Skip to main content
SpringerLink
Log in

Finite Element Methods for the Incompressible Navier-Stokes Equations

  • Chapter
Fundamental Directions in Mathematical Fluid Mechanics

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

These notes are based on lectures given in a Short Course on Theoretical and Numerical Fluid Mechanics in Vancouver, British Columbia, Canada, July 27–28, 1996, and at several other places since then. They provide an introduction to recent developments in the numerical solution of the Navier-Stokes equations by the finite element method. The material is presented in eight sections:

  1. 1.

    Introduction: Computational aspects of laminar flows

  2. 2.

    Models of viscous flow

  3. 3.

    Spatial discretization by finite elements

  4. 4.

    Time discretization and linearization

  5. 5.

    Solution of algebraic systems

  6. 6.

    A review of theoretical analysis

  7. 7.

    Error control and mesh adaptation

  8. 8.

    Extension to weakly compressible flows

Theoretical analysis is offered to support the construction of numerical methods, and often computational examples are used to illustrate theoretical results. The variational setting of the finite element Galerkin method provides the theoretical framework. The goal is to guide the development of more efficient and accurate numerical tools for computing viscous flows. A number of open theoretical problems will be formulated, and many references are made to the relevant literature.

The author acknowledges the support by the German Research Association (DFG) through the SFB 359 “Reactive Flow, Diffusion and Transport” at the University of Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. M. Ainsworth and J. T. Oden (1997), A posteriori error estimation in finite element analysis, Comput. Meth. Appl. Mech. Engrg., 142, pp. 1–88.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Apel and M. Dobrowolski (1992), Anisotropic interpolation with applications to the finite element method, Computing, 47, pp. 277–293.

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Apel (1999), Anisotropic Finite Elements: Local Estimates and Applications, Habilitation Thesis, Preprint 99-03, SFB 393, University of Magdeburg.

    Google Scholar 

  4. E. Backes (1997), Gewichtete a posteriori Fehleranalyse bei der adaptiven Finite-Elemente-Methode: Ein Vergleich zwischen Residuen- und Bank-Weiser-Schätzer, Diploma Thesis, Institute of Applied Mathematics, University of Heidelberg.

    Google Scholar 

  5. W. Bangerth and G. Kanschat (1999), deal.II Homepage, Technical Reference, Release 1.0, SFB 359, University of Heidelberg, http://gaia.iwr.uni-heidelberg.de/~deal/.

    Google Scholar 

  6. R. E. Bank, B. Weiser, and H. Yserentant (1990), A class of iterative methods for solving saddle point problems, Numer. Math., 56, 645–666.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Becker (1995), An Adaptive Finite Element Method for the Incompressible Navier-Stokes Equations on Time-Dependent Domains, Doctor Thesis, Preprint 95-44, SFB 359, Nov. 1995, University of Heidelberg.

    Google Scholar 

  8. R. Becker (1998), An adaptive finite element method for the Stokes equations including control of the iteration error, ENUMATH′95, Paris, Sept. 18–22, 1995, in: Proc. ENUMATH′97 (H. G. Bock, et al., eds.), pp. 609–620, World Scientific Publisher, Singapore.

    Google Scholar 

  9. R. Becker (1998): Weighted error estimators for finite element approximations of the incompressible Navier-Stokes equations, Preprint 98-20, SFB 359, University of Heidelberg, submitted for publication.

    Google Scholar 

  10. R. Becker, M. Braack, R. Rannacher, and C. Waguet (1998), Fast and reliable solution of the Navier-Stokes equations including chemistry, Proc. Conf. Applied Mathematics for Industrial Flow Problems (AMIF), San Feliu de Guixols (Spain), Oct. 1–3. 1998, Preprint 99-03 (SFB 359), University of Heidelberg, January 1999, to appear in Computing and Visualization in Sciences.

    Google Scholar 

  11. R. Becker, C. Johnson, and R. Rannacher (1995), Adaptive error control for multigrid finite element methods, Computing, 55, pp. 271–288.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Becker and R. Rannacher (1995), Weighted a posteriori error control in FE methods, ENUMATH′95, Paris, Sept. 18–22, 1995, Proc. ENUMATH′97 (H. G. Bock, et al., eds.), pp. 621–637, World Scientific Publishers, Singapore.

    Google Scholar 

  13. R. Becker and R. Rannacher (1994), Finite element solution of the incompressible Navier-Stokes equations on anisotropically refined meshes, Proc. Workshop “Fast Solvers for Flow Problems”, Kiel, Jan. 14–16, 1994 (W. Hackbusch and G. Wittum, eds.), pp. 52–61, NNFM, Vol. 49, Vieweg, Braunschweig.

    Google Scholar 

  14. R. Becker and R. Rannacher (1996), A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math., 4, pp. 237–264.

    MathSciNet  MATH  Google Scholar 

  15. H. Blum (1990), Asymptotic Error Expansion and Defect Correction in the Finite Element Method, Habilitation Thesis, University of Heidelberg.

    Google Scholar 

  16. M. Boman (1995), A Model Study of Hydrodynamic Stability, Masters Thesis, Chalmers University of Technology, Gothenburg, Sweden.

    Google Scholar 

  17. M. Braack (1998), An Adaptive Finite Element Method for Reactive Flow Problems, Doctor Thesis, Institute of Applied Mathematics, University of Heidelberg.

    Google Scholar 

  18. M. Braack and R. Rannacher (1999), Adaptive finite element methods for low-Mach-number flows with chemical reactions, Lecture Series 1999–03, 30th Computational Fluid Dynamics, (H. Deconinck, ed.), von Karman Institute for Fluid Dynamics, Belgium.

    Google Scholar 

  19. S. C. Brenner and R. L. Scott (1994), The Mathematical Theory of Finite Element Methods, Springer, Berlin-Heidelberg-New York.

    MATH  Google Scholar 

  20. F. Brezzi and M. Fortin (1991), Mixed and Hybrid Finite Element Methods, Springer, Berlin-Heidelberg-New York.

    Book  MATH  Google Scholar 

  21. F. Brezzi and J. Pitkäranta (1984), On the stabilization of finite element approximations of the Stokes equations, Proc. Workshop Efficient Solution of Elliptic Systems (W. Hackbusch, ed.), Vieweg, Braunschweig.

    Google Scholar 

  22. M. O. Bristeau, R. Glowinski, and J. Periaux (1987), Numerical methods for the Navier-Stokes equations: Applications to the simulation of compressible and incompressible viscous flows, Comput. Phys. Reports, 6, pp. 73–187.

    Article  Google Scholar 

  23. C. M. Chen and V. Thomée (1985), The lumped mass finite element method for a parabolic problem, J. Austral. Math. Soc, Ser. B, 26, pp. 329–354.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. J. Chorin (1968), Numerical solution of the Navier-Stokes equations, Math. Comp., 22, pp. 745–762.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. E and J. P. Liu (1998), Projection method I: Convergence and numerical boundary layers, SIAM J. Num. Anal., to appear.

    Google Scholar 

  26. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson (1995), Introduction to adaptive methods for differential equations, Acta Numerica 1995 (A. Iserles, ed.), pp. 105–158, Cambridge University Press.

    Google Scholar 

  27. M. Feistauer (1993), Mathematical Methods in Fluid Dynamics, Longman Scientific & Technical, England.

    MATH  Google Scholar 

  28. M. L. M. Giles, M. Larson, and E. Süli (1998), Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow, SIAM J. Numer. Anal, to appear.

    Google Scholar 

  29. V. Girault and P.-A. Raviart (1986), Finite Element Methods for the Navier-Stokes Equations, Springer, Heidelberg.

    Book  MATH  Google Scholar 

  30. R. Glowinski (1985), Viscous flow simulations by finite element methods and related numerical techniques, in Progress in Supercomputing in Computational Fluid Dynamics (E.M. Murman and S.S. Abarbanel, eds.), pp. 173–210, Birkhäuser, Boston.

    Google Scholar 

  31. R. Glowinski and J. Periaux (1987), Numerical methods for nonlinear problems in fluid dynamics, Proc. Int. Seminar on Scientific Supercomputers, Paris, North-Holland, Amsterdam.

    Google Scholar 

  32. P. M. Gresho (1990), On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix, Part 1: Theory, Int. J. Numer. Meth. Fluids, 11, pp. 587–620.

    Article  MathSciNet  MATH  Google Scholar 

  33. P. M. Gresho and S. T. Chan (1990), On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix, Part 2: Implementation, Int. J. Numer. Meth. Fluids, 11, pp. 621–660.

    Article  MathSciNet  MATH  Google Scholar 

  34. P. M. Gresho (1991), Incompressible fluid dynamics: Some fundamental formulation issues, Ann. Rev. Fluid Mech., 23, pp. 413–453.

    Article  MathSciNet  Google Scholar 

  35. P. M. Gresho and R. L. Sani (1998), Incompressible Flow and the Finite Element Method, John Wiley, Chichester.

    MATH  Google Scholar 

  36. W. Hackbusch (1985), Multi-Grid Methods and Applications, Springer, Heidelberg-Berlin.

    MATH  Google Scholar 

  37. P. Hanbo and C. Johnson (1995), Streamline diffusion finite element methods for fluid flow, in Finite Element Methods for Compressible and Incompressible Flow, Selected Topics from Previous VKI Lecture Series, (H. Deconinck, ed.), von Karman Institute for Fluid Dynamics, Belgium.

    Google Scholar 

  38. J. Harig (1991), Eine robuste und effiziente Finite-Elemente Methode zur Lösung der inkompressiblen 3D-Navier-Stokes Gleichungen auf Vektorrechnern, Doctor Thesis, Institute of Applied Mathematics, University of Heidelberg.

    Google Scholar 

  39. R. Hartmann (1998), A posteriori Fehlerschätzung und adaptive Schrittweiten-und Ortsgittersteuerung bei Galerkin-Verfahren für die Wärmeleitungsgleichung, Diploma Thesis, Institute of Applied Mathematics, University of Heidelberg.

    Google Scholar 

  40. J. G. Heywood (1980), The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29, pp. 639–681.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. G. Heywood and R. Rannacher (1982), Finite element approximation of the non-stationary Navier-Stokes Problem. I. Regularity of solutions and second order error estimates for spatial discretization, SIAM J. Numer. Anal., 19, pp. 275–311.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. G. Heywood and R. Rannacher (1986), Finite element approximation of the non-stationary Navier-Stokes Problem. II. Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23, pp. 750–777.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. G. Heywood and R. Rannacher (1988), Finite element approximation of the non-stationary Navier-Stokes Problem. III. Smoothing property and higher order error estimates for spatial discretization, SIAM J. Numer. Anal., 25, pp. 489–512.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. G. Heywood and R. Rannacher (1990), Finite element approximation of the non-stationary Navier-Stokes Problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27, pp. 353–384.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. G. Heywood and R. Rannacher (1986), An analysis of stability concepts for the Navier-Stokes equations, J. Reine Angew. Math., 372, pp. 1–33.

    MathSciNet  MATH  Google Scholar 

  46. J. G. Heywood, R. Rannacher, and S. Turek (1992), Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Math. Fluids, 22, pp. 325–352.

    Article  MathSciNet  Google Scholar 

  47. P. Houston, R. Rannacher, and E. Süli (1999), A posteriori error analysis for stabilised finite element approximation of transport problems, Report No. 99/04, Oxford University Computing Laboratory, Oxford, England OX1 3QD, submitted for publication.

    Google Scholar 

  48. T. J. R. Hughes and A. N. Brooks (1982), Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation, Comput. Meth. Appl. Mech. Engrg., 32, pp. 199–259.

    Article  MathSciNet  MATH  Google Scholar 

  49. T. J. R. Hughes, L. P. Franca, and M. Balestra (1986), A new finite element formulation for computational fluid mechanics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolation, Comput. Meth. Appl. Mech. Engrg., 59, pp. 85–99.

    Google Scholar 

  50. C. Johnson (1987), Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge-Lund.

    MATH  Google Scholar 

  51. C. Johnson (1989), The streamline diffusion finite element method for compressible and incompressible fluid flow, Proc. Finite Element Method in Fluids VII, Huntsville.

    Google Scholar 

  52. C. Johnson (1993), A new paradigm for adaptive finite element methods, Proc. MAFELAP′93 Conf., Brunel Univ., Uxbridge, UK, John Wiley, Chichester.

    Google Scholar 

  53. C. Johnson, S. Larsson, V. Thomée, and L. B. Wahlbin (1987), Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Math. Comp., 49, pp. 331–357.

    Article  MathSciNet  MATH  Google Scholar 

  54. C. Johnson, R. Rannacher, and M. Boman (1995), Numerics and hydrodynamic stability: Towards error control in CFD, SIAM J. Numer. Anal, 32, pp. 1058–1079.

    Article  MathSciNet  MATH  Google Scholar 

  55. C. Johnson, R. Rannacher, and M. Boman (1995), On transition to turbulence and error control in CFD, Preprint 95-06, SFB 359, University of Heidelberg.

    Google Scholar 

  56. C. Johnson and R. Rannacher (1994), On error control in CFD, Proc. Int. Workshop “Numerical Methods for the Navier-Stokes Equations”, Heidelberg, Oct. 25–28, 1993 (F.-K. Hebeker, R. Rannacher, and G. Wittum, eds.), pp. 25–28, NNFM, Vol. 47, Vieweg, Braunschweig.

    Google Scholar 

  57. S. Kracmar and J. Neustupa (1994), Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities, Z. Angew. Math. Mech., 74, pp. T637–T639.

    MATH  Google Scholar 

  58. O. Ladyshenskaya (1969), The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, London.

    Google Scholar 

  59. M. Landahl (1980), A note on an algebraic instability of viscous parallel shear flows, J. Fluid Mech., 98, p. 243.

    Article  MathSciNet  MATH  Google Scholar 

  60. P.-L. Lions (1998), Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models, Clarendon Press, Oxford.

    MATH  Google Scholar 

  61. M. Luskin and R. Rannacher (1982), On the smoothing property of the Crank-Nicolson scheme, Appl. Anal., 14, pp. 117–135.

    Article  MathSciNet  MATH  Google Scholar 

  62. M. Luskin and R. Rannacher (1981), On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal., 19, pp. 93–113.

    Article  MathSciNet  Google Scholar 

  63. K. W. Morton (1996), Numerical Solution of Convection-Diffusion Problems, Applied Mathematics and Mathematical Computation, Vol. 12, Chapman&Hall.

    MATH  Google Scholar 

  64. S. Müller-Urbaniak (1993), Eine Analyse des Zwischenschritt-θ-Verfahrens zur Lösung der instationären Navier-Stokes-Gleichungen, Doctor Thesis, Preprint 94-01, SFB 359, University of Heidelberg.

    Google Scholar 

  65. S. Müller, A. Prohl, R. Rannacher, and S. Turek (1994), Implicit time-discretization of the nonstationary incompressible Navier-Stokes equations, Proc. Workshop “Fast Solvers for Flow Problems”, Kiel, Jan. 14–16, 1994 (W. Hackbusch and G. Wittum, eds.), pp. 175-191, NNFM, Vol. 49, Vieweg, Braunschweig.

    Google Scholar 

  66. H. Oswald (1999), Lösung der instationären Navier-Stokes-Gleichungen auf Parallelrechnern, Doctor Thesis, Institute of Applied Mathematics, University of Heidelberg.

    Google Scholar 

  67. O. Pironneau (1982), On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38, pp. 309–332.

    Article  MathSciNet  MATH  Google Scholar 

  68. O. Pironneau (1983), Finite Element Methods for Fluids, John Wiley, Chichester.

    Google Scholar 

  69. A. Prohl (1995), Projektions- und Quasi-Kompressibilitätsmethoden zur Lösung der inkompressiblen Navier-Stokes-Gleichungen, Doctor Thesis, Institute of Applied Mathematics, University of Heidelberg.

    Google Scholar 

  70. A. Prohl (1998), On Quasi-Compressibility Methods and Projection Methods for Solving the Incompressible Navier-Stokes Equations, Teubner, Stuttgart.

    Google Scholar 

  71. R. Rannacher (1984), Finite element solution of diffusion problems with irregular data, Numer. Math., 43, pp. 309–327.

    Article  MathSciNet  MATH  Google Scholar 

  72. R. Rannacher (1998), Numerical analysis of nonstationary fluid flow (a survey), in “Applications of Mathematics in Industry and Technology” (V.C. Boffi and H. Neunzert, eds.), pp. 34–53, B.G. Teubner, Stuttgart.

    Google Scholar 

  73. R. Rannacher (1992), On Chorin’s projection method for the incompressible Navier-Stokes equations, Proc. Oberwolfach Conf. Navier-Stokes Equations: Theory and Numerical Methods, September 1991 (J.G. Heywood, et al., eds.), pp. 167–183, LNM, Vol. 1530, Springer, Heidelberg.

    Chapter  Google Scholar 

  74. R. Rannacher (1993), On the numerical solution of the incompressible Navier-Stokes equations, Survey lecture at the annual GAMM-Conference 1992, Leipzig, March 24–27, Z. Angew. Math. Mech., 73, pp. 203–216.

    MathSciNet  MATH  Google Scholar 

  75. R. Rannacher (1998), A posteriori error estimation in least-squares stabilized finite element schemes, Comput. Meth. Appl. Mech. Engrg., 166, pp. 99–114.

    Article  MathSciNet  MATH  Google Scholar 

  76. R. Rannacher (1999), Error control in finite element computations, Proc. NATO-Summer School “Error Control and Adaptivity in Scientific Computing” Antalya (Turkey), Aug. 1998 (H. Bulgak and C. Zenger, eds.), pp. 247–278, NATO Science Series, Series C, Vol. 536, Kluwer, Dordrecht.

    Google Scholar 

  77. R. Rannacher and S. Turek (1992), A simple nonconforming quadrilateral Stokes element, Numer. Meth. Part. Diff. Equ., 8, pp. 97–111.

    Article  MathSciNet  MATH  Google Scholar 

  78. R. Rautmann (1983), On optimal regularity of Navier-Stokes solutions at time t = 0, Math. Z., 184, pp. 141–149.

    Article  MathSciNet  MATH  Google Scholar 

  79. H. Reichert and G. Wittum (1993), On the construction of robust smoothers for incompressible flow problems, Proc. Workshop “Numerical Methods for the Navier-Stokes Equations”, Heidelberg, Oct. 25–28, 1993 (F.K. Hebeker, R. Rannacher, G. Wittum, eds.), pp. 207–216, Vieweg, Braunschweig.

    Google Scholar 

  80. M. D. Smooke (1991), Numerical Modeling of Laminar Diffusion Flames, Progress in Astronautics and Aeronautics, 135.

    Google Scholar 

  81. M. Schäfer and S. Turek (1996), The benchmark problem “flow around a cylinde”, Flow Simulation with High-Performance Computers (E.H. Hirschel, ed.), pp. 547–566, NNFM, Vol. 52, Vieweg, Braunschweig.

    Chapter  Google Scholar 

  82. F. Schieweck (1992), A parallel multigrid algorithm for solving the Navier-Stokes equations on a transputer system, Impact Comput. Sci. Engrg., 5, pp. 345–378.

    Article  MathSciNet  Google Scholar 

  83. P. Schreiber and S. Turek (1993), An efficient finite element solver for the nonsta-tionary incompressible Navier-Stokes equations in two and three dimensions, Proc. Workshop “Numerical Methods for the Navier-Stokes Equations”, Heidelberg, Oct. 25–28, 1993 (F.K. Hebeker, R. Rannacher, G. Wittum, eds.), pp. 133–144, Vieweg, Braunschweig.

    Google Scholar 

  84. J. Shen (1992), On error estimates of projection methods for the Navier-Stokes Equations: First order schemes, SIAM J. Numer. Anal., 29, pp. 57–77.

    Article  MathSciNet  MATH  Google Scholar 

  85. J. Shen (1994), Remarks on the pressure error estimates for the projection methods, Numer. Math., 67, pp. 513–520.

    Article  MathSciNet  MATH  Google Scholar 

  86. J. Shen (1996), On error estimates of the projection methods for the Navier-Stokes Equations: 2nd order schemes, Math. Comp., 65, pp. 1039–1065.

    Article  MathSciNet  MATH  Google Scholar 

  87. G. Strang (1968), On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 506–517.

    Article  MathSciNet  MATH  Google Scholar 

  88. R. Temam (1987), Navier-Stokes Equations. Theory and numerical analysis, 2nd ed., North Holland, Amsterdam.

    Google Scholar 

  89. V. Thomée (1997), Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin-Heidelberg-New York.

    MATH  Google Scholar 

  90. L. Tobiska and F. Schieweck (1989), A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation, M2AN, 23, pp. 627–647.

    MathSciNet  MATH  Google Scholar 

  91. L. Tobiska and R. Verfürth (1996), Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations, SIAM J. Numer. Anal., 33, pp. 107–127.

    Article  MathSciNet  MATH  Google Scholar 

  92. L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll (1992), A new direction in hydrodynamical stability: Beyond eigenvalues, Tech. Report CTC92TR115 12/92, Cornell Theory Center, Cornell University.

    Google Scholar 

  93. S. Turek (1994), Tools for simulating nonstationary incompressible flow via discretely divergence-free finite element models, Int. J. Numer. Meth. Fluids, 18, pp. 71–105.

    Article  MathSciNet  MATH  Google Scholar 

  94. S. Turek (1996), A comparative study of some time-stepping techniques for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 22, pp. 987–1011.

    Article  MathSciNet  MATH  Google Scholar 

  95. S. Turek (1997), On discrete projection methods for the incompressible Navier-Stokes equations: An algorithmic approach, Comput. Methods Appl. Mech. Engrg., 143, pp. 271–288.

    Article  MathSciNet  MATH  Google Scholar 

  96. S. Turek (1998), FEATFLOW: Finite Element Software for the Incompressible Navier-Stokes Equations, User Manual, Release 1.3, SFB 359, University of Heidelberg, URL: http://gaia.iwr.uni-heidelberg.de/~featflow/.

    Google Scholar 

  97. S. Turek (1999), Efficient Solvers for Incompressible Flow Problems, NNCSE, Vol. 6, Springer, Berlin-Heidelberg-New York.

    Book  MATH  Google Scholar 

  98. S. Turek, et al. (1999), Virtual Album of Fluid Motion, Preprint, Institute of Applied Mathematics, University of Heidelberg, in preparation.

    Google Scholar 

  99. M. Van Dyke (1982), An Album of Fluid Motion, The Parabolic Press, Stanford.

    Google Scholar 

  100. J. Van Kan (1986), A second-order accurate pressure-correction scheme for viscous incompressible flow, J. Sci. Stat. Comp., 7, pp. 870–891.

    Article  MATH  Google Scholar 

  101. S. P. Vanka (1986), Block-implicit multigrid solution of Navier-Stokes equations in primitive variables, J. Comp. Phys., 65, pp. 138–158.

    Article  MathSciNet  MATH  Google Scholar 

  102. R. Verfürth (1996), A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley/Teubner, New York-Stuttgart, 1996.

    MATH  Google Scholar 

  103. C. Waguet (1999) Adaptive Finite Element Computation of Chemical Flow Reactors, Doctor Thesis, SFB 359, University of Heidelberg, in preparation.

    Google Scholar 

  104. P. Wesseling (1992), An Introduction to Multigrid Methods, J. Wiley, Chichester.

    MATH  Google Scholar 

  105. G. Wittum (1990), The use of fast solvers in computational fluid dynamics, in Proc. Eighth GAMM-Conference on Numerical Methods in Fluid Mechanics (P. Wesseling, ed.), pp. 574–581. LNFM, Vol. 29, Vieweg, Braunschweig.

    Google Scholar 

  106. O. C. Zienkiewicz and J. Z. Zhu (1987), A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Meth. Engrg., 24, pp. 337–357.

    Article  MathSciNet  MATH  Google Scholar 

  107. Guohui Zhou (1997), How accurate is the streamline diffusion finite element method?, Math. Comp., 66, pp. 31–44.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, University of Heidelberg, INF 293/294, D-69120, Heidelberg, Germany

    Rolf Rannacher

Authors
  1. Rolf Rannacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Engineering Department of Mechanical Engineering, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA, 15261, USA

    Giovanni P. Galdi

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Y4

    John G. Heywood

  3. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 293/294, Germany

    Rolf Rannacher

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this chapter

Cite this chapter

Rannacher, R. (2000). Finite Element Methods for the Incompressible Navier-Stokes Equations. In: Galdi, G.P., Heywood, J.G., Rannacher, R. (eds) Fundamental Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8424-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8424-2_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9561-3

  • Online ISBN: 978-3-0348-8424-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation