On approximation of the Neumann problem by the penalty method

Michal Křížek

Applications of Mathematics (1993)

  • Volume: 38, Issue: 6, page 459-469
  • ISSN: 0862-7940

Abstract

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We prove that penalization of constraints occuring in the linear elliptic Neumann problem yields directly the exact solution for an arbitrary set of penalty parameters. In this case there is a continuum of Lagrange's multipliers. The proposed penalty method is applied to calculate the magnetic field in the window of a transformer.

How to cite

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Křížek, Michal. "On approximation of the Neumann problem by the penalty method." Applications of Mathematics 38.6 (1993): 459-469. <http://eudml.org/doc/15766>.

@article{Křížek1993,
abstract = {We prove that penalization of constraints occuring in the linear elliptic Neumann problem yields directly the exact solution for an arbitrary set of penalty parameters. In this case there is a continuum of Lagrange's multipliers. The proposed penalty method is applied to calculate the magnetic field in the window of a transformer.},
author = {Křížek, Michal},
journal = {Applications of Mathematics},
keywords = {Neumann problem; penalty method; finite elements; magnetic field; linear elliptic Neumann problem; Lagrange’s multipliers; linear elliptic Neumann problem; Lagrange's multipliers; penalty method; magnetic field},
language = {eng},
number = {6},
pages = {459-469},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On approximation of the Neumann problem by the penalty method},
url = {http://eudml.org/doc/15766},
volume = {38},
year = {1993},
}

TY - JOUR
AU - Křížek, Michal
TI - On approximation of the Neumann problem by the penalty method
JO - Applications of Mathematics
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 38
IS - 6
SP - 459
EP - 469
AB - We prove that penalization of constraints occuring in the linear elliptic Neumann problem yields directly the exact solution for an arbitrary set of penalty parameters. In this case there is a continuum of Lagrange's multipliers. The proposed penalty method is applied to calculate the magnetic field in the window of a transformer.
LA - eng
KW - Neumann problem; penalty method; finite elements; magnetic field; linear elliptic Neumann problem; Lagrange’s multipliers; linear elliptic Neumann problem; Lagrange's multipliers; penalty method; magnetic field
UR - http://eudml.org/doc/15766
ER -

References

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  7. M. Křížek W. G. Litvinov, On the methods of penalty functions and Lagrange's multipliers in the abstract Neumann problem, Z. Angew. Math. Mech. (1993). (1993) 
  8. M. Křížek Z. Milka, On a nonconventional variational method for solving the problem of linear elasticity with Neumann or periodic boundary conditions, Banach Center Publ. (1993). (1993) MR1272920
  9. M. Křížek P. Neittaanmäki M. Vondrák, A nontraditional approach for solving the Neumann problem by the finite element method, Mat. Apl. Comput. 11 (1992), 31-40. (1992) MR1185236
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  11. J. Nečas I. Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Elsevier, Amsterdam, 1981. (1981) MR0600655
  12. B. N. Pšeničnyj, Ju. M. Danilin, Numerical methods in extremum problems, (Russian), Nauka, Moscow, 1975. (1975) MR0474817
  13. L. Schwartz, Analyse mathématique, Vol. 1, Hermann, Paris, 1967. (1967) 
  14. A. E. Taylor, Introduction to functional analysis, John Wiley & Sons, New York, 1958. (1958) Zbl0081.10202MR0098966
  15. R. Temam, Navier-Stokes equations, North-Holland, Amsterdam, 3rd revised edn, 1984. (1984) Zbl0568.35002
  16. D. E. Ward, 10.1007/BF02346165, Optim. Theory Appl. 57 (1988), 485-499. (1988) Zbl0621.90081MR0944591DOI10.1007/BF02346165

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