On semiregular families of triangulations and linear interpolation

Michal Křížek

Applications of Mathematics (1991)

  • Volume: 36, Issue: 3, page 223-232
  • ISSN: 0862-7940

Abstract

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We consider triangulations formed by triangular elements. For the standard linear interpolation operator π h we prove the interpolation order to be v - π h v 1 , p C h v 2 , p for p > 1 provided the corresponding family of triangulations is only semiregular. In such a case the well-known Zlámal’s condition upon the minimum angle need not be satisfied.

How to cite

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Křížek, Michal. "On semiregular families of triangulations and linear interpolation." Applications of Mathematics 36.3 (1991): 223-232. <http://eudml.org/doc/15675>.

@article{Křížek1991,
abstract = {We consider triangulations formed by triangular elements. For the standard linear interpolation operator $\pi __h$ we prove the interpolation order to be $\left\Vert v-\{\pi __h\} v\right\Vert _\{1,p\}\le Ch\left|v\right|_\{2,p\}$ for $p>1$ provided the corresponding family of triangulations is only semiregular. In such a case the well-known Zlámal’s condition upon the minimum angle need not be satisfied.},
author = {Křížek, Michal},
journal = {Applications of Mathematics},
keywords = {finite elements; linear interpolation; maximum angle condition; Zlámal’s condition; Zlámal’s condition},
language = {eng},
number = {3},
pages = {223-232},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On semiregular families of triangulations and linear interpolation},
url = {http://eudml.org/doc/15675},
volume = {36},
year = {1991},
}

TY - JOUR
AU - Křížek, Michal
TI - On semiregular families of triangulations and linear interpolation
JO - Applications of Mathematics
PY - 1991
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 36
IS - 3
SP - 223
EP - 232
AB - We consider triangulations formed by triangular elements. For the standard linear interpolation operator $\pi __h$ we prove the interpolation order to be $\left\Vert v-{\pi __h} v\right\Vert _{1,p}\le Ch\left|v\right|_{2,p}$ for $p>1$ provided the corresponding family of triangulations is only semiregular. In such a case the well-known Zlámal’s condition upon the minimum angle need not be satisfied.
LA - eng
KW - finite elements; linear interpolation; maximum angle condition; Zlámal’s condition; Zlámal’s condition
UR - http://eudml.org/doc/15675
ER -

References

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  1. I. Babuška A. K. Aziz, 10.1137/0713021, SIAM J. Numer. Anal. 13 (1976), 214-226. (1976) MR0455462DOI10.1137/0713021
  2. R. E. Barnhill J. A. Gregory, Sard kernel theorems on triangular domains with application to finite element error bounds, Numer. Math. 25 (1976), 215-229. (1976) MR0458000
  3. P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, New York, Oxford, 1978. (1978) Zbl0383.65058MR0520174
  4. J. A. Gregory, Error bounds for linear interpolation on triangles, (in Proc. MAFELAP II, ed. J. R. Whiteman). Academic Press, London, 1976, 163-170. (1976) MR0458795
  5. P. Jamet, Estimations d'erreur pour des éléments finis droits presque dégénérés, RAIRO Anal. Numér. 10 (1976), 43-60. (1976) MR0455282
  6. P. Jamet, Estimations de l'erreur d'interpolation dans un domaine variable et applications aux éléments finis quadrilatéraux dégénérés, Méthodes Numériques en Mathématiques Appliquées, Presses de l'Université de Montreal, 55-100. MR0445863
  7. M. Křížek, On semiregular families of decompositions of a polyhedron into tetrahedra and linear interpolation, (in Proc. of the 6th Conf. Mathematical Methods in Engineering). ŠKODA, Plzeň, 1991, 269-274. (1991) 
  8. M. Křížek P. Neittaanmäki, Finite element approximation of variational problems and applications, Pitman Monographs and Surveys in Pure and Applied Mathematics vol. 50, Longman Scientific & Technical, Harlow, 1990. (1990) MR1066462
  9. J. Nečas, Les rnéthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. (1967) MR0227584
  10. G. Strang G. J. Fix, An analysis of the finite element method, Prentice-Hall, INC., New Jersey, London, 1973. (1973) MR0443377
  11. J. L. Synge, The hypercircle in mathematical physics, Cambridge University Press, Cambridge, 1957. (1957) Zbl0079.13802MR0097605
  12. A. Ženíšek, The convergence of the finite element method for boundary value problems of a system of elliptic equations, Apl. Mat. 14 (1969), 355- 377. (1969) MR0245978
  13. M. Zlámal, 10.1007/BF02161362, Numer. Math. 12 (1968), 394-409. (1968) MR0243753DOI10.1007/BF02161362

Citations in EuDML Documents

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  1. Francisco Perdomo, Ángel Plaza, Properties of triangulations obtained by the longest-edge bisection
  2. Mingxia Li, Shipeng Mao, Anisotropic interpolation error estimates via orthogonal expansions
  3. Alexander Ženíšek, Finite element variational crimes in the case of semiregular elements
  4. Aleš Prachař, On discontinuous Galerkin method and semiregular family of triangulations
  5. Alexander Ženíšek, Jana Hoderová-Zlámalová, Semiregular hermite tetrahedral finite elements
  6. Peter Oswald, Divergence of FEM: Babuška-Aziz triangulations revisited
  7. Zdeněk Milka, Finite element solution of a stationary heat conduction equation with the radiation boundary condition
  8. Jana Zlámalová, Semiregular finite elements in solving some nonlinear problems
  9. Kenta Kobayashi, Takuya Tsuchiya, A priori error estimates for Lagrange interpolation on triangles
  10. Kenta Kobayashi, Takuya Tsuchiya, Extending Babuška-Aziz's theorem to higher-order Lagrange interpolation

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