Simple finite element method in vorticity formulation for incompressible flows

Mathematics of Computation

Volumn 70, Issue 234, 2001, Pages 579-593

  Liu, Jian Guo ^a   Weinan, E ^b  

^a Bldg 142   (United States)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Error analysis; Navier stokes equations; Simple finite element method; Vorticity boundary conditions

Indexed keywords

EID: 0035635336     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-00-01239-4     Document Type: Article

References (25)

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 biharmomc 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.