Numerical Algorithms

, Volume 38, Issue 1–3, pp 155–172 | Cite as

Application of the least-squares spectral element method using Chebyshev polynomials to solve the incompressible Navier-Stokes equations

  • Michael M. J. Proot
  • Marc I. Gerritsma
Section II: Spectral Methods


In this paper the extension of the Legendre least-squares spectral element formulation to Chebyshev polynomials will be explained. The new method will be applied to the incompressible Navier-Stokes equations and numerical results, obtained for the lid-driven cavity flow at Reynolds numbers varying between 1000 and 7500, will be compared with the commonly used benchmark results. The new results reveal that the least-squares spectral element formulations based on the Legendre and Chebyshev Gauss-Lobatto Lagrange interpolating polynomials are equally accurate.


Navier-Stokes equations least-squares spectral element method Chebyshev polynomials 

AMS subject classification

41A50 65K10 65N35 65Y20 76D05 


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  1. [1]
    A.K. Aziz, R.B. Kellogg and A.B. Stephens, Least-squares methods for elliptic systems, Math. Comput. 44(169) (1985) 53–70.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    C. Bernardi and Y. Maday,Approximations spectrale de problèmes aux limites élliptiques (Springer-Verlag, 1992).Google Scholar
  3. [3]
    P.B. Bochev, Analysis of least-squares finite element methods for the Navier-Stokes equations, SIAM J. Numer. Anal. 34(5) (1997) 1817–1844MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    P.B. Bochev and M.D. Gunzburger, A least squares finite element method for the Navier-Stokes equations, App. Math. Lett. 6(2) (1993) 27–30.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P.B. Bochev and M.D. Gunzburger, Analysis of least squares finite element methods for the Stokes equations, Math. Comput. 63(208) (1994) 479–506.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    P.B. Bochev and M.D. Gunzburger, Finite element methods of leasta-squares type, SIAM Rev. 40(4) (1998) 789–837.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    O. Botella and R. Peyret, Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids 27(4) (1998) 421–433.MATHCrossRefGoogle Scholar
  8. [8]
    Z. Cai, R. Lazarov, T.A. Manteuffel and S.F. McCormick, First-order system least-squares for secondorder partial differential equations: Part I, SIAM J. Numer. Anal. 31(6) (1994) 1785–1799.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    G.F. Carey and B.-N. Jiang, Least-squares finite element method and preconditioned conjugate gradient solution, Int. J. Numer. Methods Eng. 24 (1987) 1283–1296.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    C.L. Chang, An error estimate of the least-squares finite element method for the Stokes problem in three dimensions, Math. Comput. 63(207) (1994) 41–50.MATHCrossRefGoogle Scholar
  11. [11]
    C.L. Chang and B.-N. Jiang, An error analysis of least-squares finite element method of velocity-pressure-vorticity formulation for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 84(3) (1990) 247–255.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    W. Couzy, Spectral element discretization of the unsteady Navier-Stokes equations and its iterative solution on parallel computers, PhD thesis, École Polytechnique Fédérale de Lausanne (1995).Google Scholar
  13. [13]
    J.M. Deang and M.D. Gunzburger, Issues related to least-squares finite element methods for the Stokes equations, SIAM J. Sci. Comput. 20(3) (1998) 878–906.CrossRefMathSciNetGoogle Scholar
  14. [14]
    U. Ghia, K.N. Ghia and C.T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48 (1982) 387–411.MATHCrossRefGoogle Scholar
  15. [15]
    R.D. Henderson, Adaptive spectral element methods for turbulence and transition, in:High-order Methods for Computational Physics, Leccture Notes in Computer Science, Vol. 9, (Springer, New York, 1999) pp. 225–324.Google Scholar
  16. [16]
    B.-N. Jiang, A least-squares finite element method for incompressible Navier-Stokes problems, Int. J. Numer. Methods Fluids 14(7) (1992) 843–859.MATHCrossRefGoogle Scholar
  17. [17]
    B-N. Jiang,The Least-Squares Finite Element Method, Theory and Applications in Computational Fluid Dynamics and Electromagnetics (Springer, 1998).Google Scholar
  18. [18]
    B.-N. Jiang, On the least-squares method, Comput. Methods Appl. Mech. Engrg. 152(1–2) (1998) 239–257.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    B.-N. Jiang and C.L. Chang, Least-squares finite elements for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 78(3) (1990) 297–311.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    B.-N. Jiang, T.L. Lin and L.A. Povinelli, Large-scale computation of incompressible viscous flow by least-squares finite element method. Comput. Methods Appl. Mech. Engrg. 114(3–4) (1994) 213–231.CrossRefMathSciNetGoogle Scholar
  21. [21]
    B.-N. Jiang and L. Povinelli, Least-squares finite element method for fluid dynamics, Comput. Methods. Appl. Mech. Engrg. 81(1) (1990) 13–37.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    G.E. Karniadakis and S.J. Sherwin,Spectral/hp Element Methods for CFD (Oxford University Press, 1999).Google Scholar
  23. [23]
    Y. Maday and A.T. Patera, Spectral element methods for the incompressible Navier-Stokes equations, in:State-of-the-Art Surveys in Computational Mechanics, eds. A.K. Noorr and J.T. Oden (1989). pp. 71–143.Google Scholar
  24. [24]
    J.C. Mason and D.C. Handscomb,Chebyshev Polynomials (Chapman and Hall/CRC, 2003).Google Scholar
  25. [25]
    J.P. Pontaza and J.N. Reddy, Spectral/hp least-squares finite element formulation for the Navier-Stokes equations, J. Comput. Phys. 190 (2003) 523–549.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    M.M.J. Proot, The least-squares spectral element method, PhD thesis, Delft University of Technology (2003).Google Scholar
  27. [27]
    M.M.J. Proot and M.I. Gerritsma, A least-squares spectral element formulation for the Stokes problem, J. Sci. Comput. 17(1–3) (2002) 311–322.Google Scholar
  28. [28]
    M.M.J. Proot and M.I. Gerritsma, Least-squares spectral elements applied to the Stokes problem, J. Comput. Phys. 181(2) (2002), 454–477.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    W.W. Schultz, N.Y. Lee and J.P. Boyd, Chebyshev pseudospectral method of viscous flows with corner singularities, J. Sci. Comput. 4 (1989) 1–24.MATHCrossRefGoogle Scholar
  30. [30]
    M.R. Schumack, W.W. Schultz and J.P. Boyd, Spectral method solution of the Stokes equations on nonstaggered grid, J. Comput. Phys. 94(1) (1991) 30–58.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    X.Y. Zhao and S.J. Liao, A short note on the general boundary element method for viscous flows with high Reynolds number, Internat J. Numer. Meth. Fluids 42(4) (2003) 349–359.MATHCrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Michael M. J. Proot
    • 1
  • Marc I. Gerritsma
    • 1
  1. 1.Aerospace EngineeringDelft University of TechnologyDelftThe Netherlands

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