Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics


Authors: Max D. Gunzburger, Amnon J. Meir and Janet S. Peterson
Journal: Math. Comp. 56 (1991), 523-563
MSC: Primary 76W05; Secondary 35Q99, 65N30, 76M10
MathSciNet review: 1066834
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the equations of stationary, incompressible magnetohydrodynamics posed in a bounded domain in three dimensions and treat the full, coupled system of equations with inhomogeneous boundary conditions. Under certain conditions on the data, we show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. We discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.


References [Enhancements On Off] (What's this?)

  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] E. V. Chizhonkov, A system of equations of magnetohydrodynamics type, Dokl. Akad. Nauk SSSR 278 (1984), no. 5, 1074–1077 (Russian). MR 765617
  • [3] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [4] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262
  • [5] George J. Fix, Max D. Gunzburger, and Janet S. Peterson, On finite element approximations of problems having inhomogeneous essential boundary conditions, Comput. Math. Appl. 9 (1983), no. 5, 687–700. MR 726817, 10.1016/0898-1221(83)90126-8
  • [6] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
  • [7] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • [8] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • [9] Max D. Gunzburger and Janet S. Peterson, On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations, Numer. Math. 42 (1983), no. 2, 173–194. MR 720658, 10.1007/BF01395310
  • [10] W. F. Hughes and F. J. Young, The electromagnetodynamics of fluids, Wiley, New York, 1966.
  • [11] John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
  • [12] Ohannes A. Karakashian, On a Galerkin-Lagrange multiplier method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal. 19 (1982), no. 5, 909–923. MR 672567, 10.1137/0719066
  • [13] R. A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems, SIAM J. Numer. Anal. 19 (1982), no. 2, 349–357. MR 650055, 10.1137/0719021
  • [14] Janet S. Peterson, On the finite element approximation of incompressible flows of an electrically conducting fluid, Numer. Methods Partial Differential Equations 4 (1988), no. 1, 57–68. MR 1012474, 10.1002/num.1690040105
  • [15] J. A. Shercliff, A textbook of magnetohydrodyamics, Pergamon Press, Oxford-New York-Paris, 1965. MR 0185961
  • [16] Rolf Stenberg, On some three-dimensional finite elements for incompressible media, Comput. Methods Appl. Mech. Engrg. 63 (1987), no. 3, 261–269. MR 911612, 10.1016/0045-7825(87)90072-7
  • [17] Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
  • [18] J. S. Walker, Large interaction parameter magnetohydrodynamics and applications in fusion reactor technology, Fluid Mechanics in Energy Conversion (J. Buckmaster, ed.), SIAM, Philadelphia, 1980.
  • [19] J. S. Walker, Three dimensional MHD flows in rectangular ducts with thin conducting walls and strong transverse nonuniform magnetic fields, Liquid-Metal Flows and Magnetohydrodynamics (H. Branover, P. S. Lykoudis, and A. Yakhot, eds.), AIAA, New York, 1983.
  • [20] N. S. Winowich, Magnetofluidynamic channel flow with a nonuniform magnetic field and conductive walls, Ph. D. thesis, Carnegie Mellon University, Pittsburgh, 1986.
  • [21] N. S. Winowich and W. Hughes, A finite element analysis of two dimensional mhd flow, Liquid-Metal Flows and Magnetohydrodynamics (H. Branover, P. S. Lykoudis, and A. Yakhot, eds.), AIAA, New York, 1983.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 76W05, 35Q99, 65N30, 76M10

Retrieve articles in all journals with MSC: 76W05, 35Q99, 65N30, 76M10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1066834-0
Article copyright: © Copyright 1991 American Mathematical Society