On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, p. 1-20

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations 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 a fixed bound r * . We prove the inf-sup condition uniformly with respect to the meshwidth h on general quadrilateral and hexahedral meshes with hanging nodes.

DOI : https://doi.org/10.1051/m2an:2007005
Classification:  65N30,  65N35
Keywords: inf-sup condition, higher order mixed finite element, adaptive grids, hanging nodes
@article{M2AN_2007__41_1_1_0,
     author = {Heuveline, Vincent and Schieweck, Friedhelm},
     title = {On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {1},
     year = {2007},
     pages = {1-20},
     doi = {10.1051/m2an:2007005},
     zbl = {1129.65086},
     mrnumber = {2323688},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2007__41_1_1_0}
}
Heuveline, Vincent; Schieweck, Friedhelm. On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, pp. 1-20. doi : 10.1051/m2an:2007005. http://www.numdam.org/item/M2AN_2007__41_1_1_0/

[1] M. Ainsworth and 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. | Zbl 1017.76041

[2] I. Babuška and M. Suri, The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36 (1994) 578-632. | Zbl 0813.65118

[3] C. Bernardi and Y. Maday. Approximations spectrales de problèmes aux limites elliptiques. (Spectral approximation for elliptic boundary value problems). Mathématiques & Applications, Paris, Springer-Verlag 10 (1992). | MR 1208043 | Zbl 0773.47032

[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. | 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. | 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. | 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. | 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. | Zbl 0731.76042

[9] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, Springer-Verlag 15 (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. | Zbl 0991.76040

[11] P.G. Ciarlet, The finite element method for elliptic problems. Studies in Mathematics and its Applications 4, Amsterdam - New York - Oxford: North-Holland Publishing Company (1978). | MR 520174 | Zbl 0383.65058

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

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

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

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

[16] V. Heuveline and R. Rannacher, Duality-based adaptivity in the hp-finite element method. J. Numer. Math. 11 (2003) 95-113. | Zbl 1050.65111

[17] V. Heuveline and F. Schieweck, An interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes. Technical report, University Heidelberg, SFB 359 (2004).

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

[19] 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

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

[21] Ch. Schwab, p- and hp-finite element methods. Theory and applications in solid and fluid mechanics. Numerical Mathematics and Scientific Computation, Oxford: Clarendon Press (1998). | MR 1695813 | Zbl 0910.73003

[22] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl 0696.65007

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

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

[25] L. Stupelis, Navier-Stokes equations in irregular domains. Mathematics and its Applications 326, Dordrecht: Kluwer Academic Publishers (1995). | MR 1346108 | Zbl 0837.35003