On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes

Vincent Heuveline; Friedhelm Schieweck

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 1, page 1-20
  • ISSN: 0764-583X

Abstract

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We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Qr-elements for the velocity and discontinuous P r - 1 -elements for the pressure where the order r can vary from element to element between 2 and a fixed bound r * . We prove the inf-sup condition uniformly with respect to the meshwidth h on general quadrilateral and hexahedral meshes with hanging nodes.

How to cite

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Heuveline, Vincent, and Schieweck, Friedhelm. "On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes." ESAIM: Mathematical Modelling and Numerical Analysis 41.1 (2007): 1-20. <http://eudml.org/doc/250031>.

@article{Heuveline2007,
abstract = { We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Qr-elements for the velocity and discontinuous $P_\{r-1\}$-elements for the pressure where the order r can vary from element to element between 2 and a fixed bound $r^*$. We prove the inf-sup condition uniformly with respect to the meshwidth h on general quadrilateral and hexahedral meshes with hanging nodes. },
author = {Heuveline, Vincent, Schieweck, Friedhelm},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Inf-sup condition; higher order mixed finite element; adaptive grids; hanging nodes.; inf-sup condition; adaptive grids; hanging nodes; stability; Stokes equations},
language = {eng},
month = {4},
number = {1},
pages = {1-20},
publisher = {EDP Sciences},
title = {On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes},
url = {http://eudml.org/doc/250031},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Heuveline, Vincent
AU - Schieweck, Friedhelm
TI - On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/4//
PB - EDP Sciences
VL - 41
IS - 1
SP - 1
EP - 20
AB - We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Qr-elements for the velocity and discontinuous $P_{r-1}$-elements for the pressure where the order r can vary from element to element between 2 and a fixed bound $r^*$. We prove the inf-sup condition uniformly with respect to the meshwidth h on general quadrilateral and hexahedral meshes with hanging nodes.
LA - eng
KW - Inf-sup condition; higher order mixed finite element; adaptive grids; hanging nodes.; inf-sup condition; adaptive grids; hanging nodes; stability; Stokes equations
UR - http://eudml.org/doc/250031
ER -

References

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  1. M. Ainsworth and P. Coggins, A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow. IMA J. Numer. Anal.22 (2002) 307–327.  Zbl1017.76041
  2. I. Babuška and M. Suri, The p and h - p versions of the finite element method, basic principles and properties. SIAM Rev.36 (1994) 578–632.  Zbl0813.65118
  3. C. Bernardi and Y. Maday. Approximations spectrales de problèmes aux limites elliptiques. (Spectral approximation for elliptic boundary value problems). Mathématiques & Applications, Paris, Springer-Verlag 10 (1992).  
  4. C. Bernardi and Y. Maday, Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Models Methods Appl. Sci.9 (1999) 395–414.  Zbl0944.76058
  5. D. Boffi and L. Gastaldi, On the quadrilateral Q2-P1 element for the Stokes problem. Int. J. Numer. Methods Fluids39 (2002) 1001–1011.  Zbl1101.76352
  6. J.M. Boland and R.A. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal.20 (1983) 722–731.  Zbl0521.76027
  7. S. Bönisch, V. Heuveline and P. Wittwer, Adaptive boundary conditions for exterior flow problems. J. Math. Fluid Mech.7 (2005) 85–107.  Zbl1072.76049
  8. F. Brezzi and R.S. Falk, Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal.28 (1991) 581–590.  Zbl0731.76042
  9. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, Springer-Verlag 15 (1991).  Zbl0788.73002
  10. L. Chilton and M. Suri, On the construction of stable curvilinear p version elements for mixed formulations of elasticity and Stokes flow. Numer. Math.86 (2000) 29–48.  Zbl0991.76040
  11. P.G. Ciarlet, The finite element method for elliptic problems. Studies in Mathematics and its Applications 4, Amsterdam - New York - Oxford: North-Holland Publishing Company (1978).  Zbl0383.65058
  12. M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numer.11 (1977) 341–354.  Zbl0373.65055
  13. V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Springer-Verlag, Berlin-Heidelberg-New York (1986).  Zbl0413.65081
  14. V. Heuveline and M. Hinze, Adjoint-based adaptive time-stepping for partial differential equations using proper orthogonal decomposition. Technical report, University Heidelberg, SFB 359 (2004).  
  15. V. Heuveline and R. Rannacher, A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math.15 (2001) 107–138.  Zbl0995.65111
  16. V. Heuveline and R. Rannacher, Duality-based adaptivity in the hp-finite element method. J. Numer. Math.11 (2003) 95–113.  Zbl1050.65111
  17. V. Heuveline and F. Schieweck, An interpolation operator for H1 functions on general quadrilateral and hexahedral meshes with hanging nodes. Technical report, University Heidelberg, SFB 359 (2004).  Zbl1133.65100
  18. G. Matthies, Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors. Numer. Algorithms27 (2001) 317–327.  Zbl0998.65117
  19. G. Matthies and L. Tobiska, The inf-sup condition for the mapped Qk- P k - 1 d i s c element in arbitrary space dimensions. Computing69 (2002) 119–139.  Zbl1016.65073
  20. S. Schötzau, C. Schwab and R. Stenberg, Mixed hp-fem on anisotropic meshes. II: Hanging nodes and tensor products of boundary layer meshes. Numer. Math.83 (1999) 667–697.  Zbl0958.76049
  21. Ch. Schwab, p- and hp-finite element methods. Theory and applications in solid and fluid mechanics. Numerical Mathematics and Scientific Computation, Oxford: Clarendon Press (1998).  Zbl0910.73003
  22. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  Zbl0696.65007
  23. R. Stenberg, Error analysis of some finite element methods for the Stokes problem. Math. Comp.54 (1990) 495–508.  Zbl0702.65095
  24. R. Stenberg and M. Suri, Mixed hp finite element methods for problems in elasticity and Stokes flow. Numer. Math.72 (1996) 367–389.  Zbl0855.73075
  25. L. Stupelis, Navier-Stokes equations in irregular domains. Mathematics and its Applications 326, Dordrecht: Kluwer Academic Publishers (1995).  Zbl0837.35003

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