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Preconditioning strategies for models of incompressible flow

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Abstract

We describe some new preconditioning strategies for handling the algebraic systems of equations that arise from discretization of the incompressible Navier-Stokes equations. We demonstrate how these methods adapt in a straightforward manner to decisions on implicit or explicit time discretization, explore their use on a collection of benchmark problems, and show how they relate to classical techniques such as projection methods and SIMPLE.

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References

  1. Bramble, J. H., and Pasciak, J. E. (1995). Iterative techniques for time dependent Stokes problems. In Habashi, W.,Solution Techniques for Large-Scale CFD Problems, John Wiley, New York, p. 201–216.

    Google Scholar 

  2. Cahouet, J., and Chabard, J.-P. (1988). Some fast 3D finite element solvers for the generalized Stokes problem.Int. J. Numer. Meth. Fluids,8, 869–895.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Dukowicz, J. K., and Dvinsky, A. S. (1992). Approximate factorization as a high order splitting for implicit incompressible flow equations.J. Comput. Phys. 102, 336–347.

    Article  MATH  MathSciNet  Google Scholar 

  5. Elman, H., and Silvester, D. (1996). Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations.SIAM J. Sci. Comput. 17, 33–46.

    Article  MATH  MathSciNet  Google Scholar 

  6. Elman, H. C. (1996). Multigrid and Krylov subspace methods for the discrete Stokes equations.Int. J. Numer. Meth. Fluids 227, 755–770.

    Article  Google Scholar 

  7. Elman, H. C. (1999). Preconditioning for the steady-state Navier-Stokes equations with low viscosity.SIAM J. Sci. Comput. 20, 1299–1316.

    Article  MATH  MathSciNet  Google Scholar 

  8. Elman, H. C. (2002). Preconditioners for saddle point problems arising in computational fluid dynamics.Appl. Numer. Math. 43, 75–89.

    Article  MATH  MathSciNet  Google Scholar 

  9. Elman, H. C., Howle, V. E., Shadid, J., and Tuminaro, R. (2003). A parallel block multilevel preconditioner for the 3D incompressible Navier-Stokes equations.J. Comput. Phys. 187, 505–523.

    Article  Google Scholar 

  10. Elman, H. C., Howle, V. E., Shadid, J., and Tuminaro, R. (2003). In preparation.

  11. Elman, H. C., Silvester, D. J., and Wathen, A. J. (2002). Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations.Numer. Math. 90, 665–688.

    Article  MATH  MathSciNet  Google Scholar 

  12. Faber, V., and Manteuffel, T. A. (1984). Necessary and sufficient conditions for the existence of a conjugate gradient method.SIAM J. Numer. Anal,21, 352–362.

    Article  MATH  MathSciNet  Google Scholar 

  13. Faber, V., and Manteuffel, T. A. (1987). Orthogonal error methods.SIAM J. Numer. Anal 24, 170–187.

    Article  MATH  MathSciNet  Google Scholar 

  14. Girault, V., and Raviart, P. A. (1986).Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, New York.

    Google Scholar 

  15. Greenbaum, A. (1997).Iterative Methods for Solving Linear Systems. SIAM, Philadelphia.

    MATH  Google Scholar 

  16. Henriksen, M. O., and Holmen, J. (2002). Algebraic splitting for incompressible Navier-Stokes equations.J. Comput. Phys.,175, 438–453.

    Article  MATH  MathSciNet  Google Scholar 

  17. Hughes, T. J., and Franca, L. P. (1987). 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.Comp. Meths. Appl. Mech. Engrg. 65, 85–96.

    Article  MATH  MathSciNet  Google Scholar 

  18. Karniadakis, G. E., Israeli, M., and Orszag, S. (1991). High-order splitting methods for the incompressible Navier-Stokes equations.J. Comput. Phys. 97, 414–443.

    Article  MATH  MathSciNet  Google Scholar 

  19. Kay, D., Loghin, D., and Wathen, A. (2002). A preconditioner for the steady-state Navier-Stokes equations.SIAM J. Sci. Comput. 24, 237–256.

    Article  MATH  MathSciNet  Google Scholar 

  20. Kechkar, N., and Silvester, D. (1992). Analysis of locally stabilized mixed finite element methods for the Stokes problem.Math. Comp. 58, 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  21. Loghin, D. (2001). Analysis of Preconditioned Picard Iterations for the Navier-Stokes Equations. Technical Report 01/10, Oxford University Computing Laboratory.

  22. Nicolaides, R. A. (1992). Analysis and convergence of the MAC scheme I.SIAM J. Numer. Anal. 29, 1579–1591.

    Article  MATH  MathSciNet  Google Scholar 

  23. Paige, C. C., and Saunders, M. A. (1975). Solution of sparse indefinite systems of linear equations.SIAM. J. Numer. Anal. 12, 617–629.

    Article  MATH  MathSciNet  Google Scholar 

  24. Patankar, S. V., and Spalding, D. B. (1972). A calculation proedure for heat and mass transfer in three-dimensional parabolic flows.Int. J. Heat and Mass Transfer 15, 1787–1806.

    Article  MATH  Google Scholar 

  25. Perot, J. B. (1993). An analysis of the fractional step method.J. Comput. Phys. 108, 51–58.

    Article  MATH  MathSciNet  Google Scholar 

  26. Rusten, T., and Winther, R. (1992). A preconditioned iterative method for saddle point problems.SIAM J. Matr. Anal. Appl. 13, 887–904.

    Article  MATH  MathSciNet  Google Scholar 

  27. Saad, Y., and Schultz, M. H. (1986). GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems.SIAM J. Sci. Stat. Comput. 7, 856–869.

    Article  MATH  MathSciNet  Google Scholar 

  28. Silvester, D., Elman, H., Kay, D., and Wathen, A. (2001). Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow.J. Comp. Appl. Math. 128, 261–279.

    Article  MATH  MathSciNet  Google Scholar 

  29. Silvester, D., and Wathen, A. (1994). Fast iterative solution of stabilized Stokes systems. Part II: Using block preconditioners.SIAM J. Numer. Anal. 31, 1352–1367.

    Article  MATH  MathSciNet  Google Scholar 

  30. Silvester, D. J., and Wathen, A. J. (1996). Fast and robust solvers for time-discretised incompressible Navier-Stokes equations. In Griffiths, D. F. and Watson, G. A., editors,Numerical Analysis: Proceedings of the 1995 Dundee Biennial Conference. Longman. Pitman Research Notes in Mathematics Series 344.

  31. Témam, R. (1969). Sur l’approximation de la solution des équations de Navier-Stokes par la méthod des pas fractionnaires (II).Arch. Rational Mech. Anal. 33, 377–385.

    Article  MATH  MathSciNet  Google Scholar 

  32. Verfürth, R. (1984). A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem.IMA J. Numer. Anal. 4, 441–455.

    MATH  MathSciNet  Google Scholar 

  33. Wathen, A., and Silvester, D. (1993). Fast iterative solution of stabilized Stokes systems. Part I: Using simple diagonal preconditioners.SIAM J. Numer. Anal. 30, 630–649.

    Article  MATH  MathSciNet  Google Scholar 

  34. Wathen, A. J. (1987). Realistic eigenvalue bounds for the Galerkin mass matrix.IMA J. Numer. Anal. 7, 449–457.

    Article  MATH  MathSciNet  Google Scholar 

  35. Wesseling, P. (2001).Principles of Computational Fluid Dynamics. Springer-Verlag, Berlin.

    Google Scholar 

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  1. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, 20742, College Park, MD

    H. C. Elman

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Elman, H.C. Preconditioning strategies for models of incompressible flow. J Sci Comput 25, 347–366 (2005). https://doi.org/10.1007/BF02728995

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  • DOI: https://doi.org/10.1007/BF02728995

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