Well-Posedness and Finite Element Approximation for the Stationary Magneto-Hydrodynamics Problem with Temperature-Depend
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Well-Posedness and Finite Element Approximation for the Stationary Magneto-Hydrodynamics Problem with Temperature-Dependent Parameters Hailong Qiu1 Received: 27 March 2020 / Revised: 28 October 2020 / Accepted: 3 November 2020 / Published online: 20 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this article we study a well-posedness and finite element approximation for the nonisothermal incompressible magneto-hydrodynamics flow subject to a generalized Boussinesq problem with temperature-dependent parameters. Applying some similar hypotheses in Oyar´zua et al. (IMA J Numer Anal 34:1104–1135, 2014), we prove the existence and uniqueness of weak solutions and discrete weak solutions, and derive optimal error estimates for small and smooth solutions. Finally, we provide some numerical results to confirm the rates of convergence. Keywords Incompressible magneto-hydrodynamics equations · Generalized Boussinesq problem · Well-posedness · Mixed finite element · Stability · Error estimations Mathematics Subject Classification 65N30 · 76M10 · 76W05
1 Introduction Magneto-hydrodynamics system mainly studies the dynamics of electrically conducting fluids and these magneto-hydrodynamics flows are governed by the Navier–Stokes equations and coupled with the Maxwell equations. It is widely applied in many science and engineering applications such as the design of cooling systems with liquid metals for a nuclear reactor, magneto-hydrodynamics generators, accelerators [4,19]. The complexities and the multi-physical variables needed for magneto-hydrodynamics system make efficient numerical simulation of magneto-hydrodynamics system challenging problem. In this article, we consider the stationary non-isothermal incompressible magnetohydrodynamics flow subject to a generalized Boussinesq problem with temperature-
This work is supported by the Natural Science Foundation of China (11701498, 11801492, 61877052).
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Hailong Qiu [email protected] School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China
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Journal of Scientific Computing (2020) 85:58
dependent parameters as follows: −div(2ν(θ )D(u)) + (u · ∇)u − Sc curlB × B +∇ p − α(θ )θ = f, in ,
(1a)
Sc curl(μ(θ )curlB) − Sc curl(u × B) = curl J, in ,
(1b)
divu = 0, in ,
(1c)
divB = 0, in ,
(1d)
−div(κ(θ )∇θ ) + (u · ∇)θ = g, in ,
(1e)
with following homogeneous boundary conditions: u|∂ = 0,
on ∂,
B · n|∂ = 0, n × curl B|∂ = 0, on ∂, ∂θ = 0, on ∂1 , θ = 0, on ∂2 , ∂n
(2a) (2b) (2c)
where ⊂ R d (d = 2, 3) is a polyhedral domain with boundary ∂ = ∂1 ∪ ∂2 , u, p, B and θ are the fluid velocity, the fluid pressure, the fluid magnetic and the fluid temperature, respectively. Functions f and g stand for the external force and the heat source, and function J denote the known applied current with n × J|∂ = 0. The functions ν(·), α(·), μ(·) and κ(·) are the fluid viscous diffusivity, the thermal expansion coefficient, the magnetic diffusivity and the therma
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