A Fortin operator for two-dimensional Taylor-Hood elements
Falk, Richard S.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008), p. 411-424 / Harvested from Numdam
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A standard method for proving the inf-sup condition implying stability of finite element approximations for the stationary Stokes equations is to construct a Fortin operator. In this paper, we show how this can be done for two-dimensional triangular and rectangular Taylor-Hood methods, which use continuous piecewise polynomial approximations for both velocity and pressure.

Publié le : 2008-01-01
DOI : https://doi.org/10.1051/m2an:2008008
Classification:  65N30 
@article{M2AN_2008__42_3_411_0,
     author = {Falk, Richard S.},
     title = {A Fortin operator for two-dimensional Taylor-Hood elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {42},
     year = {2008},
     pages = {411-424},
     doi = {10.1051/m2an:2008008},
     mrnumber = {2423792},
     zbl = {1143.65085},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2008__42_3_411_0}
}

Falk, Richard S. A Fortin operator for two-dimensional Taylor-Hood elements. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 411-424. doi : 10.1051/m2an:2008008. http://gdmltest.u-ga.fr/item/M2AN_2008__42_3_411_0/

[1] M. Bercovier and O. Pironneau, Error estimates for finite element solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979) 211-224. | MR 549450 | Zbl 0423.65058

[2] D. Boffi, Stability of higher-order triangular Hood-Taylor methods for the stationary Stokes equation. Math. Models Methods Appl. Sci. 4 (1994) 223-235. | MR 1269482 | Zbl 0804.76051

[3] D. Boffi, Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34 (1997) 664-670. | MR 1442933 | Zbl 0874.76032

[4] F. Brezzi and R.S. Falk, Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28 (1991) 581-590. | MR 1098408 | Zbl 0731.76042

[5] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[6] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes equations: theory and algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986). | MR 851383 | Zbl 0585.65077

[7] L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19 (1985) 111-143. | Numdam | MR 813691 | Zbl 0608.65013

[8] R. Stenberg, Error analysis of some finite element methods for the Stokes problem. Math. Comp. 54 (1990) 494-548. | MR 1010601 | Zbl 0702.65095

[9] R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numér. 18 (1984) 175-182. | Numdam | MR 743884 | Zbl 0557.76037