A finite element convergence analysis for 3D Stokes equations in case of variational crimes

Petr Knobloch

Applications of Mathematics (2000)

  • Volume: 45, Issue: 2, page 99-129
  • ISSN: 0862-7940

Abstract

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We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.

How to cite

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Knobloch, Petr. "A finite element convergence analysis for 3D Stokes equations in case of variational crimes." Applications of Mathematics 45.2 (2000): 99-129. <http://eudml.org/doc/33051>.

@article{Knobloch2000,
abstract = {We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.},
author = {Knobloch, Petr},
journal = {Applications of Mathematics},
keywords = {Stokes equations; nonstandard boundary conditions; finite element method; approximation of boundary; Stokes equations; nonstandard boundary conditions; finite element method; approximation of boundary},
language = {eng},
number = {2},
pages = {99-129},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A finite element convergence analysis for 3D Stokes equations in case of variational crimes},
url = {http://eudml.org/doc/33051},
volume = {45},
year = {2000},
}

TY - JOUR
AU - Knobloch, Petr
TI - A finite element convergence analysis for 3D Stokes equations in case of variational crimes
JO - Applications of Mathematics
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 45
IS - 2
SP - 99
EP - 129
AB - We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.
LA - eng
KW - Stokes equations; nonstandard boundary conditions; finite element method; approximation of boundary; Stokes equations; nonstandard boundary conditions; finite element method; approximation of boundary
UR - http://eudml.org/doc/33051
ER -

References

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  1. Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. (1991) MR1115205
  2. Basic error estimates for elliptic problems, In: Handbook of Numerical Analysis, v. II – Finite Element Methods (Part 1), P. G. Ciarlet, J. L. Lions (eds.), North-Holland, Amsterdam, 1991, pp. 17–351. (1991) Zbl0875.65086MR1115237
  3. The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz (ed.), Academic Press, New York, 1972, pp. 409–474. (1972) MR0421108
  4. On finite element approximations of problems having inhomogeneous essential boundary conditions, Comput. Math. Appl. 9 (1983), 687–700. (1983) MR0726817
  5. Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. (1986) MR0851383
  6. Solvability and Finite Element Discretization of a Mathematical Model Related to Czochralski Crystal Growth, PhD Thesis, Preprint MBI-96-5, Otto-von-Guericke-Universität, Magdeburg, 1996. (1996) Zbl0865.65094
  7. Variational crimes in a finite element discretization of 3D Stokes equations with nonstandard boundary conditions, East-West J. Numer. Math. 7 (1999), 133–158. (1999) Zbl0958.76043MR1699239
  8. Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Praha, 1967. (1967) MR0227584
  9. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. (1970) Zbl0207.13501MR0290095
  10. An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, 1973. (1973) MR0443377
  11. 10.1051/m2an/1987210101711, RAIRO, Modelisation Math. Anal. Numer. 21 (1987), 171–191. (1987) MR0882690DOI10.1051/m2an/1987210101711

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