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Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

Published online by Cambridge University Press:  24 May 2016

X. He  and
C. Vuik
X. He*
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, the Netherlands
C. Vuik*
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, the Netherlands
*
*Corresponding author. Email addresses: X.He-1@tudelft.nl (X. He), C.Vuik@tudelft.nl (C. Vuik)
*Corresponding author. Email addresses: X.He-1@tudelft.nl (X. He), C.Vuik@tudelft.nl (C. Vuik)
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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.

Keywords

65F1065F08incompressible Navier-Stokes equationstwo-by-two block systemspreconditioners

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 2 , May 2016 , pp. 239 - 261
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Axelsson, O.. Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.Google Scholar
[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., 17:787–810, 2010.Google Scholar
[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.CrossRefGoogle Scholar
[4]Axelsson, O. and Neytcheva, M.. Eigenvalue estimates for preconditioned saddle point matrices. Numer. Lin. Alg. Appl., 13:339–360, 2006.Google Scholar
[5]Benzi, M.. Preconditioning techniques for large linear systems: a survey. J. Comput. Phys., 182:418–477, 2002.Google Scholar
[6]Benzi, M. and Olshanskii, M.A.. An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput., 28:2095–2113, 2006.CrossRefGoogle Scholar
[7]Benzi, M., Olshanskii, M.A. and Wang, Z.. Modified augmented Lagrangian preconditioners for the incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 66:486–508, 2011.Google Scholar
[8]Benzi, M., Golub, G.H. and Liesen, J.. Numerical solution of saddle point problems. Acta Numerica, 14:1–137, 2005.Google Scholar
[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, 68:83–98, 2012.Google Scholar
[10]de Niet, A. and Wubs, F.W.. Two preconditioners for saddle point problems in fluid flows. Int. J. Numer. Meth. Fluids, 54:355–377, 2007.Google Scholar
[11]Eisenstat, S.C., Elman, H.C. and Schultz, M.H.. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 20:345–357, 1983.CrossRefGoogle Scholar
[12]Elman, H.C., Howle, V.E., Shadid, J., Shuttleworth, R. and Tuminaro, R.. Block preconditioners based on approximate commutators. SIAM J. Sci. Comput., 27:1651–1668, 2006.CrossRefGoogle Scholar
[13]Elman, H.C. and Silvester, D.J.. Fast nonsymmetric iterations and preconditioning for the Navier-Stikes equations. SIAM J. Sci. Comput., 17:33–46, 1996.Google Scholar
[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.Google Scholar
[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., 90:665–688, 2002.Google Scholar
[16]He, X., Neytcheva, M. and Serra Capizzano, S.. On an Augmented Lagrangian-Based Preconditioning of Oseen Type Problems. BIT Numerical Mathematics, 51:865–888, 2011.Google Scholar
[17]Heister, T.. A Massively Parallel Finite Element Framework with Application to Incompressible Flows. PhD Thesis, Göttingen University, 2011, available online.Google Scholar
[18]Kay, D., Loghin, D. and Wathen, A.. A preconditioner for the steady-state Navier-Stokes equations. SIAM J. Sci. Comput., 24:237–256, 2002.Google Scholar
[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, 71:830–849, 2013.CrossRefGoogle Scholar
[20]Li, C. and Vuik, C.. Eigenvalue analysis of the SIMPLE preconditioning for incompressible flow. Numer. Lin. Alg. Appl., 11:511–523, 2004.Google Scholar
[21]Murphy, M.F., Golub, G.H. and Wathen, A.J.. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput., 21:1969–1972, 2000.Google Scholar
[22]Napov, A. and Notay, Y.. An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput., 34:A1079–A1109, 2012.CrossRefGoogle Scholar
[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, 5910:108–115, 2009.Google Scholar
[24]Notay, Y.. An aggregation-based algebraic multigrid method. Electron. T. Numer. Ana., 37:123–146, 2010.Google Scholar
[25]Notay, Y.. Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput., 34:A2288–A2316, 2012.Google Scholar
[26]Notay, Y.. A new analysis of block preconditioners for saddle point problems. Report GANMN 13-01, Universite Libre de Bruxelles, Brussels, Belgium, 2013.Google Scholar
[27]Olshanskii, M.A. and Vassilevski, Y.V.. Pressure Schur complement preconditioners for the discrete Oseen problem. SIAM J. Sci. Comput., 29:2686–2704, 2007.Google Scholar
[28]Patankar, S.V.. Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York, 1980.Google Scholar
[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, 65:1180–1200, 2011.Google Scholar
[30]Rusten, T. and Winther, R.. A preconditioned iterative method for saddle point problems. SIAM J. Matrix Anal. Appl., 13:887–904, 1992.Google Scholar
[31]Saad, Y.. Iterative Methods for Sparse Linear Systems. SIAM: Philadelphia, PA, 2003.Google Scholar
[32]Saad, Y., van der Vorst, H.A.. Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math., 123:1–33. 2000.Google Scholar
[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, 42:57–77, 2003.Google Scholar
[34]Segal, A., ur Rehman, M. and Vuik, C.. Preconditioners for the incompressible Navier-Stokes equations. Numer. Math. Theor. Meth. Appl., 3:245–275, 2010.CrossRefGoogle Scholar
[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, 33:1027–1040, 2000.Google Scholar