Error estimates for finite-element Navier-Stokes solvers without standard Inf-Sup conditions

  • Jian-Guo LiuEmail author
  • Jie Liu
  • Robert L. Pego


The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C 1 elements for velocity and C 0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.


Time-dependent incompressible flow Projection method Backward facing step Driven cavity Stokes pressure Leray projection Obtuse corner Recycling 

2000 MR Subject Classification

76D05 65M15 65M60 


  1. [1]
    Armaly, B. F., Durst, F., Pereira, J. C. F., et al, Experimental and theoretical investigation of backward-facing step flow, J. Fluid Mech., 127, 1983, 473–496.CrossRefGoogle Scholar
  2. [2]
    Barth, T., Bochev, P., Gunzburger, M., et al, A taxonomy of consistently stabilized finite element methods for the Stokes problem, SIAM J. Sci. Comput., 25, 2004, 1585–1607.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Bochev, P. B., Dohrmann, C. R., and Gunzburger, M. D., Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal., 44, 2006, 82–101.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Brown, D. L., Cortez, R. and Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168, 2001, 464–499.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Botella, O. and Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids, 27, 1998, 421–433.zbMATHCrossRefGoogle Scholar
  6. [6]
    Ciarlet, P. G., The Finite Element Methods for Elliptic Problems, North Holland, Amsterdam, 1978.Google Scholar
  7. [7]
    E, W. and Liu, J.-G., Gauge method for viscous incompressible flows, Commun. Math. Sci., 1, 2003, 317–332.MathSciNetGoogle Scholar
  8. [8]
    de Veubeke, B. F., A conforming finite element for plate bending, Int. J. Solids Structures, 4, 1968, 95–108.zbMATHCrossRefGoogle Scholar
  9. [9]
    Girault, V. and Raviart, P-A., Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, 1986.zbMATHGoogle Scholar
  10. [10]
    Grubb, G. and Solonnikov, V. A., Reduction of the basic initial-boundary value problems for the Navier-Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations, J. Soviet Math., 56, 1991, 2300–2308.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Grubb, G. and Solonnikov, V. A., Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudodifferential methods, Math. Scand., 69, 1991, 217–290.zbMATHMathSciNetGoogle Scholar
  12. [12]
    Guermond, J. L., Minev, P. and Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195(44–47), 2006, 6011–6045.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Guermond, J. L. and Shen, J., A new class of truly consistent splitting schemes for incompressible flows, J. Comput. Phys., 192, 2003, 262–276.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.Google Scholar
  15. [15]
    Henshaw, W. D. and Petersson, N. A., A split-step scheme for the incompressible Navier-Stokes equations, Numerical Simulations of Incompressible Flows, World Scientific, River Edge, 2003, 108–125.Google Scholar
  16. [16]
    Johnston, H. and Liu, J.-G., Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys., 199(1), 2004, 221–259.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Karniadakis, G. E., Israeli, M. and Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97, 1991, 414–443.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 1985, 308–323.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Lai, M. J. and Schumaker, L. L., On the approximation power of splines on triangulated quadrangulations, SIAM J. Numer. Anal., 36, 1999, 143–159.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Liu, J.-G., Liu, J. and Pego, R. L., Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math., 60, 2007, 1443–1487.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Liu, J.-G. and Pego, R. L., Stable discretization of magnetohydrodynamics in bounded domains, Comm. Math. Sci., 2009, to appear.Google Scholar
  22. [22]
    Leriche, E., Perchat, E., Labrosse, G., et al, Numerical evaluation of the accuracy and stablity properties of high-order direct Stokes solvers with or without temporal splitting, J. Sci. Comput., 26, 2006, 25–43.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Liu, J., A class of efficient, stable Navier-Stokes solvers, Ph. D. thesis, University of Maryland, 2006.Google Scholar
  24. [24]
    Orszag, S. A., Israeli, M. and Deville, M., Boundary conditions for incompressible flows, J. Sci. Comput., 1, 1986, 75–111.zbMATHCrossRefGoogle Scholar
  25. [25]
    Sani, R. L., Shen, J., Pironneau, O., et al, Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 50, 2006, 673–682.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    Soane, A. M. and Rostamian, R., Variational problems in weighted Sobolev spaces on non-smooth domains, preprint.Google Scholar
  27. [27]
    Timmermans, L. J. P., Minev, P. D. and van de Vosse, F. N., An approximate projection scheme for incompressible flow using spectral elements, Int. J. Numer. Meth. Fluids, 22, 1996, 673–688.zbMATHCrossRefGoogle Scholar

Copyright information

© Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Departments of Mathematics and PhysicsDuke UniversityDurhamUSA
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore
  3. 3.Department of Mathematical Sciences and Center for Nonlinear AnalysisCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations