Skip to main content Accessibility help
×
Home

Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

  • X. He (a1) and C. Vuik (a1)

Abstract

In this paper we explore the performance of the SIMPLER, augmented Lagrangian, ‘grad-div’ preconditioners and their new variants for the two-by-two block systems arising in the incompressible Navier-Stokes equations. The lid-driven cavity and flow over a finite flat plate are chosen as the benchmark problems. For each problem the Reynolds number varies from a low to the limiting number for a laminar flow.

Copyright

Corresponding author

*Corresponding author. Email addresses: X.He-1@tudelft.nl (X. He), C.Vuik@tudelft.nl (C. Vuik)

References

Hide All
[1]Axelsson, O.. Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.
[2]Axelsson, O. and Blaheta, R.. Preconditioning of matrices partitioned in two-by-two block form: Eigenvalue estimates and Schwarz DD for mixed FEM. Numer. Lin. Alg. Appl., 17:787810, 2010.
[3]Axelsson, O. and Neytcheva, M.. A general approach to analyse preconditioners for two-by-two block matrices. Numer. Lin. Alg. Appl., article first published online: 14 DEC 2011, dpi: 10.1002/nla.830.
[4]Axelsson, O. and Neytcheva, M.. Eigenvalue estimates for preconditioned saddle point matrices. Numer. Lin. Alg. Appl., 13:339360, 2006.
[5]Benzi, M.. Preconditioning techniques for large linear systems: a survey. J. Comput. Phys., 182:418477, 2002.
[6]Benzi, M. and Olshanskii, M.A.. An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput., 28:20952113, 2006.
[7]Benzi, M., Olshanskii, M.A. and Wang, Z.. Modified augmented Lagrangian preconditioners for the incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 66:486508, 2011.
[8]Benzi, M., Golub, G.H. and Liesen, J.. Numerical solution of saddle point problems. Acta Numerica, 14:1137, 2005.
[9]Börm, S. and Le Borne, S.. factorization in preconditioners for augmented Lagrangian and grad-div stabilized saddle point systems. Int. J. Numer. Meth. Fluids, 68:8398, 2012.
[10]de Niet, A. and Wubs, F.W.. Two preconditioners for saddle point problems in fluid flows. Int. J. Numer. Meth. Fluids, 54:355377, 2007.
[11]Eisenstat, S.C., Elman, H.C. and Schultz, M.H.. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 20:345357, 1983.
[12]Elman, H.C., Howle, V.E., Shadid, J., Shuttleworth, R. and Tuminaro, R.. Block preconditioners based on approximate commutators. SIAM J. Sci. Comput., 27:16511668, 2006.
[13]Elman, H.C. and Silvester, D.J.. Fast nonsymmetric iterations and preconditioning for the Navier-Stikes equations. SIAM J. Sci. Comput., 17:3346, 1996.
[14]Elman, H.C., Silvester, D.J. and Wathen, A.J.. Finite Element and Fast Iterative Solvers: with Application in Incompressible Fluid Dynamics. Oxford Series in Numerical Mathematics and Scientific Computation, Oxford University Press: Oxford, UK, 2005.
[15]Elman, H.C., Silvester, D.J. and Wathen, A.J.. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations. Numer. Math., 90:665688, 2002.
[16]He, X., Neytcheva, M. and Serra Capizzano, S.. On an Augmented Lagrangian-Based Preconditioning of Oseen Type Problems. BIT Numerical Mathematics, 51:865888, 2011.
[17]Heister, T.. A Massively Parallel Finite Element Framework with Application to Incompressible Flows. PhD Thesis, Göttingen University, 2011, available online.
[18]Kay, D., Loghin, D. and Wathen, A.. A preconditioner for the steady-state Navier-Stokes equations. SIAM J. Sci. Comput., 24:237256, 2002.
[19]Klaij, C.M. and Vuik, C.. SIMPLE-type preconditioners for cell-centered, collocated finite volume discretization of incompressible Reynolds-averaged Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 71:830849, 2013.
[20]Li, C. and Vuik, C.. Eigenvalue analysis of the SIMPLE preconditioning for incompressible flow. Numer. Lin. Alg. Appl., 11:511523, 2004.
[21]Murphy, M.F., Golub, G.H. and Wathen, A.J.. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput., 21:19691972, 2000.
[22]Napov, A. and Notay, Y.. An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput., 34:A1079A1109, 2012.
[23]Neytcheva, M., Do-Quang, M. and He, X.. Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form. Lecture Notes in Computer Science, 5910:108115, 2009.
[24]Notay, Y.. An aggregation-based algebraic multigrid method. Electron. T. Numer. Ana., 37:123146, 2010.
[25]Notay, Y.. Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput., 34:A2288A2316, 2012.
[26]Notay, Y.. A new analysis of block preconditioners for saddle point problems. Report GANMN 13-01, Universite Libre de Bruxelles, Brussels, Belgium, 2013.
[27]Olshanskii, M.A. and Vassilevski, Y.V.. Pressure Schur complement preconditioners for the discrete Oseen problem. SIAM J. Sci. Comput., 29:26862704, 2007.
[28]Patankar, S.V.. Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York, 1980.
[29]ur Rehman, M., Geenen, T., Vuik, C., Segal, G. and MacLachlan, S.P.. On iterative methods for the incompressible Stokes problem. Int. J. Numer. Meth. Fluids, 65:11801200, 2011.
[30]Rusten, T. and Winther, R.. A preconditioned iterative method for saddle point problems. SIAM J. Matrix Anal. Appl., 13:887904, 1992.
[31]Saad, Y.. Iterative Methods for Sparse Linear Systems. SIAM: Philadelphia, PA, 2003.
[32]Saad, Y., van der Vorst, H.A.. Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math., 123:133. 2000.
[33]Sahin, M. and Owens, R.G.. A novel fully implicit finite volume method applied to the lid-driven cavity problem-part I: high Reynolds number flow calculations. Int. J. Numer. Meth. Fluids, 42:5777, 2003.
[34]Segal, A., ur Rehman, M. and Vuik, C.. Preconditioners for the incompressible Navier-Stokes equations. Numer. Math. Theor. Meth. Appl., 3:245275, 2010.
[35]Vuik, C., Saghir, A. and Boerstoel, G.P.. The Krylov accelerated SIMPLE(R) method for flow problems in industrial furnaces. Int. J. Numer. Meth. Fluids, 33:10271040, 2000.

Keywords

Related content

Powered by UNSILO

Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

  • X. He (a1) and C. Vuik (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.