Preconditioning strategies for models of incompressible flow

Journal of Scientific Computing

Volumn 25, Issue 1, 2005, Pages 347-366

  Elman, H C ^a  

^a University of Maryland ^*  (United States)

Author keywords

Incompressible fluids; Navier Stokes equations; Preconditioning; Solvers

Indexed keywords

ALGEBRA; FLUID DYNAMICS; NAVIER STOKES EQUATIONS; PROBLEM SOLVING;

DISCRETIZATION; INCOMPRESSIBLE FLUIDS; PRECONDITIONING; SOLVERS;

INCOMPRESSIBLE FLOW;

EID: 28844507541     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-004-4648-0     Document Type: Conference Paper

References (35)

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.
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.
3. Chorin, A. J. (1968). Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745-762.
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.
5. Elman, H., and Silvester, D. (1996). Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations. SIAM J. Sci. Comput. 17, 33-46.
6. Elman, H. C. (1996). Multigrid and Krylov subspace methods for the discrete Stokes equations. Int. J. Numer. Meth. Fluids 227, 755-770.
7. Elman, H. C. (1999). Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM J. Sci. Comput. 20, 1299-1316.
8. Elman, H. C. (2002). Preconditioners for saddle point problems arising in computational fluid dynamics. Appl. Numer. Math. 43, 75-89.
9. Elman, H. C., Howle, V E., Shadid, J., and Tuminaro, R. (2003). A parallel block multi-level preconditioner for the 3D incompressible Navier-Stokes equations. J. Comput. Phys. 187, 505-523.
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.
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.
13. Faber, V., and Manteuffel, T. A. (1987). Orthogonal error methods. SIAM J. Numer. Anal 24, 170-187.
14. Girault, V., and Raviart, P. A. (1986). Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, New York.
15. Greenbaum, A. (1997). Iterative Methods for Solving Linear Systems. SIAM, Philadelphia.
16. Henriksen, M. O., and Holmen, J. (2002). Algebraic splitting for incompressible Navier-Stokes equations. J. Comput. Phys., 175, 438-453.
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.
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.
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.
20. Kechkar, N., and Silvester, D. (1992). Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comp. 58, 1-10.
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.
23. Paige, C. C., and Saunders, M. A. (1975). Solution of sparse indefinite systems of linear equations. SIAM. J. Numer. Anal. 12, 617-629.
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.
25. Perot, J. B. (1993). An analysis of the fractional step method. J. Comput. Phys. 108, 51-58.
26. Rusten, T., and Winther, R. (1992). A preconditioned iterative method for saddle point problems. SIAM J. Matr. Anal. Appl. 13, 887-904.
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.
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.
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.
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 method des pas fractionnaires (II). Arch. Rational Mech. Anal. 33, 377-385.
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.
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.
34. Wathen, A. J. (1987). Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7, 449-457.
35. Wesseling, P. (2001). Principles of Computational Fluid Dynamics. Springer-Verlag, Berlin.