A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Matthies, Gunar; Skrzypacz, Piotr; Tobiska, Lutz

ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 4, pp. 713-742.

The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties 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 and projection 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 version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to 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 to much 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 multiscale approach.

Classification : 65N12, 65N30, 76D05
Mots clés : stabilised finite elements, Navier-Stokes equations, equal-order interpolation

[1] L. Alaoul and A. Ern, Nonconforming finite element methods with subgrid viscosity applied to advection-diffusion-reaction equations. Numer. Meth. Part. Diff. Equat. 22 (2006) 1106-1126. | Zbl

[2] T. Apel, Anisotropic finite elements. Local estimates and applications. Advances in Numerical Mathematics. Teubner, Leipzig (1999). | MR | Zbl

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

[4] R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173-199. | Zbl

[5] 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.

[6] R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106 (2007) 349-367. | Zbl

[7] M. Braack and E. Burman, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2006) 2544-2566. | Zbl

[8] M. Braack and T. Richter, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements. Comput. Fluids 35 (2006) 372-392.

[9] M. Braack and T. Richter, Stabilized finite elements for 3D reactive flows. Int. J. Numer. Methods Fluids 51 (2006) 981-999.

[10] 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.

[11] M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853-866. | Zbl

[12] A.N. Brooks and T.J.R. Hughes, 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. | Zbl

[13] P.G. Ciarlet, The finite element method for elliptic problems. SIAM (2002). | MR

[14] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl

[15] A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004). | MR | Zbl

[16] L.P. Franca and S.L. Frey, Stabilized finite element methods: II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 99 (1992) 209-233. | Zbl

[17] T. Gelhard, G. Lube, M.A. Olshanskii and J.-H. Starcke, Stabilized finite element schemes with LBB-stable elements for incompressible flows. J. Comput. Appl. Math. 177 (2005) 243-267. | Zbl

[18] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equation, SCM 5. Springer-Verlag, Berlin (1986). | MR | Zbl

[19] J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modelling. ESAIM: M2AN 33 (1999) 1293-1316. | Numdam | Zbl

[20] J.-L. Guermond, Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class $C^0$. Comput. Visual. Sci. 2 (1999) 131-138. | Zbl

[21] J.-L. Guermond, 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. | Zbl

[22] J.-L. Guermond, Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA J. Numer. Anal. 21 (2001) 165-197. | Zbl

[23] J.-L. Guermond, A. Marra and L. Quartapelle, Subgrid stabilized projection method for 2d unsteady flows at high Reynolds numbers. Comput. Methods Appl. Mech. Engrg. 195 (2006) 5857-5876. | Zbl

[24] T.J.R. Hughes, 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. | Zbl

[25] 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.

[26] T.J.R. Hughes and G. Sangalli, Variational multiscale analysis: The fine-scale Green's function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45 (2007) 539-367.

[27] T.J.R. Hughes, L.P. Franca and M. Balestra, 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. | Zbl

[28] V. John, On large eddy simulation and variational multiscale methods in the numerical simulation of turbulent flows. Appl. Math. 51 (2006) 321-353.

[29] V. John and S. Kaya, A finite element variational multiscale method for the Navier-Stokes equations. SIAM J. Sci. Comput. 26 (2006) 1485-1503. | Zbl

[30] V. John, S. Kaya and W.J. Layton, A two-level variational multiscale method for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4594-4603. | Zbl

[31] S. Kaya and B. Rivière, A two-grid stabilization method for solving the steady-state Navier-Stokes equations. Numer. Meth. Part. Diff. Equat. 22 (2005) 728-743. | Zbl

[32] 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.

[33] G. Lube and G. Rapin, Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16 (2006) 949-966. | Zbl

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

[35] 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).

[36] G. Matthies and L. Tobiska, The inf-sup condition for the mapped $Q_k-P_{k-1}^{disc}$ element in arbitrary space dimension. Computing 69 (2002) 119-139. | Zbl

[37] 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). | MR

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

[39] R. Stenberg, Analysis of mixed finite element methods for the Stokes problem: A unified approach. Math. Comput. 42 (1999) 9-23. | Zbl

[40] L. Tobiska, Analysis of a new stabilized higher order finite element method for advection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 196 (2006) 538-550. | Zbl

[41] L. Tobiska and R. Verfürth, Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equation. SIAM J. Numer. Anal. 33 (1996) 107-127. | Zbl