The finite element solution of parabolic equations

Josef Nedoma

Aplikace matematiky (1978)

  • Volume: 23, Issue: 6, page 408-438
  • ISSN: 0862-7940

Abstract

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In contradistinction to former results, the error bounds introduced in this paper are given for fully discretized approximate soltuions of parabolic equations and for arbitrary curved domains. Simplicial isoparametric elements in n -dimensional space are applied. Degrees of accuracy of quadrature formulas are determined so that numerical integration does not worsen the optimal order of convergence in L 2 -norm of the method.

How to cite

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Nedoma, Josef. "The finite element solution of parabolic equations." Aplikace matematiky 23.6 (1978): 408-438. <http://eudml.org/doc/15071>.

@article{Nedoma1978,
abstract = {In contradistinction to former results, the error bounds introduced in this paper are given for fully discretized approximate soltuions of parabolic equations and for arbitrary curved domains. Simplicial isoparametric elements in $n$-dimensional space are applied. Degrees of accuracy of quadrature formulas are determined so that numerical integration does not worsen the optimal order of convergence in $L_2$-norm of the method.},
author = {Nedoma, Josef},
journal = {Aplikace matematiky},
keywords = {error bounds; approximate solutions; parabolic equations; arbitrary curved domains; quadrature formulas; optimal order of convergence; error bounds; approximate solutions; parabolic equations; arbitrary curved domains; quadrature formulas; optimal order of convergence},
language = {eng},
number = {6},
pages = {408-438},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The finite element solution of parabolic equations},
url = {http://eudml.org/doc/15071},
volume = {23},
year = {1978},
}

TY - JOUR
AU - Nedoma, Josef
TI - The finite element solution of parabolic equations
JO - Aplikace matematiky
PY - 1978
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 23
IS - 6
SP - 408
EP - 438
AB - In contradistinction to former results, the error bounds introduced in this paper are given for fully discretized approximate soltuions of parabolic equations and for arbitrary curved domains. Simplicial isoparametric elements in $n$-dimensional space are applied. Degrees of accuracy of quadrature formulas are determined so that numerical integration does not worsen the optimal order of convergence in $L_2$-norm of the method.
LA - eng
KW - error bounds; approximate solutions; parabolic equations; arbitrary curved domains; quadrature formulas; optimal order of convergence; error bounds; approximate solutions; parabolic equations; arbitrary curved domains; quadrature formulas; optimal order of convergence
UR - http://eudml.org/doc/15071
ER -

References

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  1. P. G. Ciarlet, A. P. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, In A. K. Aziz: The mathematical foundations of the finite element method with applications to partial differential equations. Academic Press. New York and London. 1972. (1972) Zbl0262.65070MR0421108
  2. P. A. Raviart, The use of numerical integration in finite element methods for solving parabolic equations, Lecture presented at the Conference on Numerical Analysis. Royal Irish Academy. Dublin, August 14-18, 1972. (1972) Zbl0293.65086MR0345428
  3. Jindřich Nečas, Les Méthodes Directe en Théorie des Equations Elliptiques, Mason. Paris. 1967. (1967) MR0227584
  4. V. J. Smirnov, Kurs vyššej matěmatiki, tom V. Gosudarstvěnnoje izdatělstvo fiziko-matěmatičeskoj litěratury. Moskva. 1960. (1960) 
  5. Miloš Zlámal, 10.1090/S0025-5718-1975-0371105-2, Mathematics of Computation, 29, Nr 130 (1975), 350-359. (1975) Zbl0302.65081MR0371105DOI10.1090/S0025-5718-1975-0371105-2
  6. Miloš Zlámal, 10.1137/0710022, SIAM J. Numer. Anal., 10. No 1 (1973), 229-240. (1973) Zbl0285.65067MR0395263DOI10.1137/0710022
  7. Miloš Zlámal, 10.1137/0711031, SIAM J. Numer. Anal., 11. No 2 (1974), 347-362. (1974) Zbl0277.65064MR0343660DOI10.1137/0711031
  8. Miloš Zlámal, 10.1090/S0025-5718-1974-0388813-9, Mathematics of Computation, 28, No 126 (1974), 393-404. (1974) Zbl0296.65054MR0388813DOI10.1090/S0025-5718-1974-0388813-9
  9. T. Dupont G. Fairweather J. P. Johnson, Three-level Galerkin Methods for Parabolic Equations, SIAM J. Numer. Anal., 11, No 2 (1974). (1974) Zbl0313.65107MR0403259
  10. M. Lees, 10.1215/S0012-7094-60-02727-7, Duke Math. J., 27 (1960), 287-311. (1960) Zbl0092.32803MR0121998DOI10.1215/S0012-7094-60-02727-7
  11. Miloš Zlámal, Finite element methods for nonlinear parabolic equations, R.A.I.R.O. Analyse numérique/Numerical Analysis, 11, No 1 (1977), 93-107. (1977) Zbl0385.65049MR0502073
  12. W. Liniger, 10.1007/BF02235394, Computing, 3 (1968), 280-285. (1968) Zbl0169.19902MR0239763DOI10.1007/BF02235394

Citations in EuDML Documents

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  1. M. Vanmaele, A. Ženíšek, External finite element approximations of eigenvalue problems
  2. Josef Nedoma, The finite element solution of elliptic and parabolic equations using simplicial isoparametric elements
  3. Alexander Ženíšek, Finite element methods for coupled thermoelasticity and coupled consolidation of clay
  4. Libor Čermák, The finite element solution of second order elliptic problems with the Newton boundary condition
  5. Helena Růžičková, Alexander Ženíšek, Finite elements methods for solving viscoelastic thin plates

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