Previous |  Up |  Next


nonconforming finite element method; inf-sup condition; incompressible flow problem
It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of $n$th order accuracy in the energy norm called $P_n^{}$ elements. For $n\le 3$ we show that the stability condition holds if the velocity space is constructed using the $P_n^{}$ elements and the pressure space consists of continuous piecewise polynomial functions of degree $n$.
[1] M.  Ainsworth, 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. DOI 10.1093/imanum/22.2.307 | MR 1897411
[2] C. Bernardi, F. Hecht: More pressure in the finite element discretization of the Stokes problem. M2AN, Math. Model. Numer. Anal. 34 (2000), 953–980. DOI 10.1051/m2an:2000111 | MR 1837763
[3] J. Boland, R. Nicolaides: Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983), 722–731. DOI 10.1137/0720048 | MR 0708453
[4] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991. MR 1115205
[5] P. G. Ciarlet: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, Vol. II: Finite Element Methods (Part  1), P. G.  Ciarlet, J.-L. Lions (eds.), North-Holland, Amsterdam, 1991, pp. 17–351. MR 1115237 | Zbl 0875.65086
[6] M. Crouzeix, P.-A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations  I. Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973), 33–76. MR 0343661
[7] M. Crouzeix, R. S. Falk: Nonconforming finite elements for the Stokes problem. Math. Comput. 52 (1989), 437–456. DOI 10.1090/S0025-5718-1989-0958870-8 | MR 0958870
[8] V. Girault, P.-A.  Raviart: Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, 1986. MR 0851383
[9] V. John: Parallele Lösung der inkompressiblen Navier-Stokes Gleichungen auf adaptiv verfeinerten Gittern. PhD. Thesis, Otto-von-Guericke-Universität, Magdeburg, 1997.
[10] V. John, P. Knobloch, G. Matthies, L. Tobiska: Non-nested multi-level solvers for finite element discretisations of mixed problems. Computing 68 (2002), 313–341. DOI 10.1007/s00607-002-1444-2 | MR 1921254
[11] P. Knobloch: On the application of the $P_1^{\mathop {\mathrm mod}}$ element to incompressible flow problems. Comput. Visual. Sci. 6 (2004), 185–195. DOI 10.1007/s00791-004-0127-2 | MR 2071439
[12] P.  Knobloch: New nonconforming finite elements for solving the incompressible Navier-Stokes equations. In: Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH  2001, F.  Brezzi et al. (eds.), Springer-Verlag Italia, Milano, 2003, pp. 123–132. MR 2360713 | Zbl 1283.76033
[13] P. Knobloch: On the inf-sup condition for the  $P_3^{\mathop {\mathrm mod}}/P_2^{\mathrm disc}$ element. Computing 76 (2006), 41–54. MR 2174350
[14] P. Knobloch, L. Tobiska: The $P_1^{\mathop {\mathrm mod}}$ element: A new nonconforming finite element for convection-diffusion problems. SIAM J.  Numer. Anal. 41 (2003), 436–456. DOI 10.1137/S0036142902402158 | MR 2004183
[15] F.  Schieweck: Parallele Lösung der stationären inkompressiblen Navier-Stokes Gleichungen. Habilitationsschrift, Otto-von-Guericke-Universität, Magdeburg, 1997. (German) Zbl 0915.76051
[16] L. R. Scott, M. Vogelius: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO, Modélisation Math. Anal. Numér. 19 (1985), 111–143. DOI 10.1051/m2an/1985190101111 | MR 0813691
[17] R. Stenberg: Analysis of mixed finite element methods for the Stokes problem: a unified approach. Math. Comput. 42 (1984), 9–23. MR 0725982 | Zbl 0535.76037
[18] S. Turek: Efficient Solvers for Incompressible Flow Problems. An Algorithmic and Computational Approach. Springer-Verlag, Berlin, 1999. MR 1691839 | Zbl 0930.76002
Partner of
EuDML logo