A Preconditioned Scheme for Nonsymmetric Saddle-Point Problems

Abstract

In this paper, we present an effective preconditioning technique for solving nonsymmetric saddle-point problems. In particular, we consider those saddle-point problems that arise in the numerical simulation of particulate flows—flow of solid particles in incompressible fluids, using mixed finite element discretization of the Navier–Stokes equations.

These indefinite linear systems are solved using a preconditioned Krylov subspace method with an indefinite preconditioner. This creates an inner–outer iteration, in which the inner iteration is handled via a preconditioned Richardson scheme. We provide an analysis of our approach that relates the convergence properties of the inner to the outer iterations. Also “optimal” approaches are proposed for the implicit construction of the Richardson’s iteration preconditioner. The analysis is validated by numerical experiments that demonstrate the robustness of our scheme, its lack of sensitivity to changes in the fluid–particle system, and its “scalability”.

References

  1. 1.
    Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Nonlinear Programming. Stanford University Press, Stanford (1958) Google Scholar
  2. 2.
    Babuska, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1973) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Baggag, A.: Linear system solvers in particulate flows. Ph.D. thesis, Department of Computer Science, University of Minnesota (2003) Google Scholar
  4. 4.
    Baggag, A., Sameh, A.: A nested iterative scheme for indefinite linear systems in particulate flows. Comput. Methods Appl. Mech. Eng. 193, 1923–1957 (2004) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bank, R.E., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math. 52, 427–458 (1988) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bank, R.E., Welfert, B.D., Yserentant, H.: A class of iterative methods for solving saddle point problems. Numer. Math. 55, 645–666 (1990) MathSciNetGoogle Scholar
  7. 7.
    Barnard, S., Grote, M.: A block version of the SPAI preconditioner. In: Hendrickson, B., Yelick, K., Bishof, C. (eds.) Proceedings of the Ninth SIAM Conference on Parallel Processing for Scientific Computing, March 22–24. SIAM, Philadelphia (1999). CD-ROM Google Scholar
  8. 8.
    Benzi, M., Golub, G.H.: An iterative method for generalized saddle point problems. SIAM J. Matrix Anal. (2012, to appear) Google Scholar
  9. 9.
    Braess, D., Sarazin, R.: An efficient smoother for the stokes problem. Appl. Numer. Math. 23, 3–19 (1997) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bramble, J.H., Leyk, Z., Pasciak, J.E.: Iterative schemes for non-symmetric and indefinite elliptic boundary value problems. Math. Comput. 60, 1–22 (1993) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comput. 50, 1–18 (1988) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Bramble, J.H., Pasciak, J.E.: Iterative techniques for time dependent Stokes problems. Comput. Math. Appl. 33, 13–30 (1997) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34, 1072–1092 (1997) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Uzawa type algorithms for nonsymmetric saddle point problems. Math. Comput. 69, 667–689 (2000) MathSciNetMATHGoogle Scholar
  15. 15.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991). ISBN 3-540-97582-9 MATHCrossRefGoogle Scholar
  16. 16.
    Dyn, N., Ferguson, W.: The numerical solution of equality-constrained quadratic programming problems. Math. Comput. 41, 165–170 (1983) MathSciNetMATHGoogle Scholar
  17. 17.
    Elman, H., Silvester, D.: Fast nonsymmetric iterations and preconditioning for Navier–Stokes equations. SIAM J. Sci. Comput. 17, 33–46 (1996) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Elman, H.C.: Multigrid and Krylov subspace methods for the discrete Stokes equations. Tech. Rep. 3302, Institute for Advanced Computer Studies (1994) Google Scholar
  19. 19.
    Elman, H.C.: Perturbation of eigenvalues of preconditioned Navier–Stokes operators. SIAM J. Matrix Anal. Appl. 18, 733–751 (1997) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Elman, H.C.: Preconditioning for the steady-state Navier–Stokes equations with low viscosity. SIAM J. Sci. Comput. 20, 1299–1316 (1999) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Elman, H.C.: Preconditioners for saddle point problems arising in computational fluid dynamics. Appl. Numer. Math. 43, 75–89 (2002) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Iterative methods for problems in computational fluid dynamics. In: Chan, R., Chan, T., Golub, G. (eds.) Iterative Methods in Scientific Computing. Springer, Singapore (1997) Google Scholar
  24. 24.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier–Stokes equations. Numer. Math. 90, 641–664 (2002) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Falk, R.: An analysis of the finite element method using Lagrange multipliers for the stationary Stokes equations. Math. Comput. 30, 241–269 (1976) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Falk, R., Osborn, J.: Error estimates for mixed methods. RAIRO. Anal. Numér. 14, 249–277 (1980) MathSciNetMATHGoogle Scholar
  27. 27.
    Fischer, B., Ramage, A., Silvester, D., Wathen, A.: Minimum residual methods for augmented systems. BIT Numer. Math. 38, 527–543 (1998) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Gatica, G.N., Heuer, N.: Conjugate gradient method for dual-dual mixed formulation. Math. Comput. 71, 1455–1472 (2001) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Girault, V., Raviart, P.: Finite Element Approximation of the Navier–Stokes Equations. Lecture Notes in Math., vol. 749. Springer, New York (1981) MATHGoogle Scholar
  30. 30.
    Glowinski, R., Pan, T.W., Périaux, J.: Distributed Lagrange multiplier methods for incompressible viscous flow around moving rigid bodies. Comput. Methods Appl. Mech. Eng. 151, 181–194 (1998) MATHCrossRefGoogle Scholar
  31. 31.
    Golub, G., Wathen, A.: An iteration for indefinite systems and its application to the Navier–Stokes equations. SIAM J. Sci. Comput. 19, 530–539 (1998) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Golub, G., Wu, X., Yuan, J.Y.: SOR-like methods for augmented systems. BIT Numer. Math. 41, 71–85 (2001) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Grote, M., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18, 838–853 (1997) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Hu, H.: Direct simulation of flows of solid-liquid mixtures. Int. J. Multiph. Flow 22, 335–352 (1996) MATHCrossRefGoogle Scholar
  35. 35.
    Johnson, A.A., Tezduyar, T.E.: Simulation of multiple spheres falling in a liquid-filled tube. Comput. Methods Appl. Mech. Eng. 134, 351–373 (1996) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Johnson, A.A., Tezduyar, T.E.: 3D simulation of fluid–particle interactions with the number of particles reaching 100. Comput. Methods Appl. Mech. Eng. 145, 301–321 (1997) MATHCrossRefGoogle Scholar
  37. 37.
    Johnson, A.A., Tezduyar, T.E.: Advanced mesh generation and update methods for 3D flow simulations. Comput. Mech. 23, 130–143 (1999) MATHCrossRefGoogle Scholar
  38. 38.
    Johnson, A.A., Tezduyar, T.E.: Methods for 3D computation of fluid-object interactions in spatially-periodic flows. Comput. Methods Appl. Mech. Eng. 190, 3201–3221 (2001) MATHCrossRefGoogle Scholar
  39. 39.
    Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21, 1300–1317 (2000) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Klawonn, A.: An optimal preconditioner for a class of saddle point problems with a penalty term. SIAM J. Sci. Comput. 19, 540–552 (1998) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Klawonn, A., Starke, G.: Block triangular preconditioners for nonsymmetric saddle point problems: Field-of-values analysis. Numer. Math. 81, 577–594 (1999) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Knepley, M.: Parallel simulation of the particulate flow problem. Ph.D. thesis, Department of Computer Science, Purdue University (2000) Google Scholar
  43. 43.
    Knepley, M., Sarin, V., Sameh, A.: Parallel simulation of particulate flows. Appeared in Fifth Intl. Symp. on Solving Irregular Structured Problems in Parallel, IRREGULAR 98, LNCS, No. 1457, pp. 226–237, Springer (1998) Google Scholar
  44. 44.
    Krzyzanowski, P.: On block preconditioners for nonsymmetric saddle point problems. SIAM J. Sci. Comput. 23, 157–169 (2001) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Little, L., Saad, Y.: Block LU preconditioners for symmetric and nonsymmetric saddle point problems. Tech. Rep. 1999-104, Minnesota Supercomputer Institute, University of Minnesota (1999) Google Scholar
  46. 46.
    Lou, G.: Some new results for solving linear systems arising from computational fluid dynamics problems. Ph.D. thesis, Department of Computer Science, University of Illinois U-C (1992) Google Scholar
  47. 47.
    Luks̃an, L., Vlc̃ek, J.: Indefinitely preconditioned inexact Newton method for large sparse equality constrained nonlinear programming problems. Numer. Linear Algebra Appl. 5, 219–247 (1998) MathSciNetCrossRefGoogle Scholar
  48. 48.
    Maury, B.: Characteristics ALE method for the unsteady 3D Navier–Stokes equations with a free surface. Comput. Fluid Dyn. J. 6, 175–188 (1996) CrossRefGoogle Scholar
  49. 49.
    Maury, B.: A many-body lubrication model. C. R. Acad. Sci. Paris 325, 1053–1058 (1997) MathSciNetMATHGoogle Scholar
  50. 50.
    Maury, B.: Direct simulations of 2D fluid–particle flows in biperiodic domains. J. Comput. Phys. 156, 325–351 (1999) MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Maury, B., Glowinski, R.: Fluid–particle flow: a symmetric formulation. C. R. Acad. Sci. Paris 324, 1079–1084 (1997) MathSciNetMATHGoogle Scholar
  52. 52.
    Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21, 1969–1972 (2000) MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Ortega, J.M.: Numerical Analysis: a Second Course. Computer Science and Applied Mathematics Series. Academic Press, San Diego (1972) Google Scholar
  54. 54.
    Perugia, I., Simoncini, V.: An optimal indefinite preconditioner for mixed finite element method. Tech. Rep. 1098, Department of Mathematics, Università de Bologna, Italy (1998) Google Scholar
  55. 55.
    Perugia, I., Simoncini, V.: Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer. Linear Algebra Appl. 7, 585–616 (2000) MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Queck, W.: The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type. SIAM J. Numer. Anal. 26, 1016–1030 (1989) MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Rusten, T., Winther, R.: A preconditioned iterative method for saddle point problem. SIAM J. Matrix Anal. Appl. 13, 887–904 (1992) MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Saad, Y.: A flexible inner–outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993) MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Saad, Y., Schultz, M.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986) MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Saad, Y., Suchomel, B.: ARMS: an algebraic recursive multilevel solver for general sparse linear systems. Numer. Linear Algebra Appl. 9, 359–378 (2002) MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Sameh, A., Baggag, A.: Parallelism in iterative linear system solvers. In: Proceedings of the Sixth Japan-US Conference on Flow Simulation and Modeling, April 1, 2002 Google Scholar
  62. 62.
    Sameh, A., Baggag, A., Wang, X.: Parallel nested iterative schemes for indefinite linear systems. In: Mang, H.A., Rammerstorfer, F.G., Eberhardsteiner, J. (eds.) Proceedings of the Fifth World Congress on Computational Mechanics, (WCCM V). Vienna University of Technology, Austria, July 7–12, 2002. ISBN 3-9501554-0-6 Google Scholar
  63. 63.
    Silvester, D., Elman, H., Kay, D., Wathen, A.: Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow. J. Comput. Appl. Math. 128, 261–279 (2001) MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Silvester, D., Wathen, A.: Fast iterative solution of stabilized Stokes systems. part II: Using general block preconditioners. SIAM J. Numer. Anal. 31, 1352–1367 (1994) MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Silvester, D., Wathen, A.: Fast and robust solvers for time-discretized incompressible Navier–Stokes equations. Tech. Rep. 27, Department of Mathematics, University of Manchester (1995) Google Scholar
  66. 66.
    Simoncini, V., Szyld, D.: Flexible inner–outer Krylov subspace methods. SIAM J. Numer. Anal. 40, 2219–2239 (2003) MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Simoncini, V., Szyld, D.: Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM J. Sci. Comput. 25, 454–477 (2003) MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Stewart, G.W.: Introduction to Matrix Computations. Academic Press, San Diego (1973) MATHGoogle Scholar
  69. 69.
    Tezduyar, T.E.: Stabilized finite element formulations for incompressible flow computations. Adv. Appl. Mech. 28, 1–44 (1991) MathSciNetCrossRefGoogle Scholar
  70. 70.
    Vanderstraeten, D., Knepley, M.: Parallel building blocks for finite element simulations: Application to solid-liquid mixture flows. In: Emerson, D., Ecer, A., Periaux, J., Satofuka, N. (eds.) Proceedings of Parallel CFD’99 Conf.: Recent Developments and Advances Using Parallel Computers, pp. 133–139. Academic Press, Manchester (1997) Google Scholar
  71. 71.
    Verfürth, R.: A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem. IMA J. Numer. Anal. 4, 441–455 (1984) MathSciNetMATHGoogle Scholar
  72. 72.
    Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55, 309–325 (1989) MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Wathen, A., Silvester, D.: Fast iterative solution of stabilized Stokes systems. part I: Using simple diagonal preconditioners. SIAM J. Numer. Anal. 30, 630–649 (1993) MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Zulehner, W.: A class of smoothers for saddle point problems. Computer 65, 227–246 (2000) MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Zulehner, W.: Analysis of iterative methods for saddle point problems: a unified approach. Math. Comput. 71, 479–505 (2001) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.College of Science and EngineeringUniversité LavalQuebec CityCanada

Personalised recommendations