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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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. Computational Phys. 26 (1978), no. 2, 153–161. MR 0468597 (57 #8429)
  • 3. A. Bendali, J. M. Domínguez, 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), no. 2, 537–560. MR 787732 (86k:35121), http://dx.doi.org/10.1016/0022-247X(85)90330-0
  • 4. Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (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. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
  • 7. P. G. Ciarlet and P.-A. Raviart, A mixed finite element method for the biharmonic equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 125–145. Publication No. 33. MR 0657977 (58 #31907)
  • 8. Fadi El Dabaghi and Olivier Pironneau, Stream vectors in three-dimensional aerodynamics, Numer. Math. 48 (1986), no. 5, 561–589. MR 839617 (87h:65193), http://dx.doi.org/10.1007/BF01389451
  • 9. Weinan E and Jian-Guo Liu, Vorticity boundary condition and related issues for finite difference schemes, J. Comput. Phys. 124 (1996), no. 2, 368–382. MR 1383764 (97a:76088), http://dx.doi.org/10.1006/jcph.1996.0066
  • 10. Weinan E and Jian-Guo Liu, Essentially compact schemes for unsteady viscous incompressible flows, J. Comput. Phys. 126 (1996), no. 1, 122–138. MR 1391626 (97e:76050), http://dx.doi.org/10.1006/jcph.1996.0125
  • 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, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753 (82j:65076)
  • 13. Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383 (88b:65129)
  • 14. P. M. Gresho, Incompressible fluid dynamics: some fundamental formulation issues, Annual review of fluid mechanics, Vol.\ 23, Annual Reviews, Palo Alto, CA, 1991, pp. 413–453. MR 1090333 (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, Vol.\ 28, Adv. Appl. Mech., vol. 28, Academic Press, Boston, MA, 1992, pp. 45–140. MR 1157640 (93e:76056), http://dx.doi.org/10.1016/S0065-2156(08)70154-6
  • 16. Max D. Gunzburger, Finite element methods for viscous incompressible flows, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. A guide to theory, practice, and algorithms. MR 1017032 (91d:76053)
  • 17. Claes Johnson and Vidar Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér. 15 (1981), no. 1, 41–78 (English, with French summary). MR 610597 (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. Olivier Pironneau, Finite element methods for fluids, John Wiley & Sons, Ltd., Chichester; Masson, Paris, 1989. Translated from the French. MR 1030279 (90j:76016)
  • 23. L. Quartapelle, Numerical solution of the incompressible Navier-Stokes equations, International Series of Numerical Mathematics, vol. 113, Birkhäuser Verlag, Basel, 1993. MR 1266843 (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
Email: jliu@math.umd.edu

Weinan E
Affiliation: Courant Institute of Mathematical Sciences, New York, NY 10012
Email: weinan@cims.nyu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-00-01239-4
PII: S 0025-5718(00)01239-4
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