Efficient preconditioning techniques for finite-element quadratic discretization arising from linearized incompressible Navier-Stokes equations

International Journal for Numerical Methods in Fluids

Volumn 63, Issue 12, 2010, Pages 1394-1420

  Maliki, A El ^a   Guenette R ^a  

^a UNIVERSITÉ LAVAL   (Canada)

Author keywords

Block preconditioner; Flexible Krylov method; Hierarchical preconditioner; Mixed formulation; Navier Stokes equations; Schur complement approximation

Indexed keywords

BLOCK PRECONDITIONER; FLEXIBLE KRYLOV METHOD; MIXED FORMULATIONS; PRECONDITIONERS; SCHUR COMPLEMENT;

LINEARIZATION; NEWTON-RAPHSON METHOD; NUMERICAL ANALYSIS; VISCOUS FLOW;

NAVIER STOKES EQUATIONS;

EID: 77956596503     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.2136     Document Type: Article

References (33)

1. Turek S. Efficient Solvers for Incompressible Flow Problems. Spinger: Berlin, 1999.
2. Elman HC, Silvester DJ, Wathen AJ. Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005.
3. Benzi M, Golub GH, Leisen J. Numerical solution of saddle point problems. Acta Numerica 2005; 14:1-137.
4. Elman HC. Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM Journal on Scientific Computing 1999; 20:1299-1316.
5. Elman HC, Silvester D, Wathen AJ. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations. Numerische Mathematik 2002; 90:641-664.
6. Kay D, Loghin D, Wathen AJ. A preconditioner for the steady-state Navier-Stokes equations. SIAM Journal on Scientific Computing 2002; 24(1):237-256.
7. Eisenstat SC, Elman HC, Schultz MH. Variational iterative methods for non-symmetric systems of linear equations. SIAM Journal on Numerical Analysis 1983; 20:345-357.
8. Saad Y. A flexible inner-outer preconditioned GMRES algorithm. SIAM Journal on Scientific and Statistical Computing 1993; 14:461-469.
9. Simo J, Armero F. Unconditional stability and long-term behavior of transient Navier-Stokes and Euler equations. Computer Methods in Applied Mechanics and Engineering 1994; 111:111-154.
10. Févriere C, Laminie J, Poullet P, Angot Ph. On the penalty-projection method for the Navier-Stokes equations with the MAC mesh. Journal of Computational and Applied Mathematics 2009; 226:228-245.
11. Franca LP, Frey SL. Stabilized finite element methods: II. The incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 1992; 99:209-233.
12. Raviart PA. Les méthodes d'éléments finis en mécanique des fluides. CEA-EDF INRIA, Ecole d'Eté d'Analyse Numérique, Editions Eyrolles, 1981.
13. Olshanskii MA. A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Computer Methods in Applied Mechanics and Engineering 2002; 191:5515-5536.
14. Fortin M, Glowinsky R. Méthodes du lagrangien augmenté. Collection méthodes Mathématiques de l'Informatique. Dunod: Paris, 1992.
15. Benzi M, Olshanskii AM. An augmented Lagrangian-based approach to the Oseen problem. SIAM Journal on Scientific Computing 2006; 28(6):2095-2113.
16. Silvester DJ, Wathen AJ. Fast iterative solution of stabilised Stokes systems. Part II: using general block preconditioners. SIAM Journal on Numerical Analysis 1994; 31(5):1352-1367.
17. Murphy MF, Golub GH, Wathen AJ. A note on preconditioning for indefinite linear system. SIAM Journal on Scientific Computing 2000; 21(6):1969-1972.
18. Wathen AJ, Silvester DJ. Fast iterative solution of stabilised Stokes systems. Part I: using simple diagonal preconditioners. SIAM Journal on Numerical Analysis 1993; 30(3):630-649.
19. Elman HC. Preconditioning strategies for models of incompressible flow. Journal of Scientific Computing 2005; 25:347-366.
20. El Maliki A. Résolution de problèmes aux limites à l'aide de méthodes itératives hiérarchiques 'a préconditionneur variable. Ph.D. Thesis, Université Laval, 2007.
21. El Maliki A, Guénette R, Fortin M. An efficient hierarchical preconditioner for quadratic discretizations of finite element problems. Numerical Linear Algebra with Applications 2009; submitted.
22. Chow E, Saad Y. Approximate inverse preconditioners via sparse-sparse iterations. SIAM Journal on Scientific Computing 1998; 19(3):995-1023.
23. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer: Berlin, 1991.
24. Cahouet J, Chabard JP. Some fast 3D finite element solvers for the generalized Stokes problem. International Journal for Numerical Methods in Fluids 1988; 8:869-895.
25. Dean EJ, Glowinsky R. On some finite element methods for the numerical simulation of incompressible viscous flow. In Incompressible Computational Fluid Dynamics, Gunzburger MD, Nicolaides RY (eds). Cambridge University Press: New York, 1993; 17-65.
26. Kay D, Loghin D. A Green's function preconditioner for the steady state Navier-Stokes equations. Report 99/06, Oxford University Computing Laboratory, 1999.
27. Silvester DJ, Elman HC, Kay D, Wathen AJ. Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow. Journal of Computational and Applied Mathematics 2001; 128:261-279.
28. GIREF. Groupe Interdisciplinaire de Recherche en Éléments Finis. Available from: www.giref.ulaval.ca.
29. Balay S, Buschelman K, Gropp W, Kaushik D, Knepley M, McInnes LC, Smith B, Zhang B. PETSc Web Page. 2006. Available from: http://www.-mcs.anl.gov/petsc.
30. Fortin A, Bertrand F, Fortin M, Chamberland E, Boulanger-Nadeau PE, El Maliki A, Najeh A. An adaptive remeshing strategy for shear-thinning fluid flow simulations. Computer and Chemical Engineering 2004; 28: 2363-2375.
31. Vuik C. New insights in GMRES-like methods with variable preconditioners. Journal of Computational and Applied Mathematics 1995; 61:189-204.
32. Guénette R, Fortin A, Labbé J, Marcotte JP. Iterative solvers for quadratic discretization of the generalized Stokes problem. International Journal for Numerical Methods in Fluids 2004; 44:695-720.
33. Wathen AJ, Loghin D, Kay D, Elman HC, Silvester D. A preconditioner for the 3D Oseen equations. Report 02/04, Oxford University Computing Laboratory, 2002.