Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T20:00:17.587Z Has data issue: false hasContentIssue false
  • English
  • Français

A unified convergence analysis for local projection stabilisationsapplied to the Oseen problem

Published online by Cambridge University Press:  04 October 2007

Gunar Matthies ,
Piotr Skrzypacz  and
Lutz Tobiska
Gunar Matthies
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany. gunar.matthies@ruhr-uni-bochum.de
Piotr Skrzypacz
Affiliation:
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany. piotr.skrzypacz@mathematik.uni-magdeburg.de; tobiska@mathematik.uni-magdeburg.de
Lutz Tobiska
Affiliation:
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany. piotr.skrzypacz@mathematik.uni-magdeburg.de; tobiska@mathematik.uni-magdeburg.de
Get access

Abstract

The discretisation of the Oseen problem by finite element methods may sufferin general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi)condition can be violated. Second, spurious oscillationsoccur due to the dominating convection. One way to overcome bothdifficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation andprojection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory,we extend the results of Braack and Burman for the standard two-level versionof the local projection stabilisation to discretisations of arbitrary order onsimplices, quadrilaterals, and hexahedra. Moreover, our general theory allowsto derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads tomuch more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscaleapproach.

Keywords

Stabilised finite elementsNavier-Stokes equationsequal-orderinterpolation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 4 , July 2007 , pp. 713 - 742
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alaoul, L. and Ern, A., Nonconforming finite element methods with subgrid viscosity applied to advection-diffusion-reaction equations. Numer. Meth. Part. Diff. Equat. 22 (2006) 1106–1126. CrossRef
T. Apel, Anisotropic finite elements. Local estimates and applications. Advances in Numerical Mathematics. Teubner, Leipzig (1999).
Arnold, D.N., Boffi, D. and Falk, R.S., Approximation by quadrilateral finite elements. Math. Comput. 71 (2002) 909–922. CrossRef
Becker, R. and Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173–199. CrossRef
R. Becker and M. Braack, A two-level stabilization scheme for the Navier-Stokes equations, in Numerical mathematics and advanced applications, M. Feistauer et al. Eds., Berlin, Springer-Verlag (2004) 123–130.
Becker, R. and Vexler, B., Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106 (2007) 349–367. CrossRef
Braack, M. and Burman, E., Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2006) 2544–2566. CrossRef
Braack, M. and Richter, T., Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements. Comput. Fluids 35 (2006) 372–392. CrossRef
Braack, M. and Richter, T., Stabilized finite elements for 3D reactive flows. Int. J. Numer. Methods Fluids 51 (2006) 981–999. CrossRef
M. Braack and T. Richter, Solving multidimensional reactive flow problems with adaptive finite elements, in Reactive Flows, Diffusion and Transport, W. Jäger, R. Rannacher and J. Warnatz Eds., Springer-Verlag (2007) 93–112.
Braack, M., Burman, E., John, V. and Lube, G., Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. CrossRef
Brooks, A.N. and Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems. SIAM (2002).
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84.
A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004).
Franca, L.P. and Frey, S.L., Stabilized finite element methods: II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 99 (1992) 209–233. CrossRef
Gelhard, T., Lube, G., Olshanskii, M.A. and Starcke, J.-H., Stabilized finite element schemes with LBB-stable elements for incompressible flows. J. Comput. Appl. Math. 177 (2005) 243–267. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equation, SCM 5. Springer-Verlag, Berlin (1986).
Guermond, J.-L., Stabilization of Galerkin approximations of transport equations by subgrid modelling. ESAIM: M2AN 33 (1999) 1293–1316. CrossRef
Guermond, J.-L., Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class C 0. Comput. Visual. Sci. 2 (1999) 131–138. CrossRef
Guermond, J.-L., Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class C 0 in Hilbert spaces. Numer. Meth. Part. Diff. Equat. 17 (2001) 1–25. 3.0.CO;2-1>CrossRef
Guermond, J.-L., Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA J. Numer. Anal. 21 (2001) 165–197. CrossRef
Guermond, J.-L., Marra, A. and Quartapelle, L., Subgrid stabilized projection method for 2d unsteady flows at high Reynolds numbers. Comput. Methods Appl. Mech. Engrg. 195 (2006) 5857–5876. CrossRef
Hughes, T.J.R., Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401. CrossRef
T.J.R. Hughes and G. Sangalli, Variational multiscale analysis: Projection, optimization, the fine-scale Greens' function, and stabilized methods. USNCCM8, Austin (2005) 27–29.
Hughes, T.J.R. and Sangalli, G., Variational multiscale analysis: The fine-scale Green's function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45 (2007) 539–367. CrossRef
Hughes, T.J.R., Franca, L.P. and Balestra, M., A new finite element formulation for computational fluid dynamics. V: Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accomodating equal-order interpolations. Comput. Methods Appl. Mech. Engrg. 59 (1986) 85–99. CrossRef
John, V., On large eddy simulation and variational multiscale methods in the numerical simulation of turbulent flows. Appl. Math. 51 (2006) 321–353. CrossRef
John, V. and Kaya, S., A finite element variational multiscale method for the Navier-Stokes equations. SIAM J. Sci. Comput. 26 (2006) 1485–1503. CrossRef
John, V., Kaya, S. and Layton, W.J., A two-level variational multiscale method for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4594–4603. CrossRef
Kaya, S. and Rivière, B., A two-grid stabilization method for solving the steady-state Navier-Stokes equations. Numer. Meth. Part. Diff. Equat. 22 (2005) 728–743. CrossRef
G. Lube, Stabilized FEM for incompressible flow. Critical review and new trends, in European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006, P. Wesseling, E. Onate and J. Périaux Eds., The Netherlands (2006) 1–20 TU Delft.
Lube, G. and Rapin, G., Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16 (2006) 949–966. CrossRef
Matthies, G., Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors. Numer. Algorithms 27 (2001) 317–327. CrossRef
G. Matthies and G. Lube, On streamline-diffusion methods of inf-sup stable discretisations of the generalised Oseen problem. Preprint 2007-02, Institut für Numerische und Angewandte Mathematik, Georg-August-Universiät Göttingen (2007).
Matthies, G. and Tobiska, L., The inf-sup condition for the mapped $Q_k-P_{k-1}^{disc}$ element in arbitrary space dimension. Computing 69 (2002) 119–139. CrossRef
H.-G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems, SCM 24. Springer-Verlag, Berlin (1996).
Scott, L. R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. CrossRef
Stenberg, R., Analysis of mixed finite element methods for the Stokes problem: A unified approach. Math. Comput. 42 (1999) 9–23.
Tobiska, L., Analysis of a new stabilized higher order finite element method for advection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 196 (2006) 538–550. CrossRef
Tobiska, L. and Verfürth, R., Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equation. SIAM J. Numer. Anal. 33 (1996) 107–127. CrossRef