Abstract
We consider a recently introduced triangular nonconforming finite element of third-order accuracy in the energy norm called Pmod3 element. We show that this finite element is appropriate for approximating the velocity in incompressible flow problems since it satisfies an inf-sup condition for discontinuous piecewise quadratic pressures.
Similar content being viewed by others
References
J. Boland R. Nicolaides (1983) ArticleTitleStability of finite elements under divergence constraints SIAM J. Numer. Anal. 20 722–731 Occurrence Handle10.1137/0720048
F. Brezzi M. Fortin (1991) Mixed and hybrid finite-elements methods Springer New York
Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of numerical analysis (Ciarlet, P. G., Lions, J. L., eds.), vol. 2 : Finite-element methods (pt. 1), pp. 17–351. Amsterdam: North-Holland 1991.
Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite-element methods for solving the stationary Stokes equations I. RAIRO 7(R-3), 33–76 (1973).
M. Crouzeix R. S. Falk (1989) ArticleTitleNonconforming finite elements for the Stokes problem Math. Comput. 52 437–456
V. Girault P.-A. Raviart (1986) Finite-element methods for Navier–Stokes equations Springer Berlin
John, V.: Parallele Lösung der inkompressiblen Navier–Stokes Gleichungen auf adaptiv verfeinerten Gittern. PhD thesis, Otto-von-Guericke-Universität Magdeburg, 1997.
V. John P. Knobloch G. Matthies L. Tobiska (2002) ArticleTitleNon-nested multi-level solvers for finite element discretisations of mixed problems Computing 68 313–341 Occurrence Handle10.1007/s00607-002-1444-2
P. Knobloch (2004) ArticleTitleOn the application of the P1mod element to incompressible flow problems Comput. Visual. Sci. 6 185–195 Occurrence Handle10.1007/s00791-004-0127-2
Knobloch, P.: New nonconforming finite elements for solving the incompressible Navier–Stokes equations. In: Numerical mathematics and advanced applications (Brezzi, F. et al., eds.). Proc. of ENUMATH 2001, pp. 123–132. Milano: Springer 2003.
P. Knobloch L. Tobiska (2003) ArticleTitleThe P1mod element a new nonconforming finite element for convection–diffusion problems SIAM J. Numer. Anal. 41 436–456 Occurrence Handle10.1137/S0036142902402158
Matthies, G., Tobiska, L.: Inf–sup stable non-nonforming finite elements of arbitrary order on triangles. Preprint Nr. 12, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, 2004.
Schieweck, F.: Parallele Lösung der stationären inkompressiblen Navier–Stokes Gleichungen. Habilitationsschrift, Otto-von-Guericke-Universität Magdeburg, 1997.
Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. M2AN 19, 111–143 (1985).
R. Stenberg (1984) ArticleTitleAnalysis of mixed finite element methods for the Stokes problem a unified approach Math. Comput. 42 9–23
S. Turek (1999) Efficient solvers for incompressible flow problems. An algorithmic and computational approach Springer Berlin
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Knobloch, P. On the inf–sup Condition for the Pmod 3 / Pdisc 2 Element. Computing 76, 41–54 (2006). https://doi.org/10.1007/s00607-005-0125-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-005-0125-3