| Title:
|
On approximation of the Neumann problem by the penalty method (English) |
| Author:
|
Křížek, Michal |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
38 |
| Issue:
|
6 |
| Year:
|
1993 |
| Pages:
|
459-469 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Neumann problem |
| Keyword:
|
penalty method |
| Keyword:
|
finite elements |
| Keyword:
|
magnetic field |
| Keyword:
|
linear elliptic Neumann problem |
| Keyword:
|
Lagrange’s multipliers |
| MSC:
|
35J05 |
| MSC:
|
35J25 |
| MSC:
|
35J50 |
| MSC:
|
35Q60 |
| MSC:
|
65N30 |
| MSC:
|
78-08 |
| MSC:
|
78A25 |
| idZBL:
|
Zbl 0795.65075 |
| idMR:
|
MR1241449 |
| DOI:
|
10.21136/AM.1993.104568 |
| . |
| Date available:
|
2008-05-20T18:46:33Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104568 |
| . |
| Reference:
|
[1] I. Babuška: Uncertainties in engineering design: mathematical theory and numerical experience.In the Optimal Shape, (J. Bennet and M. M. Botkin eds.), Plenum Press (1986), also in Technical Note BN-1044, Univ. of Maryland (1985), 1-35. |
| Reference:
|
[2] J. Céa: Optimization, théorie et algorithmes.Dunod, Paris, 1971. MR 0298892 |
| Reference:
|
[3] P. G. Ciarlet: The finite element method for elliptic problems.North-Holland, Amsterdam, 1978. Zbl 0383.65058, MR 0520174 |
| Reference:
|
[4] I. Doležel: Numerical calculation of the leakage field in the window of a transformer with magnetic shielding.Acta Tech. ČSAV (1981), 563-588. |
| Reference:
|
[5] M. Feistauer: Mathematical methods in fluid dynamics.Longman Scientific & Technical, Harlow, 1993. Zbl 0819.76001 |
| Reference:
|
[6] I. Hlaváček J. Nečas: On inequalities of Korn's type.Arch. Rational Mech. Anal. 36 (1970), 305-334. MR 0252844, 10.1007/BF00249518 |
| Reference:
|
[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). |
| Reference:
|
[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). MR 1272920 |
| Reference:
|
[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. MR 1185236 |
| Reference:
|
[10] W. G. Litvinov: Optimization in elliptic boundary value problems with applications to mechanics.(in Russian), Nauka, Moscow, 1987. Zbl 0688.49003, MR 0898435 |
| Reference:
|
[11] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: an introduction.Elsevier, Amsterdam, 1981. MR 0600655 |
| Reference:
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[12] B. N. Pšeničnyj, Ju. M. Danilin: Numerical methods in extremum problems.(Russian), Nauka, Moscow, 1975. MR 0474817 |
| Reference:
|
[13] L. Schwartz: Analyse mathématique, Vol. 1.Hermann, Paris, 1967. |
| Reference:
|
[14] A. E. Taylor: Introduction to functional analysis.John Wiley & Sons, New York, 1958. Zbl 0081.10202, MR 0098966 |
| Reference:
|
[15] R. Temam: Navier-Stokes equations.North-Holland, Amsterdam, 3rd revised edn, 1984. Zbl 0568.35002 |
| Reference:
|
[16] D. E. Ward: Exact penalties and sufficient conditions for optimality in nonsmooth optimization.Optim. Theory Appl. 57 (1988), 485-499. Zbl 0621.90081, MR 0944591, 10.1007/BF02346165 |
| . |
