Respectively scaled splitting iteration method for a class of block 4-by-4 linear systems from eddy current electromagne
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Respectively scaled splitting iteration method for a class of block 4‑by‑4 linear systems from eddy current electromagnetic problems Min‑Li Zeng1,2 Received: 11 June 2020 / Revised: 26 September 2020 / Accepted: 4 October 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract In this paper, we present a respectively scaled splitting (RSS) iteration method for the block 4-by-4 linear system from eddy current electromagnetic problems. Unconditional convergence properties of the RSS iteration method are established. Theoretical results show that the quasi-optimal iterative parameter that minimizes the spectral radius is 𝛼opt = 1 and the corresponding convergence factor is no more than 1 . The validity of theoretical analysis and the effectiveness of RSS methods are veri2 fied by numerical experiments. Keywords Iteration method · Block 4-by-4 linear system · Quasi-optimal parameter · Convergence · Eddy current electromagnetic problems Mathematics Subject Classification 65F10 · 65F50 · 65W05
1 Introduction Consider the eddy current electromagnetic problem [1, 11–13]: find the state y and control u that minimizes the cost functional,
This work was supported by the National Natural Science Foundation of China (No. 11901324), the Natural Science Foundation of Fujian Province (No. 2020J01906) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (No. 2018-39) * Min‑Li Zeng [email protected] 1
School of Mathematics and Finance, Putian University, Putian 351100, Fujian, People’s Republic of China
2
Key Laboratory of Financial Mathematics (Putian University), Fujian Province University, Putian 351100, Fujian, People’s Republic of China
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M.-L. Zeng
J(y, u) =
𝛽 1 |y − yd |2 dxdt + |u|2 dxdt, ∫ 2 Ω×[0,T] 2 ∫Ω×(0,T)
(1)
subject to the regularized state equation of the form
⎧ 𝜕y ⎪ 𝜎 + curl(𝜈curly) + 𝜀y = u in Ω × (0, T), ⎪ 𝜕t ⎨y × n = 0 on 𝜕Ω × (0, T), ⎪ in Ω. ⎪ y(0) = y(T) ⎩
(2)
Here yd is the desired state and 𝛽 > 0 is a cost regularization parameter. We assume that Ω ⊂ ℝ3 is a bounded Lipschitz domain. The reluctivity 𝜈 ∈ L∞ (Ω) = {s(x)| infm(Ω0 )=0 {supx∈Ω∕Ω0 |s(x)|} < ∞} , where s(x) is a measurable function in Ω , is uniformly positive and independent of |curly| , i.e., the eddy current problem is linear. We also suppose that the conductivity 𝜎 ∈ L∞ (Ω) is constant, positive in conducting and zero in nonconducting subdomains. Applying a Lagrange multiplier w to impose the state equation, we can obtain the Lagrangian functional as ) ( 𝜕y 𝜎 + curl(𝜈curly) + 𝜀y − u wdxdt. 𝐋(y, u, w) = J(y, u) + ∫Ω×[0,T] 𝜕t The first order necessary conditions ∇(y,u,w) 𝐋(y, u, w) leads to both the relation 𝛽u − w = 0 in Ω × (0, T) and the reduced optimality system
⎧ 𝜎 𝜕y + curl(𝜈curly) + 𝜀y − 𝛽 −1 w = 0 ⎪ 𝜕t ⎪ 𝜕w + curl(𝜈curlw) + 𝜀w + y = yd ⎪ −𝜎 𝜕t ⎪ ⎨y × n = 0 ⎪ ⎪w × n = 0 ⎪ y(0) = y(T) ⎪ ⎩w = 0
in
Ω × (0, T),
in
Ω × (0, T),
on 𝜕Ω × (0, T), on 𝜕Ω × (0, T), in Ω, on 𝜕Ω × {T}.
(3)
If the problem (1) w
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