Abstract
In this work, we are concerned with the local and parallel finite element algorithm based on the Oseen-type iteration for solving the stationary incompressible magnetohydrodynamics. Under the uniqueness condition, the error estimates with respect to iterative step m and small mesh sizes H and \(h\ll H\) of the proposed method are derived. Finally, some numerical experiments are provided to show the high efficiency of our algorithm.
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Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Math. Comput. 56, 523–563 (1991)
Moreau, R.: Magneto-hydrodynamics. Kluwer Academic Publishers, Dordrecht (1990)
Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)
Schötzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004)
Badia, S., Codina, R., Planas, R.: On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics. J. Comput. Phys. 234, 399–416 (2013)
Li, F.Y., Xu, L.W.: Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations. J. Comput. Phys. 231, 2655–2675 (2012)
Li, F.Y., Xu, L.W., Yakovlev, S.: Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field. J. Comput. Phys. 230, 4828–4847 (2011)
Dong, X.J., He, Y.N.: Two-level newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics. J. Sci. Comput. 63, 426–451 (2015)
Dong, X.J., He, Y.N.: Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D/3D stationary incompressible magnetohydrodynamics. Sci. China Math. 59, 589–608 (2016)
Layton, W.J., Meir, A.J., Schmidt, P.G.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198–210 (1997)
Salah, N.B., Soulaimani, A., Habashi, W.G.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)
Gerbeau, J.-F., Bris, C.L., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)
Yuksel, G., Ingram, R.: Numerical analysis of a finite element, Crank–Nicolson discretization for MHD flows at small magnetic Reynolds numbers. Int. J. Numer. Anal. Model. 10, 74–98 (2013)
Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. M2AN. Math. Model. Numer. Anal. 42, 1065–1087 (2008)
Huang, Y.Q., Shi, Z.C., Tang, T., Xue, W.M.: A multilevel successive iteration method for nonlinear elliptic problems. Math. Comput. 73, 525–539 (2004)
Huang, Y.Q., Chen, Y.P.: A multi-level iterative method for solving finite element equations of nonlinear singular two-point boundary value problems. Nat. Sci. J. Xiangtan Univ. 16, 23–26 (1994)
Huang, Y.Q., Xue, W.M.: Convergence of finite element approximations and multilevel linearization for Ginzburg–Landau model of \(d\)-wave superconductors. Adv. Comput. Math. 17, 309–330 (2002)
Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881–909 (2000)
He, Y.N., Xu, J.C., Zhou, A.H., Li, J.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109, 415–434 (2008)
Shang, Y.Q., He, Y.N.: A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 209, 172–183 (2012)
He, Y.N., Mei, L.Q., Shang, Y.Q., Cui, J.: Newton iterative parallel finite element algorithm for the steady Navier–Stokes equations. J. Sci. Comput. 44, 92–106 (2010)
Xu, J.C.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)
Chen, Y.P., Liu, H.W., Liu, S.: Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods. Int. J. Numer. Methods Eng. 69, 408–422 (2007)
Huang, Y.Q., Kornhuber, R., Widlund, O., Xu, J.C.: Domain Decomposition Methods in Sicience and Engineering XIX. Springer, Berlin (2011)
Huang, Y.Q., Xu, J.C.: A conforming finite element method for overlapping and nonmatching grids. Math. Comput. 72, 1057–1066 (2002)
Dong, X.J., He, Y.N., Zhang, Y.: Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 276, 287–311 (2014)
Hasler, U., Schneebeli, A., Schötzau, D.: Mixed finite element approximation of incompressible MHD problems based on weighted regularization. Appl. Numer. Math. 51, 19–25 (2004)
Sermane, M., Temam, R.: Some mathematics questions related to the MHD equations. Commun. Pure Appl. Math. XXXIV, 635–664 (1984)
Brenner, S.C., Cui, J., Li, F.Y., Sung, L.-Y.: A nonconforming finite element method for a two-dimensional curl–curl and grad–div problem. Numer. Math. 109, 509–533 (2008)
Girault, V., Raviart, P.A.: Finite Element Approximation of Navier–Stokes Equations. Springer, Berlin (1986)
Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21, 337–344 (1984)
Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem I: regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)
He, Y.N.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2015)
He, Y.N., Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for the Navier–Stokes problem. J. Comput. Math. 24, 227–238 (2006)
Hecht, F., Pironneau, O., Hyaric, A., Ohtsuka, K.: http://www.freefem.org
Acknowledgments
The authors sincerely thank the reviewers and editor for their helpful suggestions. The first author is supported by NSFC (No. 11401174). The second author is supported by the Major Research Plan of NSFC (No. 91430213).
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Tang, Q., Huang, Y. Local and Parallel Finite Element Algorithm Based on Oseen-Type Iteration for the Stationary Incompressible MHD Flow. J Sci Comput 70, 149–174 (2017). https://doi.org/10.1007/s10915-016-0246-1
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DOI: https://doi.org/10.1007/s10915-016-0246-1
Keywords
- Local and parallel algorithm
- Finite element
- Oseen iteration
- Stationary incompressible magnetohydrodynamics