Skip to main content
Erschienen in: Journal of Scientific Computing 3/2017

03.09.2016

Two-Level Penalty Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics Equations

verfasst von: Haiyan Su, Xinlong Feng, Jianping Zhao

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2017

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This work is concerned with the study of two-level penalty finite element method for the 2D/3D stationary incompressible magnetohydrodynamics equations. The new method is an interesting combination of the Newton iteration and two-level penalty finite element algorithm with two different finite element pairs \(P_{1}b\)-\(P_{1}\)-\(P_{1}b\) and \(P_{1}\)-\(P_{0}\)-\(P_{1}\). Moreover, the rigorous analysis of stability and error estimate for the proposed method are given. Numerical results verify the theoretical results and show the applicability and effectiveness of the presented scheme.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Gunzburger, M., Meir, A., Peterson, J.: On the existence, uniquess and finite element approximation of solutions of the equations of sationary, incompressible magnetohydrodynamic. Math. Comput. 56, 523–563 (1991)CrossRefMATH Gunzburger, M., Meir, A., Peterson, J.: On the existence, uniquess and finite element approximation of solutions of the equations of sationary, incompressible magnetohydrodynamic. Math. Comput. 56, 523–563 (1991)CrossRefMATH
2.
Zurück zum Zitat Moreau, R.: Magneto-hydrodynamics. Kluwer Academic Publishers, Dordrecht (1990) Moreau, R.: Magneto-hydrodynamics. Kluwer Academic Publishers, Dordrecht (1990)
4.
Zurück zum Zitat He, C., Wang, Y.: On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 238, 1–17 (2007)MathSciNetCrossRefMATH He, C., Wang, Y.: On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 238, 1–17 (2007)MathSciNetCrossRefMATH
5.
Zurück zum Zitat Schonbek, M., Schonbek, T., Solli, E.: Large-time behaviour of solutions to the magnetohydrodynamics equations. Math. Ann. 304, 717–756 (1996)MathSciNetCrossRefMATH Schonbek, M., Schonbek, T., Solli, E.: Large-time behaviour of solutions to the magnetohydrodynamics equations. Math. Ann. 304, 717–756 (1996)MathSciNetCrossRefMATH
6.
Zurück zum Zitat Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)MathSciNetCrossRefMATH Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)MathSciNetCrossRefMATH
7.
Zurück zum Zitat Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)MathSciNetCrossRefMATH Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)MathSciNetCrossRefMATH
8.
Zurück zum Zitat Codina, R., Silva, N.: Stabilized finite element approximation of the stationary magnetohydrodynamics equations. Comput. Mech. 38, 344–355 (2006)MathSciNetCrossRefMATH Codina, R., Silva, N.: Stabilized finite element approximation of the stationary magnetohydrodynamics equations. Comput. Mech. 38, 344–355 (2006)MathSciNetCrossRefMATH
9.
Zurück zum Zitat Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)MathSciNetCrossRefMATH Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)MathSciNetCrossRefMATH
10.
Zurück zum Zitat Gerbeau, J.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000)MathSciNetCrossRefMATH Gerbeau, J.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000)MathSciNetCrossRefMATH
11.
Zurück zum Zitat Ravindran, S.: Linear feedback control and approximation for a system governed by unsteady MHD equations. Comput. Methods Appl. Mech. Eng. 198, 524–541 (2008)MathSciNetCrossRefMATH Ravindran, S.: Linear feedback control and approximation for a system governed by unsteady MHD equations. Comput. Methods Appl. Mech. Eng. 198, 524–541 (2008)MathSciNetCrossRefMATH
12.
Zurück zum Zitat Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)MathSciNetCrossRefMATH Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)MathSciNetCrossRefMATH
13.
Zurück zum Zitat Meir, A., Schmidt, P.: Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary. SIAM J. Numer. Anal. 36, 1304–1332 (1999)MathSciNetCrossRefMATH Meir, A., Schmidt, P.: Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary. SIAM J. Numer. Anal. 36, 1304–1332 (1999)MathSciNetCrossRefMATH
14.
Zurück zum Zitat He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351–1359 (2009)MathSciNetCrossRefMATH He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351–1359 (2009)MathSciNetCrossRefMATH
15.
Zurück zum Zitat Xu, H., He, Y.: Some iterative finite element methods for steady Navier–Stokes equations with different viscosities. J. Comput. Phys. 232, 123–152 (2013)MathSciNetCrossRefMATH Xu, H., He, Y.: Some iterative finite element methods for steady Navier–Stokes equations with different viscosities. J. Comput. Phys. 232, 123–152 (2013)MathSciNetCrossRefMATH
16.
Zurück zum Zitat Dong, X., He, Y., 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)MathSciNetCrossRef Dong, X., He, Y., 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)MathSciNetCrossRef
17.
Zurück zum Zitat Dong, X., He, Y.: Two-level Newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics. J. Sci. Comput. 63, 426–451 (2015)MathSciNetCrossRefMATH Dong, X., He, Y.: Two-level Newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics. J. Sci. Comput. 63, 426–451 (2015)MathSciNetCrossRefMATH
18.
Zurück zum Zitat Su, H., Feng, X., Huang, P.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Methods Appl. Mech. Eng. 304, 521–545 (2016)MathSciNetCrossRef Su, H., Feng, X., Huang, P.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Methods Appl. Mech. Eng. 304, 521–545 (2016)MathSciNetCrossRef
19.
Zurück zum Zitat Shen, J.: On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations. Numer. Math. 62, 49–73 (1992)MathSciNetCrossRefMATH Shen, J.: On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations. Numer. Math. 62, 49–73 (1992)MathSciNetCrossRefMATH
20.
Zurück zum Zitat Shen, J.: On error estimates of the penalty method for unsteady Navier–Stokes equations. SIAM J. Numer. Anal. 32, 386–403 (1995)MathSciNetCrossRefMATH Shen, J.: On error estimates of the penalty method for unsteady Navier–Stokes equations. SIAM J. Numer. Anal. 32, 386–403 (1995)MathSciNetCrossRefMATH
21.
Zurück zum Zitat He, Y.: Optimal error estimate of the penalty finite element method for the time-dependent Navier–Stokes equations. Math. Comput. 74, 1201–1216 (2005)MathSciNetCrossRefMATH He, Y.: Optimal error estimate of the penalty finite element method for the time-dependent Navier–Stokes equations. Math. Comput. 74, 1201–1216 (2005)MathSciNetCrossRefMATH
22.
Zurück zum Zitat He, Y., Li, J., Yang, X.: Two-level penalized finite element methods for the stationary Navier–Stoke equations. Int. J. Inf. Syst. Sci. 2, 131–143 (2006)MathSciNetMATH He, Y., Li, J., Yang, X.: Two-level penalized finite element methods for the stationary Navier–Stoke equations. Int. J. Inf. Syst. Sci. 2, 131–143 (2006)MathSciNetMATH
24.
25.
Zurück zum Zitat Layton, W., Lenferink, H., Peterson, J.: A two-level Newton finite element algorithm for approximating electrically conducting incompressible fluid flows. Comput. Math. Appl. 28, 21–31 (1994)MathSciNetCrossRefMATH Layton, W., Lenferink, H., Peterson, J.: A two-level Newton finite element algorithm for approximating electrically conducting incompressible fluid flows. Comput. Math. Appl. 28, 21–31 (1994)MathSciNetCrossRefMATH
26.
Zurück zum Zitat Layton, W., Meir, A., Schmidtz, P.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198–210 (1997)MathSciNetMATH Layton, W., Meir, A., Schmidtz, P.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198–210 (1997)MathSciNetMATH
27.
Zurück zum Zitat Zhang, G., Zhang, Y., He, Y.: Two-level coupled and decoupled parallel correction methods for stationary incompressible magnetohydrodynamics. J. Sci. Comput. 65, 920–939 (2015)MathSciNetCrossRefMATH Zhang, G., Zhang, Y., He, Y.: Two-level coupled and decoupled parallel correction methods for stationary incompressible magnetohydrodynamics. J. Sci. Comput. 65, 920–939 (2015)MathSciNetCrossRefMATH
28.
Zurück zum Zitat He, Y.: Two-level method based on fnite element and Crank–Nicolson extrapolation for the time-dependent Navier–Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)MathSciNetCrossRefMATH He, Y.: Two-level method based on fnite element and Crank–Nicolson extrapolation for the time-dependent Navier–Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)MathSciNetCrossRefMATH
29.
Zurück zum Zitat Gerbeau, J., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)CrossRefMATH Gerbeau, J., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)CrossRefMATH
30.
Zurück zum Zitat He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2014)CrossRefMATH He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2014)CrossRefMATH
Metadaten
Titel
Two-Level Penalty Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics Equations
verfasst von
Haiyan Su
Xinlong Feng
Jianping Zhao
Publikationsdatum
03.09.2016
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-016-0276-8

Weitere Artikel der Ausgabe 3/2017

Journal of Scientific Computing 3/2017 Zur Ausgabe