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Uniformly stable mixed hp-finite elementson multilevel adaptive grids with hanging nodes

Published online by Cambridge University Press:  03 April 2008

Friedhelm Schieweck
Friedhelm Schieweck*
Affiliation:
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany. schiewec@ovgu.de
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Abstract

We consider a family of quadrilateral or hexahedral mixed hp-finite elements for an incompressible flow problem 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 an arbitrary bound.For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size,we prove the inf-sup condition uniformly with respect to the meshsize and the polynomial degree.

Keywords

Stokes probleminf-sup conditionmixed hp-FEMquadrilateral and hexahedral finite elementsmultilevel adaptive gridshanging nodes.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 3 , May 2008, pp. 493 - 505
Copyright
© EDP Sciences, SMAI, 2008

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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
Arnold, D.N., Boffi, D. and Falk, R.S., Approximation by quadrilateral finite elements. Math. Comput. 71 (2002) 909–922. CrossRef
Babuška, I. and Miller, A., A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Methods Appl. Mech. Engrg. 61 (1987) 1–40. CrossRef
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 15. Springer-Verlag (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
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, Germany, 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
Heuveline, V. and Schieweck, F., H 1-interpolation on quadrilateral and hexahedral meshes with hanging nodes. Computing 80 (2007) 203–220. CrossRef
Heuveline, V. and Schieweck, F., On the inf-sup condition for higher order mixed fem on meshes with hanging nodes. ESAIM: M2AN 41 (2007) 1–20. CrossRef
Matthies, G., Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors. Numer. Algorithms 27 (2001) 317–327. CrossRef
G. Matthies, Finite element methods for free boundary value problems with capillary surfaces. Ph.D. thesis, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Germany (2002). [Published at Shaker-Verlag Aachen].
Matthies, G. and Schieweck, F., On the reference mapping for quadrilateral and hexahedral finite elements on multilevel adaptive grids. Computing 80 (2007) 95–119. CrossRef
Matthies, G. and Tobiska, L., The inf-sup condition for the mapped Q k - $P_{k-1}^{{\rm disc}}$ 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
C. Schwab, p- and hp-Finite Element Methods, Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation. Oxford Science Publications, Clarendon Press (1998).
Stenberg, R., Error analysis of some finite element methods for the Stokes problem. Math. Comput. 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
Toselli, A. and Schwab, C., Mixed hp-finite element approximations on geometric edge and boundary layer meshes in three dimensions. Numer. Math. 94 (2003) 771–801. CrossRef