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On the inf-sup condition for higher order mixed FEMon mesheswith hanging nodes

Published online by Cambridge University Press:  26 April 2007

Vincent Heuveline  and
Friedhelm Schieweck
Vincent Heuveline
Affiliation:
Institute for Applied Mathematics II, University Karlsruhe (TH), Postfach 6980, 76128 Karlsruhe, Germany. [email protected] Scientific Supercomputing Center, University of Karlsruhe (TH), Zirkel 2, 76128 Karlsruhe, Germany. [email protected]
Friedhelm Schieweck
Affiliation:
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany. [email protected]
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Abstract

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Q r -elements for the velocity anddiscontinuous $P_{r-1}$ -elements for the pressure where the orderr can vary from element to elementbetween 2 and a fixed bound $r^*$ .We prove the inf-sup condition uniformly with respect to the meshwidth hon general quadrilateral and hexahedral meshes with hanging nodes.

Keywords

Inf-sup conditionhigher order mixed finite elementadaptive gridshanging nodes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 1 , January 2007 , pp. 1 - 20
Copyright
© EDP Sciences, SMAI, 2007

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References

Ainsworth, M. and Coggins, P., A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow. IMA J. Numer. Anal. 22 (2002) 307–327. CrossRef
Babuška, I. and Suri, M., The p and h - p versions of the finite element method, basic principles and properties. SIAM Rev. 36 (1994) 578–632. CrossRef
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).
Bernardi, C. and Maday, Y., Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Models Methods Appl. Sci. 9 (1999) 395–414. CrossRef
Boffi, D. and Gastaldi, L., On the quadrilateral Q2-P1 element for the Stokes problem. Int. J. Numer. Methods Fluids 39 (2002) 1001–1011. CrossRef
Boland, J.M. and Nicolaides, R.A., Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722–731. CrossRef
Bönisch, S., Heuveline, V. and Wittwer, P., Adaptive boundary conditions for exterior flow problems. J. Math. Fluid Mech. 7 (2005) 85–107. CrossRef
Brezzi, F. and Falk, R.S., Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28 (1991) 581–590. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, Springer-Verlag 15 (1991).
Chilton, L. and Suri, M., On the construction of stable curvilinear p version elements for mixed formulations of elasticity and Stokes flow. Numer. Math. 86 (2000) 29–48. CrossRef
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).
Fortin, M., An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numer. 11 (1977) 341–354. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Springer-Verlag, Berlin-Heidelberg-New York (1986).
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).
Heuveline, V. and Rannacher, R., A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15 (2001) 107–138. CrossRef
Heuveline, V. and Rannacher, R., Duality-based adaptivity in the hp-finite element method. J. Numer. Math. 11 (2003) 95–113. CrossRef
V. Heuveline and F. Schieweck, An interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes. Technical report, University Heidelberg, SFB 359 (2004).
Matthies, G., Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors. Numer. Algorithms 27 (2001) 317–327. CrossRef
Matthies, G. and Tobiska, L., The inf-sup condition for the mapped Q k - $P_{k-1}^{disc}$ element in arbitrary space dimensions. Computing 69 (2002) 119–139. CrossRef
Schötzau, S., Schwab, C. and Stenberg, R., Mixed hp-fem on anisotropic meshes. II: Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83 (1999) 667–697. CrossRef
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).
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. CrossRef
Stenberg, R., Error analysis of some finite element methods for the Stokes problem. Math. Comp. 54 (1990) 495–508. CrossRef
Stenberg, R. and Suri, M., Mixed hp finite element methods for problems in elasticity and Stokes flow. Numer. Math. 72 (1996) 367–389. CrossRef
L. Stupelis, Navier-Stokes equations in irregular domains. Mathematics and its Applications 326, Dordrecht: Kluwer Academic Publishers (1995).