On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition

Ivan Hlaváček; Michal Křížek

Aplikace matematiky (1987)

  • Volume: 32, Issue: 2, page 131-154
  • ISSN: 0862-7940

Abstract

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Second order elliptic systems with Dirichlet boundary conditions are solved by means of affine finite elements on regular uniform triangulations. A simple averagign scheme is proposed, which implies a superconvergence of the gradient. For domains with enough smooth boundary, a global estimate O ( h 3 / 2 ) is proved in the L 2 -norm. For a class of polygonal domains the global estimate O ( h 2 ) can be proven.

How to cite

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Hlaváček, Ivan, and Křížek, Michal. "On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition." Aplikace matematiky 32.2 (1987): 131-154. <http://eudml.org/doc/15485>.

@article{Hlaváček1987,
abstract = {Second order elliptic systems with Dirichlet boundary conditions are solved by means of affine finite elements on regular uniform triangulations. A simple averagign scheme is proposed, which implies a superconvergence of the gradient. For domains with enough smooth boundary, a global estimate $O(h^\{3/2\})$ is proved in the $L^2$-norm. For a class of polygonal domains the global estimate $O(h^2)$ can be proven.},
author = {Hlaváček, Ivan, Křížek, Michal},
journal = {Aplikace matematiky},
keywords = {post-processing; averaged gradient; Galerkin method; system; finite elements; superconvergence; global estimate; elliptic systems; post-processing; averaged gradient; Galerkin method; system; finite elements; superconvergence; global estimate},
language = {eng},
number = {2},
pages = {131-154},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition},
url = {http://eudml.org/doc/15485},
volume = {32},
year = {1987},
}

TY - JOUR
AU - Hlaváček, Ivan
AU - Křížek, Michal
TI - On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition
JO - Aplikace matematiky
PY - 1987
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 32
IS - 2
SP - 131
EP - 154
AB - Second order elliptic systems with Dirichlet boundary conditions are solved by means of affine finite elements on regular uniform triangulations. A simple averagign scheme is proposed, which implies a superconvergence of the gradient. For domains with enough smooth boundary, a global estimate $O(h^{3/2})$ is proved in the $L^2$-norm. For a class of polygonal domains the global estimate $O(h^2)$ can be proven.
LA - eng
KW - post-processing; averaged gradient; Galerkin method; system; finite elements; superconvergence; global estimate; elliptic systems; post-processing; averaged gradient; Galerkin method; system; finite elements; superconvergence; global estimate
UR - http://eudml.org/doc/15485
ER -

References

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  1. S. Agmon A. Douglis L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964), 35-92. (1964) Zbl0123.28706MR0162050
  2. A. B. Andreev, Superconvergence of the gradient for linear triangle elements for elliptic and parabolic equations, C. R. Acad. Bulgare Sci. 37 (1984), 293 - 296. (1984) Zbl0575.65106MR0758156
  3. I. Babuška A. Miller, The post-processing technique in the finite element method, Parts I-III, Internat. J. Numer. Methods Engrg. 20 (1984), 1085-1109, 1111-1129. (1984) Zbl0535.73053
  4. C. M. Chen, Optimal points of the stresses for triangular linear element, Numer. Math. J. Chinese Univ. 2 (1980), 12-20. (1980) Zbl0534.73057MR0619174
  5. C. M. Chen, W 1 , -interior estimates for finite element method on regular mesh, J. Comput. Math. 3 (1985), 1-7. (1985) Zbl0603.34024MR0815405
  6. P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, New York, Oxford, 1978. (1978) Zbl0383.65058MR0520174
  7. I. Hlaváček M. Hlaváček, On the existence and uniqueness of solutions and some variational principles in linear theories of elasticity with couple-stresses, Apl. Mat. 14 (1969), 387-410. (1969) Zbl0195.27003MR0250537
  8. V. P. Iljin, Svojstva někotorych klassov differenciruemych funkcij mnogich peremennych, zadannych v n-mernoj oblasti, Trudy Mat. Inst. Steklov. 66 (1962), 227-363. (1962) 
  9. M. Křížek P. Neittaanmäki, 10.1007/BF01379664, Numer. Math. 45 (1984), 105-116. (1984) Zbl0575.65104MR0761883DOI10.1007/BF01379664
  10. M. Křížek P. Neittaanmäki, On Superconvergence techniques, Preprint n. 34, Univ. of Jyväskylä, 1984, 1 - 43 (to appear in Acta Appl. Math.). (1984) Zbl0624.65107MR0900263
  11. N. Levine, 10.1093/imanum/5.4.407, IMA J. Numer. Anal. 5 (1985), 407-427. (1985) Zbl0584.65067MR0816065DOI10.1093/imanum/5.4.407
  12. Q. Lin J. Ch. Xu, Linear finite elements with high accuracy, J. Comput. Math. 3 (1985), 115-133. (1985) Zbl0577.65094MR0854355
  13. A. Louis, 10.1007/BF01396494, Numer. Math. 33 (1979), 43-53. (1979) Zbl0435.65090MR0545741DOI10.1007/BF01396494
  14. L. A. Oganesjan V. J. Rivkind L. A. Ruchovec, Variational-difference methods for the solution of elliptic equations. Part I, (Proc. Sem., Issue 5, Vilnius, 1973), Inst. of Phys. and Math., Vilnius, 1973, 3-389. (1973) 
  15. L. A. Oganesjan L. A. Ruchovec, An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional regions with smooth boundary, Ž. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 1102-1120. (1969) MR0295599
  16. L. A. Oganesjan L. A. Ruchovec, Variational-difference methods for the solution of elliptic equations, Izd. Akad. Nauk Armjanskoi SSR, Jerevan, 1979. (1979) 
  17. J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. (1967) MR0227584
  18. J. Nečas I. Hlaváček, 10.1007/BF00249518, Arch. Rational Mech. Anal. 36 (1970), 305-334. (1970) Zbl0193.39002MR0252844DOI10.1007/BF00249518
  19. J. Nečas I. Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Elsevier, Amsterdam, Oxford, New York, 1981. (1981) Zbl0448.73009MR0600655
  20. V. Thomée, High order local approximations to derivatives in the finite element method, Math. Соmр. 31 (1977), 652-660. (1977) Zbl0367.65055MR0438664
  21. B. Westergren, Interior estimates for elliptic systems of difference equations, (Thesis). Univ. of Goteborg, 1982. (1982) 
  22. Q. D. Zhu, Natural inner Superconvergence for the finite element method, (Proc. China-France Sympos. on the Finite Element method, Beijing, 1982), Science Press, Beijing, Gordon and Breach, New York, 1983, 935-960. (1982) MR0754041
  23. M. Zlámal, Superconvergence and reduced integration in the finite element method, Math. Соmр. 32 (1978), 663-685. (1978) MR0495027

Citations in EuDML Documents

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  1. Ivan Hlaváček, Michal Křížek, On a superconvergent finite element scheme for elliptic systems. II. Boundary conditions of Newton's or Neumann's type
  2. Ivan Hlaváček, Michal Křížek, On a superconvergent finite element scheme for elliptic systems. III. Optimal interior estimates
  3. Balázs Kovács, On the numerical performance of a sharp a posteriori error estimator for some nonlinear elliptic problems
  4. Wei Chen, Michal Křížek, What is the smallest possible constant in Céa's lemma?

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