Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



An error estimate for two-dimensional Stokes driven cavity flow

Authors: Zhiqiang Cai and Yanqiu Wang
Journal: Math. Comp. 78 (2009), 771-787
MSC (2000): Primary 65N15, 65N30, 76D07
Published electronically: October 1, 2008
MathSciNet review: 2476559
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Discontinuous velocity boundary data for the lid driven cavity flow has long been causing difficulties in both theoretical analysis and numerical simulations. In finite element methods, the variational form for the driven cavity flow is not valid since the velocity is not in $ \boldsymbol{H}^1$. Hence standard error estimates do not work. By using only $ \mathbf{W}^{1,r}$ $ (1<r< 2)$ regularity and constructing a continuous approximation to the boundary data, here we present error estimates for both the velocity-pressure formulation and the pseudostress-velocity formulation of the two-dimensional Stokes driven cavity flow.

References [Enhancements On Off] (What's this?)

  • 1. C. Amrouche and V. Girault, On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan Acad. 67 (1991), 171-175. MR 1114965 (92i:35098)
  • 2. Douglas N. Arnold and Ragnar Winther, Mixed finite element for elasticity, Numer. Math. 92 (2002), 401-419. MR 1930384 (2003i:65103)
  • 3. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Springer-Verlag, Berlin, Heidelberg, 1994. MR 1278258 (95f:65001)
  • 4. Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 5. Z. Cai and G. Starke, First-order system least squares for the stress-displacement formulation: Linear elasticity, SIAM J. Numer. Anal. 41 (2003), 715-730. MR 2004196 (2005e:65180)
  • 6. Z. Cai, C. Tong, P.S. Vassilevski, and C. Wang, Mixed finite element methods for incompressible flow: Stationary Stokes equations, Preprint.
  • 7. Z. Cai and Y. Wang, A multigrid method for the pseudostress formulation of Stokes problems, SIAM J. Sci. Comp. 29 (2007), 2078-2095. MR 2350022
  • 8. L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Univ. Padova 31 (1961), 308-340. MR 0138894 (25:2334)
  • 9. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • 10. M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, R.A.I.R.O. R3 (1973), 33-76. MR 0343661 (49:8401)
  • 11. H. Elman, D. Silvester, and A. Wathen, Finite elements and fast iterative solvers: With applications in incompressible fluid dynamics, Oxford University Press, New York, 2005. MR 2155549 (2006f:65002)
  • 12. R.S. Falk and J.E. Osborn, Error estimates for mixed methods, RAIRO. Numer. Anal. 14 (1980), 249-277. MR 592753 (82j:65076)
  • 13. G.J. Fix and M.D. Gunzburger, On finite element approximations of problems having inhomogeneous essential boundary conditions, Comp. Math. Appl. 9 (1983), 687-700. MR 726817 (85b:65102)
  • 14. G.P. Galdi, C.G. Simader, and H. Sohr, On the Stokes problem in Lipschitz domains, Annali di Matematica pura ed applicata CLXVII (1994), 147-163. MR 1313554 (95m:35142)
  • 15. Vivette Girault and Pierre Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, no. 5, Springer-Verlag, New York, 1986. MR 851383 (88b:65129)
  • 16. P. Grisvard, Singularités des solutions du problème de Stokes dans un polygone, Univ. de Nice., 1979.
  • 17. -, Elliptic problems in nonsmooth domains, Pitman, Boston, 1985. MR 775683 (86m:35044)
  • 18. M.D. Gunzburger and S.L. Hou, Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses, SIAM J. Numer. Anal. 29 (1992), 390-424. MR 1154272 (93d:76039)
  • 19. P. Hood and C. Taylor, A numerical solution of the Navier-Stokes equations using the finite element technique, Comp. and Fluids 1 (1973), 73-100. MR 0339677 (49:4435)
  • 20. D.D. Joseph and L. Sturges, The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. II, SIAM J. Appl. Math. 34 (1978), 7-27. MR 0475146 (57:14765)
  • 21. H.K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18 (1964), 1-18.
  • 22. P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical aspects of the finite element method (I. Galligani and E. Magenes, eds.), Lecture notes in Mathematics, Vol. 606, Springer-Verlag, 1977. MR 0483555 (58:3547)
  • 23. P.N. Shankar, The eddy structure in Stokes flows in a cavity, J. Fluid Mech. 250 (1993), 371-383.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N15, 65N30, 76D07

Retrieve articles in all journals with MSC (2000): 65N15, 65N30, 76D07

Additional Information

Zhiqiang Cai
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Yanqiu Wang
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Received by editor(s): September 13, 2007
Received by editor(s) in revised form: May 7, 2008
Published electronically: October 1, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society