Citation: | Jingke Wu, Rong An, Yuan Li. OPTIMAL $ H^1 $ ERROR ANALYSIS OF A FRACTIONAL STEP FINITE ELEMENT SCHEME FOR A HYBRID MHD SYSTEM[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1535-1556. doi: 10.11948/20200277 |
This paper presents a fractional step finite element scheme for a hybrid MHD system coupled by the nonstationary Navier-Stokes equations and the steady Maxwell equations, which can be viewed that the magnetic phenomena reach their steady state "infinitely" faster than the fluid hydrodynamics phenomena. The proposed fractional step scheme has the following features: the first one is that the proposed scheme is a decoupled scheme, which means the magnetic field and velocity field can be solved independently at the same time discrete level. The second one is that the nonlinearity and the divergence-free of the Navier-Stokes equations are splitted by introducing an intermediate velocity field. We focus on a rigorous error analysis and obtain the optimal $ {\textbf{H}}^1 $ convergence order $ \mathcal O(\Delta t+h) $ for the magnetic and the velocity under the time step condition $ \Delta t = \mathcal O (h) $, where $ h $ is the mesh size. Finally, numerical results are shown to illustrate the theoretical convergence analysis.
[1] | R. An, Error analysis of a new fractional-step method for the incompressible Navier-Stokes equations with variable density, J. Sci. Comput., 2020, 84, article number: 3. |
[2] | R. An and Y. Li, Error analysis of first-order projection method for time-dependent magnetohydrodynamics equations, Appl. Numer. Math., 2017, 112, 167-181. doi: 10.1016/j.apnum.2016.10.010 |
[3] | R. An and C. Zhou, Error analysis of a fractional-step method for magnetohydrodynamics equations, J. Comput. Appl. Math., 2017, 313, 168-184. doi: 10.1016/j.cam.2016.09.005 |
[4] | J. Blasco and R. Codina, Error estimates for an operator-splitting method for incompressible flows, Appl. Numer. Math., 2004, 51, 1-17. doi: 10.1016/j.apnum.2004.02.004 |
[5] | H. Gao and W. Qiu, A semi-implict energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations, Comput. Methods Appl. Mech. Engrg., 2019, 346, 982-1001. doi: 10.1016/j.cma.2018.09.037 |
[6] | J. Gerbeau, A stabilized finite elemenet method for the incompressible magnetohydrodynamic equations, Numer. Math., 2000, 87, 83-111. doi: 10.1007/s002110000193 |
[7] | J. Gerbeau and C. Le Bris, Mathematical study of a coupled system arising in magnetohydrodynamics, Technical Report CERMICS. |
[8] | J. Gerbeau and C. Le Bris, A coupled system arising in magnetohydrodynamics, Appl. Math. Lett., 1999, 12, 53-57. |
[9] | J. Gerbeau and C. Le Bris, T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006. |
[10] | F. Guillén-González and M. V. Dedondo-Neble, Spatial error estimates for a finite element viscosity-splitting scheme for the Navier-Stokes equations, Inter. J. Numer. Anal. Model., 2013, 10, 826-844. |
[11] | M. Gunzburger, A. Meir and J. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompresible magnetohydrodynamics, Math. Comp., 1991, 56, 523-563. doi: 10.1090/S0025-5718-1991-1066834-0 |
[12] | Y. He, Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations, IMA J. Numer. Anal., 2015, 35, 767-801. doi: 10.1093/imanum/dru015 |
[13] | J. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem Part IV: error analysis for second-order time discretization, SIAM J. Numer. Anal., 1990, 27, 353-384. doi: 10.1137/0727022 |
[14] | Y. Li, Y. Ma and R. An, Decoupled, semi-implicit scheme for a coupled system arising in magnetohydrodynamics problem, Appl. Numer. Math., 2018, 127, 142-163. doi: 10.1016/j.apnum.2018.01.005 |
[15] | Y. Li and X. Luo, Second-order semi-implicit Crank-Nicolson scheme for a coupled magnetohydrodynamics system, Appl. Numer. Math., 2019, 145, 48-68. doi: 10.1016/j.apnum.2019.06.001 |
[16] | A. Prohl, Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamic system, ESAIM: M2AN, 2008, 42, 1065-1087. doi: 10.1051/m2an:2008034 |
[17] | D. Schötzau, Mixed finite element methods for stationary incompressible magnetohydrodynamics, Numer. Math., 2004, 96, 771-800. doi: 10.1007/s00211-003-0487-4 |
[18] | R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977. |
[19] | X. Yang, G. Zhang and X. He, Convergence analysis of an Unconditionally energy stable projection scheme for magnetohydrodynamic equations, Appl. Numer. Math., 2019, 136, 235-256. doi: 10.1016/j.apnum.2018.10.013 |
[20] | G. Zhang, X. He and X. Yang, A decoupled, linear and unconditionally energy stable scheme with finite element discretizations for magnetohydrodynamic equations, J. Sci. Comput., 2019, 81, 1678-1711. doi: 10.1007/s10915-019-01059-1 |
[21] | G. Zhang, X. He and X. Yang, Fully decoupled, linear and unconditionally energy stable time discretization for solving the magnetohydrodynamic equations, J. Comput. Appl. Math., 2020, 369, 112636. doi: 10.1016/j.cam.2019.112636 |
[22] | G. Zhang and Y. He, Decoupled schemes for unsteady MHD equations Ⅱ: finite element spatial discretization and numerical implementation, Comput. Math. Appl., 2015, 69, 1390-1406. doi: 10.1016/j.camwa.2015.03.019 |
[23] | Y. Zhang, Y. Hou and L. Shan, Numerical analysis of the Crank-Nicolson extrapolation time discrete scheme for magnetohydrodynamics flows, Numer. Methods Part. Diff. Equa., 2015, 31, 2169-2208. doi: 10.1002/num.21989 |