Uniformly stable mixed hp-finite elements on multilevel adaptive grids with hanging nodes

Schieweck, Friedhelm

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008), p. 493-505 / Harvested from Numdam

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Résumé

We consider a family of quadrilateral or hexahedral mixed $h p$ -finite elements for an incompressible flow problem with $Q_{r}$ -elements for the velocity and discontinuous $P_{r - 1}$ -elements for the pressure where the order $r$ can vary from element to element between $2$ and an arbitrary bound. For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size, we prove the inf-sup condition uniformly with respect to the mesh size and the polynomial degree.

Publié le : 2008-01-01

MR 2423796 | Zbl 1154.76034

DOI : https://doi.org/10.1051/m2an:2008014
Classification: 65N30, 65N35, 76D07

@article{M2AN_2008__42_3_493_0,
     author = {Schieweck, Friedhelm},
     title = {Uniformly stable mixed hp-finite elements on multilevel adaptive grids with hanging nodes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {42},
     year = {2008},
     pages = {493-505},
     doi = {10.1051/m2an:2008014},
     mrnumber = {2423796},
     zbl = {1154.76034},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2008__42_3_493_0}
}

Schieweck, Friedhelm. Uniformly stable mixed hp-finite elements on multilevel adaptive grids with hanging nodes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 493-505. doi : 10.1051/m2an:2008014. http://gdmltest.u-ga.fr/item/M2AN_2008__42_3_493_0/

Bibliographie

[1] M. Ainsworth and P. Coggins, A uniformly stable family of mixed $h p$ -finite elements with continuous pressures for incompressible flow. IMA J. Numer. Anal. 22 (2002) 307-327. | MR 1897411 | Zbl 1017.76041

[2] D.N. Arnold, D. Boffi and R.S. Falk, Approximation by quadrilateral finite elements. Math. Comput. 71 (2002) 909-922. | MR 1898739 | Zbl 0993.65125

[3] I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Methods Appl. Mech. Engrg. 61 (1987) 1-40. | MR 880421 | Zbl 0593.65064

[4] C. Bernardi and Y. Maday, Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Models Methods Appl. Sci. 9 (1999) 395-414. | MR 1686546 | Zbl 0944.76058

[5] D. Boffi and L. Gastaldi, On the quadrilateral Q $_{2}$ -P $_{1}$ element for the Stokes problem. Int. J. Numer. Methods Fluids 39 (2002) 1001-1011. | MR 1918982 | Zbl 1101.76352

[6] J.M. Boland and R.A. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. | MR 708453 | Zbl 0521.76027

[7] S. Bönisch, V. Heuveline and P. Wittwer, Adaptive boundary conditions for exterior flow problems. J. Math. Fluid Mech. 7 (2005) 85-107. | MR 2127743 | Zbl 1072.76049

[8] 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

[9] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[10] L. Chilton and M. Suri, On the construction of stable curvilinear $p$ version elements for mixed formulations of elasticity and Stokes flow. Numer. Math. 86 (2000) 29-48. | MR 1774009 | Zbl 0991.76040

[11] 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

[12] V. Heuveline and M. Hinze, Adjoint-based adaptive time-stepping for partial differential equations using proper orthogonal decomposition. Technical report, University Heidelberg, Germany, SFB 359 (2004).

[13] V. Heuveline and R. Rannacher, A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15 (2001) 107-138. | MR 1887731 | Zbl 0995.65111

[14] V. Heuveline and R. Rannacher, Duality-based adaptivity in the $h p$ -finite element method. J. Numer. Math. 11 (2003) 95-113. | MR 1987590 | Zbl 1050.65111

[15] V. Heuveline and F. Schieweck, $H^{1}$ -interpolation on quadrilateral and hexahedral meshes with hanging nodes. Computing 80 (2007) 203-220. | MR 2328915 | Zbl 1133.65100

[16] V. Heuveline and F. Schieweck, On the inf-sup condition for higher order mixed fem on meshes with hanging nodes. ESAIM: M2AN 41 (2007) 1-20. | Numdam | MR 2323688 | Zbl 1129.65086

[17] G. Matthies, Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors. Numer. Algorithms 27 (2001) 317-327. | MR 1880917 | Zbl 0998.65117

[18] G. Matthies, Finite element methods for free boundary value problems with capillary surfaces. Ph.D. thesis, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Germany (2002). [Published at Shaker-Verlag Aachen].

[19] G. Matthies and F. Schieweck, On the reference mapping for quadrilateral and hexahedral finite elements on multilevel adaptive grids. Computing 80 (2007) 95-119. | MR 2318501 | Zbl 1123.65114

[20] 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. | MR 1954791 | Zbl 1016.65073

[21] S. Schötzau, C. Schwab and R. Stenberg, Mixed $h p$ -fem on anisotropic meshes II: Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83 (1999) 667-697. | MR 1728222 | Zbl 0958.76049

[22] C. Schwab, $p$ - and $h p$ -Finite Element Methods, Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation. Oxford Science Publications, Clarendon Press (1998). | MR 1695813 | Zbl 0910.73003

[23] R. Stenberg, Error analysis of some finite element methods for the Stokes problem. Math. Comput. 54 (1990) 495-508. | MR 1010601 | Zbl 0702.65095

[24] R. Stenberg and M. Suri, Mixed $h p$ finite element methods for problems in elasticity and Stokes flow. Numer. Math. 72 (1996) 367-389. | MR 1367655 | Zbl 0855.73075

[25] A. Toselli and C. Schwab, Mixed $h p$ -finite element approximations on geometric edge and boundary layer meshes in three dimensions. Numer. Math. 94 (2003) 771-801. | MR 1990592 | Zbl 1051.65111