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Algebraic pressure segregation methods for the incompressible Navier-Stokes equations

  • S. Badia
  • R. Codina
Article

Abstract

This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled.

Keywords

pressure segregation pressure correction fractional step velocity correction predictor-corrector incompressible Navier-Stokes equations preconditioners 

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References

  1. 1.
    F. Armero and J.C. Simo. Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with application to MHD and Navier-Stokes equations.Computer Methods in Applied Mechanics and Engineering, 131:41–90, 1996.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    K. Arrow, L. Hurwicz, and H. Uzawa. Studies in Nonlinear Programming. Stanford University Press Stanford,CA, 1958.Google Scholar
  3. 3.
    I. Babuŝka. Error bounds for the finite element method.Numerische Mathematik, 16:322–333, 1971.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. Badia. Stabilized Pressure Segregation Methods and their Application to Fluid-Structure Interaction Problems. PhD thesis, Escola Tècnica Superior d’Enginyers de Camins,CanalsiPorts, Universitat Politècnica de Catalunya, Barcelona, 2006.Google Scholar
  5. 5.
    S. Badia and R. Codina. Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition.Numerische Mathematik, 107(4):533–557, 2007.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. Badia and R. Codina. On some fluid-structure iterative algorithms using pressure segregation methods. Application to aeroelasticity.International Journal for Numerical Methods in Engineering, 72:46–71, 2007.CrossRefMathSciNetGoogle Scholar
  7. 7.
    S.Badia and R. Codina. Pressure segregation methods based on a discrete pressure Poisson equation. An algebraic approach. International Journal for Numerical Methods in Fluids,In press.Google Scholar
  8. 8.
    G.A. Baker, V.A. Dougalis, and A. Karakashian. On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations.Mathematics of Computation, 39:339–375, 1982.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J.B. Bell, P. Colella, and H.M. Glaz. A second-order projection method for the incompressible Navier-Stokes equations.Journal of Computational Physics, 85:257–283, 1989.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Blasco and R. Codina. Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier-Stokes equations.Applied Numerical Mathematics, 38:475–497, 2001.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Blasco and R. Codina. Error estimates for an operator splitting method for incompressible flows.Applied Numerical Mathematics, 51:1–17, 2004.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Blasco, R. Codina, and A. Huerta. A fractional step method for the incompressible Navier-Stokes equations related to a predictor-multicorrector algorithm.International Journal for Numerical Methods in Fluids, 28:1391–1419, 1998.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J.M. Boland and R.A. Nicolaides. Stability of finite elements under divergence constraints.SIAM Journal on Numerical Analysis, 20:722–731, 1983.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S.C. Brenner and L.R. Scott.The mathematical theory of finite element methods. Springer-Verlag, 1994.Google Scholar
  15. 15.
    F. Brezzi. On the existence,uniqueness and approximation of saddle point problems arising from lagrange multipliers.RAIRO Anal.Numer., 8:129–151, 1974.MathSciNetGoogle Scholar
  16. 16.
    F. Brezzi and K.J. Bathe. A discourse on the stability conditions for mixed finite element formulations.Computer Methods in Applied Mechanics and Engineering, 82:27–57, 1990.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    F. Brezzi and J. Douglas. Stabilized mixed methods for the Stokes problem.Numerische Mathematik, 53:225–235, 1988.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer Verlag, 1991.Google Scholar
  19. 19.
    F. Brezzi, J. Rappaz, and P.A. Raviart. Finite dimensional approximation of nonlinear problems. Part I: Branches of non-singular solutions.Numerische Mathematik, 36:1–25, 1981.CrossRefGoogle Scholar
  20. 20.
    F. Brezzi, J. Rappaz, and P.A. Raviart. Finite dimensional approximation of nonlinear problems. Part II: Limit points.Numerische Mathematik, 37:1–28, 1981.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    F. Brezzi, J. Rappaz, and P.A. Raviart. Finite dimensional approximation of nonlinear problems. Part III: Simple bifurcation points.Numerische Mathematik, 38:1–30, 1981.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A.N. Brooks and T.J.R. Hughes. Streamline upwind / Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation.Computer Methods in Applied Mechanics and Engineering, 32:199–259, 1982MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    J. Cahouet and J.-P. Chabard. Some fast 3D finite element solvers for the generalized Stokes problem.International Journal for Numerical Methods in Fluids, 8:869–895, 1988.CrossRefMathSciNetGoogle Scholar
  24. 24.
    A.J. Chorin. A numerical method for solving incompressible viscous problems.Journal of Computational Physics, 2:12–26, 1967.MATHCrossRefGoogle Scholar
  25. 25.
    A.J. Chorin. The numerical solution of the Navier-Stokes equations for an incompressible fluid. AEC Research and Development Report, NYO-1480-82. New York University, New York, 1967.Google Scholar
  26. 26.
    A.J. Chorin. Numerical solution of the Navier-Stokes equations.Mathematics of Computation, 22:745–762, 1968.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    A.J. Chorin. On the convergence of discrete approximation to the Navier-Stokes equations.Mathematics of Computation, 23, 1969.Google Scholar
  28. 28.
    P.G. Ciarlet. The finite element method for elliptic problems. North-Holland, Amsterdam, 1978.MATHCrossRefGoogle Scholar
  29. 29.
    Clay Mathematics Institute.http://www.claymath.org/millenium/. Millenium Problems, 2000.Google Scholar
  30. 30.
    R. Codina. Comparison of some finite element methods for solving the diffusion-convection-reaction equation.Computer Methods in Applied Mechanics and Engineering, 156:185–210, 1998.MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    R. Codina. Stabilization of incompressibility and convection through orthogonal subscales in finite element methods.Computer Methods in Applied Mechanics and Engineering, 190:1579–1599, 2000.MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    R. Codina. Pressure stability in fractional step finite element methods for incompressible flows.Journal of Computational Physics, 170:112–140, 2001.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    R. Codina. Stabilized finite element approximation of transient incompressible flows using orthogonal subscales.Computer Methods in Applied Mechanics and Engineering, 191:4295–4321, 2002.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    R. Codina and S. Badia. Second order fractional step schemes for the incompressible Navier-Stokes equations. Inherent pressure stability and pressure stabilization. InProceedings of WCCM VI, Beijing, China, 2004.Google Scholar
  35. 35.
    R. Codina and S. Badia. On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems.Computer Methods in Applied Mechanics and Engineering, 195:2900–2918, 2006.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    R. Codina and J. Blasco. A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation.Computer Methods in Applied Mechanics and Engineering, 143:373–391, 1997.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    R. Codina and J. Blasco. Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations.Numerische Mathematik, 87:59–81, 2000.MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    R. Codina and J. Blasco. Analysis of a stabilized finite element approximation of the transient convection-difusion-reaction equation using orthogonal subscales.Computing and Visualization in Science, 4:167–174, 2002.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    R. Codina and A. Folch. A stabilized finite element predictor-corrector scheme for the incompressible Navier-Stokes equations using a nodal based implementation.International Journal for Numerical Methods in Fluids, 44:483–503, 2004.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    R. Codina, J. Principe, O. Guasch, and S. Badia. Time dependent subscales in the stabilized finite element approximation of incompressible flow problems.Computer Methods in Applied Mechanics and Engineering, 196:24132–2430, 2007.CrossRefMathSciNetGoogle Scholar
  41. 41.
    R. Codina and O. Soto. Approximation of the incompressible Navier-Stokes equations using orthogonal-subscale stabilization and pressure segregation on anisotropic finite element meshes.Computer Methods in Applied Mechanics and Engineering, 193:1403–1419, 2004.MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    G. de Rham.Variétés différentiables formes, courants, formes harmoniques. Paris Hermann, 1973.Google Scholar
  43. 43.
    W.E and J.G. Liu. Projection method I:Convergence and numerical boundary layers.SIAM Journal on Numerical Analysis, 32:1017–1057, 1995.CrossRefMathSciNetGoogle Scholar
  44. 44.
    H.C. Elman. Preconditioners for saddle point problems arising in computational fluid dynamics.Applied Numerical Mathematics, 43:75–89, 2002.MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    H.C. Elman, V.E. Howle, J.N. Shadid, and R.S. Tuminaro. A parallel block multi-level preconditioner for the 3D incompressible Navier-Stokes equations.Journal of Computational Physics, 187:504–523, 2003.MATHCrossRefGoogle Scholar
  46. 46.
    H.C. Elman, D.J. Silvester, and A.J. Wathen. Block preconditioners for the discrete incompressible navier-stokes equations.International Journal for Numerical Methods in Fluids, 40:333–344, 2002.MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    A. Ern and J.L. Guermond.Theory and Practice of Finite Elements. Springer-Verlag, 2004.Google Scholar
  48. 48.
    L. Franca and R. Stenberg. Error analysis of some Galerkin least-squares methods for the elasticity equations.SIAM Journal on Numerical Analysis, 28:1680–1697, 1991.MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    I. Fried and D.S. Malkus. Finite element mass matrix lumping by numerical integration with no convergence rate loss.International Journal of Solids and Structures, 11:461–466, 1975.MATHCrossRefGoogle Scholar
  50. 50.
    P. Gervasio. Convergence analysis of high order algebraic fractional step schemes for time-dependent Stokes equations. Technical report, Quaderno del Seminario Matematico di Brescia, 2006.Google Scholar
  51. 51.
    P. Gervasio and F. Saleri. Algebraic fractional-step schemes for time-dependent incompressible Navier-Stokes equations.Journal of Scientific Computating, 27(1-3):257–269, 2006.MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    P. Gervasio, F. Saleri, and A. Veneziani. Algebraic fractional-step schemes with spectral methods for the incompressible Navier-Stokes equations.Journal of Computational Physics, 214(1):347–365, 2006.MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    V. Girault and P.A. Raviart.Finite element methods for Navier-Stokes equations. Springer-Verlag, 1986.Google Scholar
  54. 54.
    P.M. Gresho and R.L. Sani.Incompressible flow and the finite element method. John Wiley & Sons, 2000.Google Scholar
  55. 55.
    P.M. Gresho. 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 I: Theory.International Journal for Numerical Methods in Fluids, 11:587–620, 1990.MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    P.M. Gresho, S.T. Chan, M.A. Christon, and A.C. Hindmarsh. A little more on stabilizedq 1 q 1 for transient viscous incompressible flow.International Journal for Numerical Methods in Fluids, 21:837–856, 1995.MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    J.L. Guermond. Remarques sur les méthodes de projection pour l’approximation des éequations de Navier-Stokes.Numerische Mathematik, 67:465–473, 1994.MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    J.L. Guermond, P. Minev, and J. Shen. An overview of projection methods for incompressible flows.Computer Methods in Applied Mechanics and Engineering, to appear.Google Scholar
  59. 59.
    J.L. Guermond and L. Quartapelle. On stability and convergence of projection methods based on pressure Poisson equation.International Journal for Numerical Methods in Fluids, 26:1039–1053, 1998.MATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    J.L. Guermond and L. Quartapelle. On the approximation of the unsteady Navier-Stokes equations by finite element projection methods.Numerische Mathematik, 80:207–238, 1998.MATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    J.L. Guermond and J. Shen. A new class of truly consistent splitting schemes for incompressible flows.Journal of Computational Physics, 192:262–276, 2003.MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    J.L. Guermond and J. Shen. Velocity-correction projection methods for incompressible flows.SIAM Journal on Numerical Analysis, 41:112–134, 2003.MATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    J.L. Guermond and J. Shen. On the error estimates for the rotational pressure-correction projection methods.Mathematics of Computation, 73:1719–1737, 2004.MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    J.G. Heywood and R. Rannacher. Finite element approximation of the nonstationary Navier-Stokes problem. I: Regularity of solutions and second-order error estimates for spatial discretization.SIAM Journal on Numerical Analysis, 19:275–311, 1982.MATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    E. Hinton, T. Rock, and O.C. Zienkiewicz. A note on mass lumping and related processes in the finite element method.Earthquake Engineering and Structural Dynamics, 4:245–249, 1976.CrossRefGoogle Scholar
  66. 66.
    T.J.R. Hughes. Multiscale phenomena: Green’s function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized formulations.Computer Methods in Applied Mechanics and Engineering, 127:387–401, 1995.MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces.Computer Methods in Applied Mechanics and Engineering, 65:85–96, 1987.MATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations.Computer Methods in Applied Mechanics and Engineering, 73:173–189, 1989.MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    W. Hundsdorfer and J.G. Verwer.Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, 2003.Google Scholar
  70. 70.
    G.E. Karniadakis, M. Israeli, and S.E. Orszag. High order splitting methods for the incompressible Navier-Stokes equations.Journal of Computational Physics, 59:414–443, 1991.CrossRefMathSciNetGoogle Scholar
  71. 71.
    J. Kim and P. Moin. Application of the fractional step method to incompressible Navier-Stokes equations.Journal of Computational Physics, 59:308–323, 1985.MATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    O. Ladyzhenskaya.The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1969.MATHGoogle Scholar
  73. 73.
    J. Leray. Essai sur les mouvements d’un liquide visqueux emplissant l’espace.Acta Mathematica, 63:193–248, 1934.MATHCrossRefMathSciNetGoogle Scholar
  74. 74.
    D. Loghin and A.J. Wathen. Schur complement preconditioners for the Navier-Stokes equations.International Journal for Numerical Methods in Fluids, 40:403–412, 2002.MATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Oden and L.Z. Demkowicz.Applied Funtional Analysis. CRC Press,1996.Google Scholar
  76. 76.
    S.A. Orszag, M. Israeli, and M. Deville. Boundary conditions for incompressible flows.J. Sci. Comput., 1:75–111, 1986.MATHCrossRefGoogle Scholar
  77. 77.
    J.B. Perot. An analysis of the fractional step method.Journal of Computational Physics, 108:51–58, 1993.MATHCrossRefMathSciNetGoogle Scholar
  78. 78.
    A. Prohl.Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations. B.G. Teubner Stuttgart, 1997.MATHGoogle Scholar
  79. 79.
    A. Quarteroni, G. Sacchi Landriani, and A. Valli. Coupling viscous and inviscid Stokes equations via a domain decomposition method for finite elements.Numerische Mathematik, 59:831–859, 1991.MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    A. Quarteroni, F. Saleri, and A. Veneziani. Analysis of the Yosida method for the incompressible Navier-Stokes equations.Journal de Mathématiques Pures et Appliquées, 78:473–503, 1999.MATHCrossRefMathSciNetGoogle Scholar
  81. 81.
    A. Quarteroni, F. Saleri, and A. Veneziani. Factorization methods for the numerical approximation of Navier-Stokes equations.Computer Methods in Applied Mechanics and Engineering, 188:505–526, 2000.MATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    R. Rannacher.On Chorin’s projection method for incompressible Navier-Stokes equations, Lecture Notes in Mathematics, volume 1530, pages 167–183. Springer, Berlin, 1992.Google Scholar
  83. 83.
    Y. Saad.Iterative methods for sparse linear systems. PWS Publishing, Boston, MA, 1996.MATHGoogle Scholar
  84. 84.
    F. Saleri and A. Veneziani. Pressure correction algebraic splitting methods for the incompressible Navier-Stokes equations.SIAM Journal on Numerical Analysis, 43(1):174–194, 2005.MATHCrossRefMathSciNetGoogle Scholar
  85. 85.
    J. Shen. On error estimates for some higher order projection and penalty-projection methods for Navier-Stokes equations.Numerische Mathematik, 62:49–73, 1992.CrossRefMathSciNetGoogle Scholar
  86. 86.
    J. Shen. On error estimates of projection methods for Navier-Stokes equations: first order squemes.SIAM Journal on Numerical Analysis, 29:57–77, 1992.MATHCrossRefMathSciNetGoogle Scholar
  87. 87.
    J. Shen. A remark on the projection-3 method.International Journal for Numerical Methods in Fluids, 16:249–253, 1993.MATHCrossRefMathSciNetGoogle Scholar
  88. 88.
    J. Shen. Remarks on the pressure error estimates for the projection methods.Numerische Mathematik, 67:513–520, 1994.MATHCrossRefMathSciNetGoogle Scholar
  89. 89.
    J. Shen. On error estimates of the projection methods for the Navier-Stokes equations: second-order squemes.Mathematics of Computation, 65:1039–1065, 1996.MATHCrossRefMathSciNetGoogle Scholar
  90. 90.
    J.C. Simo and F. Armero. Unconditional stability and long term behavior of transient algorithms for the incompressible Navier-Stokes equations.Computer Methods in Applied Mechanics and Engineering, 111:111–154, 1994.MATHCrossRefMathSciNetGoogle Scholar
  91. 91.
    G. Strang and J. Fix.An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, 1973.MATHGoogle Scholar
  92. 92.
    R. Temam. Sur la stabilité et la convergence de la méthode des pas fractionaires.Ann. Mat. Pura Appl., LXXIV:191–380, 1968.MathSciNetGoogle Scholar
  93. 93.
    R. Temam. Une méthode d’approximations de la solution des equations de Navier-Stokes.Bull. Soc. Math. France, 98:115–152, 1968.MathSciNetGoogle Scholar
  94. 94.
    R. Temam. Sur l ’approximation de la solution des equations de Navier-Stokes par la méthode des pas fractionaires (I).Archives for Rational Mechanics and Analysis, 32:135–153, 1969.MATHCrossRefMathSciNetGoogle Scholar
  95. 95.
    R. Temam. Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionaires (II).Archives for Rational Mechanics and Analysis, 33:377–385, 1969.MATHCrossRefMathSciNetGoogle Scholar
  96. 96.
    R. Temam.Navier-Stokes equations. North-Holland, 1984.Google Scholar
  97. 97.
    R. Temam. Remark on the pressure boundary condition for the projection method.Theoretical Computational Fluid Dynamics, 3:181–184, 1991.MATHCrossRefGoogle Scholar
  98. 98.
    L.J.P. Timmermans, P.D. Minev, and F.N. Van de Vosse. An approximate projection scheme for incompressible flow using espectral elements.International Journal for Numerical Methods in Fluids, 22:673–688, 1996.MATHCrossRefGoogle Scholar
  99. 99.
    S. Turek.Effcient Solvers for Incompressible Flow Problems. Lecture Notes in Computational Science and Engineering. Springer, 1999.Google Scholar
  100. 100.
    J.van Kan. A second-order accurate pressure correction scheme for viscous incompressible flow.SIAM Journal of Sci. Stat. Comp., 7:870–891, 1986.MATHCrossRefGoogle Scholar

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© CIMNE 2007

Authors and Affiliations

  • S. Badia
    • 1
  • R. Codina
    • 1
  1. 1.CIMNEUniversitat Politécnica de CatalunyaBarcelonaSpain

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