Abstract
In this article we study a well-posedness and finite element approximation for the non-isothermal incompressible magneto-hydrodynamics flow subject to a generalized Boussinesq problem with temperature-dependent parameters. Applying some similar hypotheses in Oyarźua 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.
Similar content being viewed by others
References
Armero, F., Simo, J.: Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 131, 41–90 (1996)
Badia, S., Codina, R., Planas, R.: Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics. Arch. Comput. Methods Eng. 22, 621–636 (2015)
Davis, T.A.: Algorithm 832: UMFPACK V4.3-an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 196–199 (2004)
Davidson, P.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)
Cimatti, G.: A plane problem of incompressible magneto-hydrodynamics with viscosity and resistivity depending on the temperature. Rend. Mat. Acc. Lincei 15, 137–146 (2004)
Farhloul, M., Zine, A.: A dual mixed formulation for non-isothermal Oldroyd–Stokes problem. Math. Model. Nat. Phenom. 6, 130–156 (2011)
Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations, Series in Computational Mathematics. Springer, New York (1986)
Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence and uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comput. 56, 523–563 (1991)
Getling, A.V.: Rayleigh-Benard Convection: Structures and Dynamics. World Scientific, Singapore (1998)
Gerbeau, J.F.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000)
Gerbeau, J.F., Le Bris, C., Lelièvre, T.: Mathematical methods for the magnetohydrodynamics of liquid metals. In: Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)
Greif, C., Li, D., Schötzau, D., Wei, X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 199, 2840–2855 (2010)
Heywood, J.G.: The Navier–Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)
Hasler, U., Schneebeli, A., Schötzau, D.: Mixed finite element approximation of incompressible MHD problems based on weighted regularization. Appl. Numer. Math. 51, 19–45 (2004)
Hecht, F.: New development in freefem++. J. Numer. Math. 20, 251–265 (2012)
He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2015)
Lorca, S.A., Boldrini, J.L.: Stationary solutions for generalized Boussinesq models. J. Differ. Equ. 124, 389–406 (1996)
Layton, W., Tran, W., Trenchea, H.: Stability of partitioned methods for magnetohydrodynamics flows at small magnetic Reynolds number. Contemp. Math. 586, 231–238 (2013)
Moreau, R.: Magneto-Hydrodynamics. Kluwer Academic Publishers, Dordrecht (1990)
Meir, A.J.: Thermally coupled, stationary, incompressible MHD flow, existence, uniqueness, and finite element approximation. Numer. Methods Partial Differ. Equ. 11, 311–337 (1995)
Meir, A.J., Schmidt, P.G.: Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary. SIAM J. Numer. Anal. 36, 1304–1332 (1999)
Ni, L., Roussev, I.I., Lin, J., Ziegler, U.: Impact of temperature-dependent resistivity and thermal conduction on plasmoid instabilities in current sheets in the solar corona. Astrophys. J. 758, 1–11 (2012)
Oyarźua, R., Qin, T., Schötzau, D.: An exactly divergence-free finite element method for a generalized Boussinesq problem. IMA J. Numer. Anal. 34, 1104–1135 (2014)
Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamic system. Math. Model. Numer. Anal. 42, 1065–1087 (2008)
Ravindran, S.S.: Partitioned time-stepping scheme for an MHD system with temperature-dependent coefficients. IMA J. Numer. Anal. 39, 1860–1887 (2018)
Schötzau, D.: Mixed finite element methods for incompressible magnetohydrodynamics. Numer. Math. 96, 771–800 (2004)
Temam, R.: Navier–Stokes Equations, Theory and Numerical, 3rd edn. North-Holland, Amsterdam (1983)
Tabata, M., Tagami, D.: Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients. Numer. Math. 100, 351–372 (2005)
Wiedmer, M.: Finite element approximation for equations of magnetohydrodynamics. Math. Comp. 69, 83–101 (2000)
Wang, D.: Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM J. Appl. Math. 63, 1424–1441 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the Natural Science Foundation of China (11701498, 11801492, 61877052).
Rights and permissions
About this article
Cite this article
Qiu, H. Well-Posedness and Finite Element Approximation for the Stationary Magneto-Hydrodynamics Problem with Temperature-Dependent Parameters. J Sci Comput 85, 58 (2020). https://doi.org/10.1007/s10915-020-01361-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01361-3
Keywords
- Incompressible magneto-hydrodynamics equations
- Generalized Boussinesq problem
- Well-posedness
- Mixed finite element
- Stability
- Error estimations