆t)fori =1, 2, λ n ≈ λ(n∆t).
The explicit in time semidiscrete approximation to the problem (5), (6) reads
as follows:
u 0 i = u 1 i =0
for n =1, 2,...,N − 1. Find u n+1
1 ∈ H 1 ( ˜Ω), u n+1
2 ∈ H 1 (R) andλ n+1 ∈
H −1/2 (γ) such that
∫
u n+1
1 − 2u n 1 + u n−1 ∫
1
ε 1
˜Ω ∆t 2 w 1 dx + µ −1
1 ∇un 1 ·∇w 1 dx+
˜Ω
∫
+ ε(x) un+1 2 − 2u n 2 + u n−1 ∫
2
R
∆t 2 w 2 dx + µ −1 (x)∇u n 2 ·∇w 2 dx+
R
√
+ ε 1 µ −1 u
1
∫Γ n+1
1 − u n−1 ∫
1
w 1 dΓ + λ n+1 (w 2 − w 1 )dγ =
ext
2∆t
γ
∫
∫
= f1 n w 1 dx + f2 n w 2 dx for all w 1 ∈ H 1 ( ˜Ω),w 2 ∈ H 1 (R), (7)
∫
˜Ω
R
ζ(u n+1
2 − u n+1
1 )dγ = 0 for all ζ ∈ H −1/2 (γ). (8)
γ
Remark 2. The integral over γ is written formally; the exact formulation requires
the use of the duality pairing 〈·, ·〉 between H −1/2 (γ) andH 1/2 (γ).