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Inf-sup stable nonconforming finite elements of higher order onquadrilaterals and hexahedra

Published online by Cambridge University Press:  23 October 2007

Gunar Matthies
Gunar Matthies*
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany. [email protected]
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Abstract

We present families of scalar nonconforming finite elements of arbitraryorder $r\ge 1$ with optimal approximation properties on quadrilaterals andhexahedra. Their vector-valued versions together with a discontinuouspressure approximation of order $r-1$ form inf-sup stable finite element pairsof order r for the Stokes problem. The well-known elements by Rannacherand Turek are recovered in the case r=1. A numerical comparison betweenconforming and nonconforming discretisations will be given. Since higherorder nonconforming discretisation on quadrilaterals and hexahedra have lessunknowns and much less non-zero matrix entries compared to correspondingconforming methods, these methods are attractive for numerical simulations.

Keywords

Nonconforming finite elementsinf-sup stabilityquadrilateralshexahedra.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 5 , September 2007 , pp. 855 - 874
Copyright
© EDP Sciences, SMAI, 2007

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References

Bercovier, M. and Pironneau, O., Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979) 211–224. CrossRef
Braess, D. and Sarazin, R., An efficient smoother for the Stokes problem. Appl. Numer. Math. 23 (1997) 3–19. CrossRef
Bramble, J.H. and Hilbert, S.R., Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112–124. CrossRef
Cai, Z., Douglas, J., Jr. and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36 (1999) 215–232. CrossRef
Cai, Z., Douglas, J., Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming quadrilateral finite elements: a correction. Calcolo 37 (2000) 253–254. CrossRef
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO. Anal. Numér. 7 (1973) 33–76.
Douglas, J., Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM: M2AN 33 (1999) 747–770. CrossRef
Fortin, M., An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 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).
Han, H.D., Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984) 223–233.
Hennart, J.P., Jaffré, J. and Roberts, J.E., A constructive method for deriving finite elements of nodal type. Numer. Math. 53 (1988) 701–738. CrossRef
V. John, Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES Models . Lecture Notes in Computational Science and Engineering 34, Springer-Verlag, Berlin, Heidelberg, New York (2003).
John, V. and Matthies, G., Higher-order finite element discretizations in a benchmark problem for incompressible flows. Int. J. Num. Meth. Fluids 37 (2001) 885–903. CrossRef
John, V. and Matthies, G., MooNMD—a program package based on mapped finite element methods. Comput. Vis. Sci. 6 (2004) 163–169. CrossRef
John, V., Knobloch, P., Matthies, G. and Tobiska, L., Non-nested multi-level solvers for finite element discretisations of mixed problems. Computing 68 (2002) 313–341. 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
Matthies, G. and Tobiska, L., Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293–309. CrossRef
J. Maubach and P. Rabier, Nonconforming finite elements of arbitrary degree over triangles. RANA report 0328, Technical University of Eindhoven (2003).
Rannacher, R. and Turek, S., Simple nonconforming quadrilateral Stokes element. Numer. Meth. Part. Diff. Equ. 8 (1992) 97–111. CrossRef
F. Schieweck, A general transfer operator for arbitrary finite element spaces. Preprint 00-25, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg (2000).
Vanka, S., Block-implicit multigrid calculation of two-dimensional recirculating flows. Comp. Meth. Appl. Mech. Engrg. 59 (1986) 29–48. CrossRef
Verfürth, R., Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numér. 18 (1984) 175–182. CrossRef