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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics
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by Max D. Gunzburger, Amnon J. Meir and Janet S. Peterson PDF
Math. Comp. 56 (1991), 523-563 Request permission

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
  • Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
  • E. V. Chizhonkov, A system of equations of magnetohydrodynamics type, Dokl. Akad. Nauk SSSR 278 (1984), no. 5, 1074–1077 (Russian). MR 765617
  • Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
  • G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Grundlehren der Mathematischen Wissenschaften, vol. 219, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John. MR 0521262
  • 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, DOI 10.1016/0898-1221(83)90126-8
  • 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
  • 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, DOI 10.1007/978-3-642-61623-5
  • P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • 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, DOI 10.1007/BF01395310
    • W. F. Hughes and F. J. Young, The electromagnetodynamics of fluids, Wiley, New York, 1966.
  • John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
  • 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, DOI 10.1137/0719066
  • R. A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems, SIAM J. Numer. Anal. 19 (1982), no. 2, 349–357. MR 650055, DOI 10.1137/0719021
  • 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, DOI 10.1002/num.1690040105
  • J. A. Shercliff, A textbook of magnetohydrodyamics, Pergamon Press, Oxford-New York-Paris, 1965. MR 0185961
  • Rolf Stenberg, On some three-dimensional finite elements for incompressible media, Comput. Methods Appl. Mech. Engrg. 63 (1987), no. 3, 261–269. MR 911612, DOI 10.1016/0045-7825(87)90072-7
  • 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
    • J. S. Walker, Large interaction parameter magnetohydrodynamics and applications in fusion reactor technology, Fluid Mechanics in Energy Conversion (J. Buckmaster, ed.), SIAM, Philadelphia, 1980. 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. N. S. Winowich, Magnetofluidynamic channel flow with a nonuniform magnetic field and conductive walls, Ph. D. thesis, Carnegie Mellon University, Pittsburgh, 1986. 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.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Math. Comp. 56 (1991), 523-563
  • MSC: Primary 76W05; Secondary 35Q99, 65N30, 76M10
  • DOI: https://doi.org/10.1090/S0025-5718-1991-1066834-0
  • MathSciNet review: 1066834