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Simple finite element method in vorticity formulation for incompressible flows

Authors: Jian-Guo Liu and Weinan E
Journal: Math. Comp. 70 (2001), 579-593
MSC (2000): Primary 65M60, 76M10
Published electronically: March 3, 2000
MathSciNet review: 1710644
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Abstract | References | Similar Articles | Additional Information

Abstract: A very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation. This method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work. Optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements. The method is convectively stable and is particularly suited for moderate to high Reynolds number flows.

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  • 1. E. Barragy and G.F. Carey, Stream function vorticity solution using high-$p$ element-by-element techniques, Commun. in Applied Numer. Methods, 9 (1993) 387-395.
  • 2. K.E. Barrett, A variational principle for the stream function-vorticity formulation of the Navier-Stokes equations incorporating no-slip conditions, J. Comput. Phys. 26 (1978) 153-161. MR 57:8429
  • 3. A. Bendali, J.M. Dominguez and S. Gallic, A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three dimensional domains, J. Math. Anal. Appl. 107 (1985) 537-560. MR 86k:35121
  • 4. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991. MR 92d:65187
  • 5. A. Campion-Renson and M. J. Crochet, On the stream function-vorticity finite element solutions of Navier-Stokes equations, Int. J. Num. Meth. Engng. 12 (1978) 1809-1818.
  • 6. P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 58:25001
  • 7. P. Ciarlet and P. Raviart, A mixed finite element method for the biharmonic equation, in ``Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations'', C. de Boor, ed, Academic Press, New York, (1974) 125-143. MR 58:31907
  • 8. F. El Dabaghi and O. Pironneau, Stream vector in three dimensional aerodynamics, Numer. Math. 48 (1986) 561-589. MR 87h:65193
  • 9. Weinan E and J.-G. Liu, Vorticity boundary condition and related issues for finite difference schemes, J. Comput. Phys., 124 (1996), 368-382. MR 97a:76088
  • 10. Weinan E and J.-G. Liu, Essentially compact schemes for unsteady viscous incompressible flows, J. Comput. Phys., 126 (1996), 122-138. MR 97e:76050
  • 11. Weinan E and J.-G. Liu, Finite difference methods in vorticity-vector potential formulation on 3D non-staggered grids, J. Comput. Phys., 138 (1997) 57-82.
  • 12. R.S. Falk and J.E. Osborn, Error estimates for mixed methods, R.A.I.R.O. Anal. Numer. 14 (1980) 249-277. MR 82j:65076
  • 13. V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, Berlin, (1986). MR 88b:65129
  • 14. P.M. Gresho, Incompressible fluid dynamics: some fundamental formulation issues, Ann. Rev. Fluid Mech. 23 (1991), 413-453. MR 92e:76017
  • 15. P.M. Gresho, Some interesting issues in incompressible fluid dynamics, both in the continuum and in numerical simulation, Advances in Applied Mechanics, 28 (1992), 45-140. MR 93e:76056
  • 16. M. Gunzburger, Finite element methods for viscous incompressible flows, Academic Press, Boston, 1989 MR 91d:76053
  • 17. C. Johnson and V. Thomée, Error estimates for some mixed finite element methods for parabolic type problems, R.A.I.R.O. Anal. Numer. 15 (1981) 41. MR 83c:65239
  • 18. M. Ikegawa, A new finite element technique for the analysis of steady viscous flow problems, Int. J. Num. Meth. Engng. 14 (1979) 103-113.
  • 19. J.-G. Liu and C.-W. Shu, A high order discontinuous Galerkin method for incompressible flows, J. Comput. Phys., (to appear).
  • 20. J.-G. Liu and J. Xu, An efficient finite element method for viscous incompressible flows, in preparation.
  • 21. S.A. Orszag and M. Israeli, Numerical simulation of viscous incompressible flows, Ann. Rev. Fluid Mech., 6 (1974), 281-318.
  • 22. O.A. Pironneau, Finite Element Methods for Fluids, John Wiley and Sons (1989) MR 90j:76016
  • 23. L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations, Birkhäuser, Berlin, (1993). MR 95i:76060
  • 24. R. Schulz, A mixed method for fourth order problems using linear elements, R.A.I.R.O. Anal. Numer. 12 (1978) 85-90.
  • 25. W.N.R. Stevens, Finite element, stream function-vorticity solution of steady laminar natural convection, Int. J. Num. Meth. Fluids 2 (1982) 349-366.

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Additional Information

Jian-Guo Liu
Affiliation: Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, MD 20742

Weinan E
Affiliation: Courant Institute of Mathematical Sciences, New York, NY 10012

Keywords: Simple finite element method, Navier-Stokes equations, vorticity boundary conditions, error analysis
Received by editor(s): June 8, 1999
Published electronically: March 3, 2000
Additional Notes: JGL was supported in parts by NSF grant DMS-9805621.
Article copyright: © Copyright 2000 American Mathematical Society

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