Finite element analysis of primal and dual variational formulations of semicoercive elliptic problems with nonhomogeneous obstacles on the boundary

Van Bon Tran

Aplikace matematiky (1988)

  • Volume: 33, Issue: 1, page 1-21
  • ISSN: 0862-7940

Abstract

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The Poisson equation with non-homogeneous unilateral condition on the boundary is solved by means of finite elements. The primal variational problem is approximated on the basis of linear triangular elements, and O ( h ) -convergence is proved provided the exact solution is regular enough. For the dual problem piecewise linear divergence-free approximations are employed and O ( h 3 / 2 ) -convergence proved for a regular solution. Some a posteriori error estimates are also presented.

How to cite

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Tran, Van Bon. "Finite element analysis of primal and dual variational formulations of semicoercive elliptic problems with nonhomogeneous obstacles on the boundary." Aplikace matematiky 33.1 (1988): 1-21. <http://eudml.org/doc/15519>.

@article{Tran1988,
abstract = {The Poisson equation with non-homogeneous unilateral condition on the boundary is solved by means of finite elements. The primal variational problem is approximated on the basis of linear triangular elements, and $O(h)$-convergence is proved provided the exact solution is regular enough. For the dual problem piecewise linear divergence-free approximations are employed and $O(h^\{3/2\})$-convergence proved for a regular solution. Some a posteriori error estimates are also presented.},
author = {Tran, Van Bon},
journal = {Aplikace matematiky},
keywords = {semi-coercive elliptic problems; Poisson equation; finite elements; convergence; dual problem; a posteriori error estimates; variational inequalities; semi-coercive elliptic problems; Poisson equation; finite elements; convergence; dual problem; a posteriori error estimates},
language = {eng},
number = {1},
pages = {1-21},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finite element analysis of primal and dual variational formulations of semicoercive elliptic problems with nonhomogeneous obstacles on the boundary},
url = {http://eudml.org/doc/15519},
volume = {33},
year = {1988},
}

TY - JOUR
AU - Tran, Van Bon
TI - Finite element analysis of primal and dual variational formulations of semicoercive elliptic problems with nonhomogeneous obstacles on the boundary
JO - Aplikace matematiky
PY - 1988
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 33
IS - 1
SP - 1
EP - 21
AB - The Poisson equation with non-homogeneous unilateral condition on the boundary is solved by means of finite elements. The primal variational problem is approximated on the basis of linear triangular elements, and $O(h)$-convergence is proved provided the exact solution is regular enough. For the dual problem piecewise linear divergence-free approximations are employed and $O(h^{3/2})$-convergence proved for a regular solution. Some a posteriori error estimates are also presented.
LA - eng
KW - semi-coercive elliptic problems; Poisson equation; finite elements; convergence; dual problem; a posteriori error estimates; variational inequalities; semi-coercive elliptic problems; Poisson equation; finite elements; convergence; dual problem; a posteriori error estimates
UR - http://eudml.org/doc/15519
ER -

References

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  1. I. Hlaváček, Dual finite element analysis for semi-coercive unilateral boundary value problems, Apl. Mat. 23 (1978), 52-71. (1978) Zbl0407.65048MR0480160
  2. I. Hlaváček, Dual finite element analysis for elliptic problems with obstacles on the boundary, Apl. Mat. 22 (1977), 244-255. (1977) Zbl0422.65065MR0440958
  3. J. Haslinger I. Hlaváček, Convergence of a finite element method based on the dual variational formulation, Apl. Mat. 21 (1976), 43-65. (1976) Zbl0326.35020MR0398126
  4. R. S. Falk, 10.1090/S0025-5718-1974-0391502-8, Math. Comp. 28 (1974), 963-971. (1974) Zbl0297.65061MR0391502DOI10.1090/S0025-5718-1974-0391502-8
  5. F. Brezzi W. W. Hager P. A. Raviart, 10.1007/BF01404345, Numer. Math. 28 (1977), 431-443. (1977) Zbl0369.65030MR0448949DOI10.1007/BF01404345
  6. J. Haslinger, Finite element analysis for unilateral problem with obstacles on the boundary, Apl. Mat. 22 (1977), 180-188. (1977) Zbl0434.65083MR0440956
  7. I. Hlaváček, Dual finite element analysis for unilateral boundary value problems, Apl. Mat. 22 (1977), 14-51. (1977) Zbl0416.65070MR0426453
  8. I. Hlaváček, Convergence of dual finite element approximations for unilateral boundary value problems, Apl. Mat. 25 (1980), 375-386. (1980) Zbl0462.65064MR0590491
  9. J. Céa, Optimisation, théorie et algorithmes, Dunod, Paris 1971. (1971) Zbl0211.17402MR0298892

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