Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 5, p. 855-874

We present families of scalar nonconforming finite elements of arbitrary order $r\ge 1$ with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order $r-1$ form inf-sup stable finite element pairs of order $r$ for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case $r=1$. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order nonconforming discretisation on quadrilaterals and hexahedra have less unknowns and much less non-zero matrix entries compared to corresponding conforming methods, these methods are attractive for numerical simulations.

DOI : https://doi.org/10.1051/m2an:2007034
Classification:  65N12,  65N30
Keywords: nonconforming finite elements, inf-sup stability, quadrilaterals, hexahedra
@article{M2AN_2007__41_5_855_0,
author = {Matthies, Gunar},
title = {Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {41},
number = {5},
year = {2007},
pages = {855-874},
doi = {10.1051/m2an:2007034},
zbl = {1147.65094},
zbl = {pre05289352},
mrnumber = {2363886},
language = {en},
url = {http://www.numdam.org/item/M2AN_2007__41_5_855_0}
}
Matthies, Gunar. Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 5, pp. 855-874. doi : 10.1051/m2an:2007034. http://www.numdam.org/item/M2AN_2007__41_5_855_0/

[1] M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979) 211-224. | Zbl 0423.65058

[2] D. Braess and R. Sarazin, An efficient smoother for the Stokes problem. Appl. Numer. Math. 23 (1997) 3-19. | Zbl 0874.65095

[3] J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112-124. | Zbl 0201.07803

[4] Z. Cai, J. Douglas, Jr. and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36 (1999) 215-232. | Zbl 0947.76047

[5] Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming quadrilateral finite elements: a correction. Calcolo 37 (2000) 253-254. | Zbl 1012.65124

[6] 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. | Numdam | Zbl 0302.65087

[7] J. Douglas, 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. | Numdam | Zbl 0941.65115

[8] M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341-354. | Numdam | Zbl 0373.65055

[9] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes equations. Springer-Verlag, Berlin-Heidelberg-New York (1986). | MR 851383 | Zbl 0585.65077

[10] H.D. Han, Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984) 223-233. | Zbl 0573.65083

[11] J.P. Hennart, J. Jaffré and J.E. Roberts, A constructive method for deriving finite elements of nodal type. Numer. Math. 53 (1988) 701-738. | Zbl 0677.65101

[12] 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). | MR 2018955 | Zbl 1035.76001

[13] V. John and G. Matthies, Higher-order finite element discretizations in a benchmark problem for incompressible flows. Int. J. Num. Meth. Fluids 37 (2001) 885-903. | Zbl 1007.76040

[14] V. John and G. Matthies, MooNMD-a program package based on mapped finite element methods. Comput. Vis. Sci. 6 (2004) 163-169. | Zbl 1061.65124

[15] V. John, P. Knobloch, G. Matthies and L. Tobiska, Non-nested multi-level solvers for finite element discretisations of mixed problems. Computing 68 (2002) 313-341. | Zbl 1006.65137

[16] G. Matthies and L. Tobiska, The inf-sup condition for the mapped ${Q}_{k}/{P}_{k-1}^{disc}$ element in arbitrary space dimensions. Computing 69 (2002) 119-139. | Zbl 1016.65073

[17] G. Matthies and L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293-309. | Zbl 1089.65123

[18] J. Maubach and P. Rabier, Nonconforming finite elements of arbitrary degree over triangles. RANA report 0328, Technical University of Eindhoven (2003).

[19] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Meth. Part. Diff. Equ. 8 (1992) 97-111. | Zbl 0742.76051

[20] F. Schieweck, A general transfer operator for arbitrary finite element spaces. Preprint 00-25, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg (2000).

[21] S. Vanka, Block-implicit multigrid calculation of two-dimensional recirculating flows. Comp. Meth. Appl. Mech. Engrg. 59 (1986) 29-48. | Zbl 0604.76025

[22] R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numér. 18 (1984) 175-182. | Numdam | Zbl 0557.76037