Nonhomogeneous boundary conditions and curved triangular finite elements

Alexander Ženíšek

Aplikace matematiky (1981)

  • Volume: 26, Issue: 2, page 121-141
  • ISSN: 0862-7940

Abstract

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Approximation of nonhomogeneous boundary conditions of Dirichlet and Neumann types is suggested in solving boundary value problems of elliptic equations by the finite element method. Curved triangular elements are considered. In the first part of the paper the convergence of the finite element method is analyzed in the case of nonhomogeneous Dirichlet problem for elliptic equations of order 2 m + 2 , in the second part of the paper in the case of nonhomogeneous mixed boundary value problem for second order elliptic equations. In both parts of the paper of numerical integration is studied.

How to cite

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Ženíšek, Alexander. "Nonhomogeneous boundary conditions and curved triangular finite elements." Aplikace matematiky 26.2 (1981): 121-141. <http://eudml.org/doc/15188>.

@article{Ženíšek1981,
abstract = {Approximation of nonhomogeneous boundary conditions of Dirichlet and Neumann types is suggested in solving boundary value problems of elliptic equations by the finite element method. Curved triangular elements are considered. In the first part of the paper the convergence of the finite element method is analyzed in the case of nonhomogeneous Dirichlet problem for elliptic equations of order $2m+2$, in the second part of the paper in the case of nonhomogeneous mixed boundary value problem for second order elliptic equations. In both parts of the paper of numerical integration is studied.},
author = {Ženíšek, Alexander},
journal = {Aplikace matematiky},
keywords = {nonhomogeneous boundary conditions; Dirichlet; Neumann; finite element method; curved triangular elements; convergence; nonhomogeneous boundary conditions; Dirichlet; Neumann; finite element method; curved triangular elements; convergence},
language = {eng},
number = {2},
pages = {121-141},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonhomogeneous boundary conditions and curved triangular finite elements},
url = {http://eudml.org/doc/15188},
volume = {26},
year = {1981},
}

TY - JOUR
AU - Ženíšek, Alexander
TI - Nonhomogeneous boundary conditions and curved triangular finite elements
JO - Aplikace matematiky
PY - 1981
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 26
IS - 2
SP - 121
EP - 141
AB - Approximation of nonhomogeneous boundary conditions of Dirichlet and Neumann types is suggested in solving boundary value problems of elliptic equations by the finite element method. Curved triangular elements are considered. In the first part of the paper the convergence of the finite element method is analyzed in the case of nonhomogeneous Dirichlet problem for elliptic equations of order $2m+2$, in the second part of the paper in the case of nonhomogeneous mixed boundary value problem for second order elliptic equations. In both parts of the paper of numerical integration is studied.
LA - eng
KW - nonhomogeneous boundary conditions; Dirichlet; Neumann; finite element method; curved triangular elements; convergence; nonhomogeneous boundary conditions; Dirichlet; Neumann; finite element method; curved triangular elements; convergence
UR - http://eudml.org/doc/15188
ER -

References

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  1. J. J. Blair, Higher order approximations to the boundary conditions for the finite element method, Math. Соmр. 30 (1976), 250-262. (1976) Zbl0342.65068MR0398123
  2. J. H. Bramble S. R. Hilbert, 10.1137/0707006, SIAM J. Numer. Anal. 7 (1970), 112-124. (1970) MR0263214DOI10.1137/0707006
  3. P. G. Ciarlet P. A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), pp. 409-474, Academic Press, New York 1972. (1972) MR0421108
  4. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam 1978. (1978) Zbl0383.65058MR0520174
  5. S. Koukal, Piecewise polynomial interpolations in the finite element method, Apl. Mat. 18 (1973), 146-160. (1973) MR0321318
  6. L. Mansfield, 10.1137/0715037, SIAM J. Numer. Anal. 15 (1978), 568-579. (1978) Zbl0391.65047MR0471373DOI10.1137/0715037
  7. J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague 1967. (1967) MR0227584
  8. R. Scott, 10.1137/0712032, SIAM J. Numer. Anal. 12 (1975), 404-427. (1975) Zbl0357.65082MR0386304DOI10.1137/0712032
  9. G. Strang, 10.1007/BF01395933, Numer. Math. 19 (1972), 81-98. (1972) Zbl0221.65174MR0305547DOI10.1007/BF01395933
  10. M. Zlámal, 10.1137/0710022, SIAM J. Numer. Anal. 10 (1973), 229-240. (1973) MR0395263DOI10.1137/0710022
  11. M. Zlámal, 10.1137/0711031, SIAM J. Numer. Anal. 11 (1974), 347-362. (1974) MR0343660DOI10.1137/0711031
  12. A. Ženíšek, Curved triangular finite C m -elements, Apl. Mat. 23 (1978), 346-377. (1978) MR0502072
  13. A. Ženíšek, Discrete forms of Friedrichs' inequalities in the finite element method, (To appear.) Zbl0475.65072MR0631681

Citations in EuDML Documents

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  1. Alexander Ženíšek, Discrete forms of Friedrichs' inequalities in the finite element method
  2. Libor Čermák, A note on a discrete form of Friedrichs' inequality
  3. Alexander Ženíšek, How to avoid the use of Green's theorem in the Ciarlet-Raviart theory of variational crimes
  4. Jiří Hřebíček, Numerical analysis of the general biharmonic problem by the finite element method
  5. Libor Čermák, The finite element solution of second order elliptic problems with the Newton boundary condition
  6. Miloslav Feistauer, Veronika Sobotíková, Finite element approximation of nonlinear elliptic problems with discontinuous coefficients
  7. Miloslav Feistauer, Karel Najzar, Karel Švadlenka, On a parabolic problem with nonlinear Newton boundary conditions

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