# Internal finite element approximations in the dual variational method for second order elliptic problems with curved boundaries

Aplikace matematiky (1984)

- Volume: 29, Issue: 1, page 52-69
- ISSN: 0862-7940

## Access Full Article

top## Abstract

top## How to cite

topHlaváček, Ivan, and Křížek, Michal. "Internal finite element approximations in the dual variational method for second order elliptic problems with curved boundaries." Aplikace matematiky 29.1 (1984): 52-69. <http://eudml.org/doc/15333>.

@article{Hlaváček1984,

abstract = {Using the stream function, some finite element subspaces of divergence-free vector functions, the normal components of which vanish on a part of the piecewise smooth boundary, are constructed. Applying these subspaces, an internal approximation of the dual problem for second order elliptic equations is defined.
A convergence of this method is proved without any assumption of a regularity of the solution. For sufficiently smooth solutions an optimal rate of convergence is proved. The internal approximation can be obtained by solving a system of linear algebraic equations with a positive definite matrix.},

author = {Hlaváček, Ivan, Křížek, Michal},

journal = {Aplikace matematiky},

keywords = {dual variational methods; stream function; finite element; piecewise smooth boundary; dual problem; optimal rate of convergence; dual variational methods; stream function; finite element; piecewise smooth boundary; dual problem; optimal rate of convergence},

language = {eng},

number = {1},

pages = {52-69},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Internal finite element approximations in the dual variational method for second order elliptic problems with curved boundaries},

url = {http://eudml.org/doc/15333},

volume = {29},

year = {1984},

}

TY - JOUR

AU - Hlaváček, Ivan

AU - Křížek, Michal

TI - Internal finite element approximations in the dual variational method for second order elliptic problems with curved boundaries

JO - Aplikace matematiky

PY - 1984

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 29

IS - 1

SP - 52

EP - 69

AB - Using the stream function, some finite element subspaces of divergence-free vector functions, the normal components of which vanish on a part of the piecewise smooth boundary, are constructed. Applying these subspaces, an internal approximation of the dual problem for second order elliptic equations is defined.
A convergence of this method is proved without any assumption of a regularity of the solution. For sufficiently smooth solutions an optimal rate of convergence is proved. The internal approximation can be obtained by solving a system of linear algebraic equations with a positive definite matrix.

LA - eng

KW - dual variational methods; stream function; finite element; piecewise smooth boundary; dual problem; optimal rate of convergence; dual variational methods; stream function; finite element; piecewise smooth boundary; dual problem; optimal rate of convergence

UR - http://eudml.org/doc/15333

ER -

## References

top- P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, New York, Oxford, 1978. (1978) Zbl0383.65058MR0520174
- P. G. Ciarlet P. A. Raviart, 10.1016/0045-7825(72)90006-0, Comput. Methods Appl. Mech. Engrg. 1 (1972), 217-249. (1972) MR0375801DOI10.1016/0045-7825(72)90006-0
- P. Doktor, On the density of smooth functions in certain subspaces of Sobolev spaces, Comment. Math. Univ. Carolin. 14, 4 (1973), 609-622. (1973) MR0336317
- B. M. Fraeijs de Veubeke M. Hogge, 10.1002/nme.1620050107, Internat. J. Numer. Methods Engrg. 5 (1972), 65-82. (1972) DOI10.1002/nme.1620050107
- V. Girault P. A. Raviart, Finite element approximation of the Navier-Stokes equations, Springer-Verlg, Berlin, Heidelberg, New York, 1979. (1979) MR0548867
- J. Haslinger I. Hlaváček, Contact between elastic perfectly plastic bodies, Apl. Mat. 27 (1982), 27-45. (1982) MR0640138
- J. Haslinger I. Hlaváček, Convergence of a finite element method based on the dual variational formulation, Apl. Mat. 21 (1976), 43 - 65. (1976) MR0398126
- I. Hlaváček, The density of solenoidal functions and the convergence of a dual finite element method, Apl. Mat. 25 (1980), 39-55. (1980) MR0554090
- M. Křížek, Conforming equilibrium finite element methods for some elliptic plane problems, RAIRO Anal. Numer. 17 (1983), 35--65. (1983) MR0695451
- O. A. Ladyzenskaya, The mathematical theory of viscous incompressible flow, Gordon & Breach, New York, 1969. (1969) MR0254401
- J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. (1967) MR0227584
- J. Nečas I. Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York, 1981. (1981) MR0600655
- P. Neittaanmäki J. Saranen, 10.1007/BF01400312, Numer. Math. 37 (1981), 333-337. (1981) MR0627107DOI10.1007/BF01400312
- J. Penman J. R. Fraser, 10.1109/TMAG.1982.1061883, IEEE Trans. on Magnetics 18 (1982), 319-324. (1982) DOI10.1109/TMAG.1982.1061883
- G. Strang G. J. Fix, An analysis of the finite element method, Prentice Hall, New Jersey, 1973. (1973) MR0443377
- J. M. Thomas, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes, Thesis, Université Paris VI, 1977. (1977)
- M. Zlámal, Curved elements in the finite element method, Čislennyje metody mechaniki splošnoj sredy, SO AN SSSR, 4 (1973), No. 5, 25-49. (1973)
- M. Zlámal, 10.1137/0710022, SIAM J. Numer. Anal. 10 (1973), 229-240. (1973) MR0395263DOI10.1137/0710022

## Citations in EuDML Documents

top- Sergey Korotov, On equilibrium finite elements in three-dimensional case
- Juraj Weisz, A posteriori error estimate of approximate solutions to a mildly nonlinear elliptic boundary value problem
- Ivan Hlaváček, Michal Křížek, Internal finite element approximation in the dual variational method for the biharmonic problem
- Van Bon Tran, Dual finite element analysis for contact problem of elastic bodies with an enlarging contact zone

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.