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A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

Published online by Cambridge University Press:  12 August 2008

Malte Braack
Malte Braack*
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. [email protected]
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Abstract

It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instancestreamline upwind Petrov-Galerkin (SUPG), are optimal on isotropicmeshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. Wedescribe the new methodand prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropicstabilization methods.The capability of the methodis illustrated by means of two numerical test problems.

Keywords

Incompressible flowNavier-Stokes equationsstabilized finiteelementsanisotropic meshes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 6 , November 2008 , pp. 903 - 924
Copyright
© EDP Sciences, SMAI, 2008

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References

T. Apel, Anisotropic finite elements: Local estimates and applications, Advances in Numerical Mathematics. Teubner, Stuttgart (1999).
R. Becker, An adaptive finite element method for the incompressible Navier-Stokes equation on time-dependent domains. Ph.D. Dissertation, SFB-359 Preprint 95-44, Universität Heidelberg, Germany (1995).
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, ENUMATH 2003, E.A.M. Feistauer Ed., Springer (2004) 123–130.
Becker, R., Braack, M. and Vexler, B., Numerical parameter estimaton for chemical models in multidimensional reactive flows. Combust. Theory Model. 8 (2004) 661–682.
Becker, R., Braack, M. and Vexler, B., Parameter identification for chemical models in combustion problems. Appl. Numer. Math. 54 (2005) 519–536. CrossRef
M. Braack, Anisotropic H 1-stable projections on quadrilateral meshes, in Numerical Mathematics and Advanced Applications, Enumath Proc. 2005, B. de Castro Ed., Springer (2006) 495–503.
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
M. Braack and T. Richter, Local projection stabilization for the Stokes system on anisotropic quadrilateral meshes, in Numerical Mathematics and Advanced Applications, Enumath Proc. 2005, B. de Castro Ed., Springer (2006) 770–778.
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 flow. Int. J. Numer. Methods Fluids 51 (2006) 981–999. CrossRef
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. and Hughes, T., Streamline upwind Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. CrossRef
Burman, E., Fernandez, M. and Hansbo, P., Edge stabilization for the incompressible Navier-Stokes equations: a continuous interior penalty finite element method. SIAM J. Numer. Anal. 44 (2006) 1248–1274. CrossRef
P. Ciarlet, Finite Element Methods for Elliptic Problems. North-Holland, Amsterdam (1978).
Codina, R., Stabilization of incompressibility and convection through orthogonal subscales in finite element methods. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1579–1599. CrossRef
Codina, R. and Soto, O., Approximation of the incompressible Navier-Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1403–1419. CrossRef
Formaggia, L. and Perotto, S., Anisotropic error estimates for elliptic problems. Numer. Math. 94 (2003) 67–92. CrossRef
Formaggia, L., Micheletti, S. and Perotto, S., Anisotropic mesh adaptation in computational fluid dynamics: Application to the advection-diffusion-reaction and the Stokes problems. Appl. Numer. Math. 51 (2004) 511–533. CrossRef
Franca, L. and Frey, S., Stabilized finite element methods: II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 99 (1992) 209–233. CrossRef
Guermond, J.-L., Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM: M2AN 33 (1999) 1293–1316. CrossRef
Hansbo, P. and Szepessy, A., A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 84 (1990) 175–192. CrossRef
Hughes, T., Franca, L. and Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumvent the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation for the Stokes problem accommodating equal order interpolation. Comput. Methods Appl. Mech. Engrg. 59 (1986) 89–99. CrossRef
John, V. and Kaya, S., A finite element variational multiscale method for the Navier-Stokes equations. SIAM J. Sci. Comp. 26 (2005) 1485–1503. CrossRef
John, V., Kaya, S. and Layton, W., A two-level variational multiscale method for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 26 (2005) 4594–4603.
Kunisch, K. and Vexler, B., Optimal vortex reduction for instationary flows based on translation invariant cost functionals. SIAM J. Contr. Opt. 46 (2007) 1368–1397. CrossRef
Linss, T., Anisotropic meshes and streamline-diffusion stabilization for convection-diffusion problems. Comm. Numer. Methods Engrg. 21 (2005) 515–525. CrossRef
Lube, G. and Apel, T., Anisotropic mesh refinement in stabilized Galerkin methods. Numer. Math. 74 (1996) 261–282.
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
G. Lube, T. Knopp and R. Gritzki, Stabilized FEM with anisotropic mesh refinement for the Oseen problem, in Proceedings ENUMATH 2005, Springer (2006) 799–806.
Matthies, G., Skrzypacz, P. and Tobiska, L., A unified convergence analysis for local projection stabilisations applied ro the Oseen problem. ESAIM: M2AN 41 (2007) 713–742. CrossRef
Micheletti, S., Perotto, S. and Picasso, M., Stabilized finite elements on anisotropic meshes: A priori estimate for the advection-diffusion and the Stokes problem. SIAM J. Numer. Anal. 41 (2003) 1131–1162. CrossRef
Paillere, H., Le Quéré, P., Weisman, C., Vierendeels, J., Dick, E., Braack, M., Dabbene, F., Beccantini, A., Studer, E., Kloczko, T., Corre, C., Heuveline, V., Darbandi, M. and Hosseinizadeh, S., Modelling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part 2. Contributions to the June 2004 conference. ESAIM: M2AN 39 (2005) 617–621. CrossRef
Tobiska, L. and Lube, G., A modified streamline diffusion method for solving the stationary Navier-Stokes equations. Numer. Math. 59 (1991) 13–29. CrossRef