Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek

Applications of Mathematics (1996)

  • Volume: 41, Issue: 5, page 367-398
  • ISSN: 0862-7940

Abstract

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The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain Ω whose boundary ∂ Ω is formed by two circles Γ 1 , Γ 2 with the same center S 0 and radii R 1 , R 2 = R 1 + ϱ , where ϱ ≪ R 1 . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for u = 0 are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus triangles obeying only the maximum angle condition and narrow quadrilaterals are used. The restrictions of test functions on triangles are linear functions while on quadrilaterals they are four-node isoparametric functions. Both the effect of numerical integration and that of approximation of the boundary are analyzed. The rate of convergence O ( h ) in the norm of the Sobolev space H 1 is proved under the following conditions: 1. the

How to cite

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Ženíšek, Alexander. "Finite element variational crimes in the case of semiregular elements." Applications of Mathematics 41.5 (1996): 367-398. <http://eudml.org/doc/32956>.

@article{Ženíšek1996,
abstract = {The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega $ whose boundary $\partial \Omega $ is formed by two circles $\Gamma _1$, $\Gamma _2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho $, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus triangles obeying only the maximum angle condition and narrow quadrilaterals are used. The restrictions of test functions on triangles are linear functions while on quadrilaterals they are four-node isoparametric functions. Both the effect of numerical integration and that of approximation of the boundary are analyzed. The rate of convergence $O(h)$ in the norm of the Sobolev space $H^1$ is proved under the following conditions: 1. the},
author = {Ženíšek, Alexander},
journal = {Applications of Mathematics},
keywords = {finite element method; elliptic problems; semiregular elements; maximum angle condition; variational crimes; finite element method; elliptic problems; maximum angle condition; variational crimes},
language = {eng},
number = {5},
pages = {367-398},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finite element variational crimes in the case of semiregular elements},
url = {http://eudml.org/doc/32956},
volume = {41},
year = {1996},
}

TY - JOUR
AU - Ženíšek, Alexander
TI - Finite element variational crimes in the case of semiregular elements
JO - Applications of Mathematics
PY - 1996
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 41
IS - 5
SP - 367
EP - 398
AB - The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega $ whose boundary $\partial \Omega $ is formed by two circles $\Gamma _1$, $\Gamma _2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho $, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus triangles obeying only the maximum angle condition and narrow quadrilaterals are used. The restrictions of test functions on triangles are linear functions while on quadrilaterals they are four-node isoparametric functions. Both the effect of numerical integration and that of approximation of the boundary are analyzed. The rate of convergence $O(h)$ in the norm of the Sobolev space $H^1$ is proved under the following conditions: 1. the
LA - eng
KW - finite element method; elliptic problems; semiregular elements; maximum angle condition; variational crimes; finite element method; elliptic problems; maximum angle condition; variational crimes
UR - http://eudml.org/doc/32956
ER -

References

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  1. 10.1137/0713021, SIAM J. Numer. Anal. 13 (1976), 214–226. (1976) MR0455462DOI10.1137/0713021
  2. 10.1007/BF01396664, Numer. Math. 50 (1987), 451–475. (1987) MR0875168DOI10.1007/BF01396664
  3. Estimations d’erreur pour des éléments finis presque dégénérés, RAIRO Anal. Numér. 10 (1976), 43–61. (1976) MR0455282
  4. On semiregular families of triangulations and linear interpolation, Appl. Math. 36 (1991), 223–232. (1991) MR1109126
  5. Les Méthodes Directes en Théorie des Equations Elliptiques, Academia-Masson, Prague-Paris, 1967. (1967) MR0227584
  6. Variational-Difference Methods for the Solution of Elliptic Problems, Izd. Akad. Nauk ArSSR, Jerevan, 1979. (Russian) (1979) 
  7. Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations, Academic Press, London, 1990. (1990) MR1086876
  8. 10.1007/s002110050163, Numer. Math. 72 (1995), 123–141. (1995) MR1359711DOI10.1007/s002110050163

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