How to recover the gradient of linear elements on nonuniform triangulations

Ivan Hlaváček; Michal Křížek; Vladislav Pištora

Applications of Mathematics (1996)

  • Volume: 41, Issue: 4, page 241-267
  • ISSN: 0862-7940

Abstract

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We propose and examine a simple averaging formula for the gradient of linear finite elements in R d whose interpolation order in the L q -norm is 𝒪 ( h 2 ) for d < 2 q and nonuniform triangulations. For elliptic problems in R 2 we derive an interior superconvergence for the averaged gradient over quasiuniform triangulations. A numerical example is presented.

How to cite

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Hlaváček, Ivan, Křížek, Michal, and Pištora, Vladislav. "How to recover the gradient of linear elements on nonuniform triangulations." Applications of Mathematics 41.4 (1996): 241-267. <http://eudml.org/doc/32949>.

@article{Hlaváček1996,
abstract = {We propose and examine a simple averaging formula for the gradient of linear finite elements in $R^d$ whose interpolation order in the $L^q$-norm is $\mathcal \{O\}(h^2)$ for $d<2q$ and nonuniform triangulations. For elliptic problems in $R^2$ we derive an interior superconvergence for the averaged gradient over quasiuniform triangulations. A numerical example is presented.},
author = {Hlaváček, Ivan, Křížek, Michal, Pištora, Vladislav},
journal = {Applications of Mathematics},
keywords = {weighted averaged gradient; linear elements; nonuniform triangulations; superapproximation; superconvergence; weighted averaged gradient; linear elements; nonuniform triangulations; superapproximation; superconvegence},
language = {eng},
number = {4},
pages = {241-267},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {How to recover the gradient of linear elements on nonuniform triangulations},
url = {http://eudml.org/doc/32949},
volume = {41},
year = {1996},
}

TY - JOUR
AU - Hlaváček, Ivan
AU - Křížek, Michal
AU - Pištora, Vladislav
TI - How to recover the gradient of linear elements on nonuniform triangulations
JO - Applications of Mathematics
PY - 1996
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 41
IS - 4
SP - 241
EP - 267
AB - We propose and examine a simple averaging formula for the gradient of linear finite elements in $R^d$ whose interpolation order in the $L^q$-norm is $\mathcal {O}(h^2)$ for $d<2q$ and nonuniform triangulations. For elliptic problems in $R^2$ we derive an interior superconvergence for the averaged gradient over quasiuniform triangulations. A numerical example is presented.
LA - eng
KW - weighted averaged gradient; linear elements; nonuniform triangulations; superapproximation; superconvergence; weighted averaged gradient; linear elements; nonuniform triangulations; superapproximation; superconvegence
UR - http://eudml.org/doc/32949
ER -

References

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  1. 10.1007/BF01385730, Numer. Math. 60 (1992), 429–463. (1992) MR1142306DOI10.1007/BF01385730
  2. 10.1002/nme.1620200611, Internat. J. Numer. Methods Engrg. 20 (1984), 1085–1109, 1111–1129. (1984) DOI10.1002/nme.1620200611
  3. Taschenbuch mathematischer Formeln, VEB Fachbuchverlag, Leipzig, 1979. (1979) MR1246330
  4. 10.1007/BF01447854, Appl. Math. Optim. 2 (1975), 130–169. (1975) MR0443372DOI10.1007/BF01447854
  5. 10.1137/0707006, SIAM J. Numer. Anal. 7 (1970), 112–124. (1970) MR0263214DOI10.1137/0707006
  6. Estimates for spline projections, RAIRO Anal. Numér. 10 (1976), 5–37. (1976) MR0436620
  7. Basic error estimates for elliptic problems, Handbook of Numerical Analysis II (P. G. Ciarlet, J. L. Lions eds.), North-Holland, Amsterdam, 1991. (1991) MR1115237
  8. 10.1007/BF01385773, Numer. Math. 59 (1991), 107–127. (1991) MR1106377DOI10.1007/BF01385773
  9. 10.1002/num.1690060105, Numer. Methods Partial Differential Equations 6 (1990), 59–74. (1990) MR1034433DOI10.1002/num.1690060105
  10. 10.1002/num.1690070106, Numer. Methods Partial Differential Equations 7 (1991), 61–83. (1991) MR1088856DOI10.1002/num.1690070106
  11. Design sensitivity analysis of structural systems, Academic Press, London, 1986. (1986) MR0860040
  12. On a superconvergent finite element scheme for elliptic systems, Parts I–III, Apl. Mat. 32 (1987), 131–154, 200–213, 276–289. (1987) 
  13. Optimal interior and local error estimates of a recovered gradient of linear elements on nonuniform triangulations, J. Comput. Math (to appear). (to appear) MR1414854
  14. Superconvergence of the gradient for linear finite elements for nonlinear elliptic problems, Proc. of the ISNA Conf., Prague, 1987, Teubner, Leipzig, 1988, 199–204. Zbl0677.65107MR1171706
  15. Superconvergence of the gradient of linear elements for 3D Poisson equation, Proc. Internat. Conf. Optimal Algorithms (ed. B. Sendov), Blagoevgrad, 1986, Izd. Bulg. Akad. Nauk, Sofia, 1986, 172–182. 
  16. An equilibrium finite element method in three-dimensional elasticity, Apl. Mat. 27 (1982), 46–75. (1982) MR0640139
  17. 10.1007/BF01379664, Numer. Math. 45 (1984), 105–116. (1984) MR0761883DOI10.1007/BF01379664
  18. 10.1016/0377-0427(87)90018-5, J. Comput. Appl. Math. 18 (1987), 221–233. (1987) MR0896426DOI10.1016/0377-0427(87)90018-5
  19. 10.1007/BF00047538, Acta Appl. Math. 9 (1987), 175–198. (1987) MR0900263DOI10.1007/BF00047538
  20. 10.1093/imanum/5.4.407, IMA J. Numer. Anal. 5 (1985), 407–427. (1985) Zbl0584.65067MR0816065DOI10.1093/imanum/5.4.407
  21. A rectangle test for singular solution with irregular meshes, Proc. of Systems Sci. & Systems Engrg., Great Wall (H.K.) Culture Publ. Co., 1991, 236–237. 
  22. Asymptotic expansion for the derivative of finite elements, J. Comput. Math. 2 (1984), 361–363. (1984) MR0869509
  23. 10.1007/BF01396494, Numer. Math. 33 (1979), 43–53. (1979) Zbl0435.65090MR0545741DOI10.1007/BF01396494
  24. An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with a smooth boundary (Russian), Ž. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 1102–1120. (1969) MR0295599
  25. 10.1137/0726033, SIAM J. Numer. Anal. 26 (1989), 553–573. (1989) MR0997656DOI10.1137/0726033
  26. Local behavior in finite element methods, Handbook of Numerical Analysis II (P. G. Ciarlet, J. L. Lions eds.), North-Holland, Amsterdam, 1991, 353–522. (1991, 353–522) MR1115238
  27. Superconvergence in Galerkin finite element methods (Lecture notes), Cornell Univ., 1994, 1–243. MR1439050
  28. 10.1002/num.1690030106, Numer. Methods Partial Differential Equations 3 (1987), 65–82. (1987) MR1012906DOI10.1002/num.1690030106
  29. Superkonvergenz des Gradienten im Postprocessing von FiniteElemente-Methoden, Preprint Nr. 94, Tech. Univ. Chemnitz, 1989, 1–15. 
  30. 10.1002/nme.1620330702, Part 1, Internat. J. Numer. Methods Engrg. 33 (1992), 1331–1364. (1992) MR1161557DOI10.1002/nme.1620330702

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