Solvers for the Coupled Linear Systems of Equations

  • Volker John
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 51)


Remark 9.1 (Motivation) Many methods for the simulation of incompressible flow problems require the simulation of coupled linear problems for velocity and pressure of the form
$$\displaystyle{ \mathcal{A}\underline{x} = \left (\begin{array}{*{10}c} A& D\\ B &-C \end{array} \right )\left (\begin{array}{*{10}c} \underline{u}\\ \underline{p} \end{array} \right ) = \left (\begin{array}{*{10}c} \underline{f}\\ \underline{f_{ p}} \end{array} \right ) =\underline{ y}, }$$
$$\displaystyle\begin{array}{rcl} & & A \in \mathbb{R}^{dN_{v}\times dN_{v} },\ D \in \mathbb{R}^{dN_{v}\times N_{p} },\ B \in \mathbb{R}^{N_{p}\times dN_{v} },\ C \in \mathbb{R}^{N_{p}\times N_{p} }, {}\\ & & \underline{u},\underline{f} \in \mathbb{R}^{dN_{v} },\ \underline{p},\underline{f_{p}} \in \mathbb{R}^{N_{p} }, {}\\ \end{array}$$
such that
$$\displaystyle{\mathcal{A}\in \mathbb{R}^{(dN_{v}+N_{p})\times (dN_{v}+N_{p})},\quad \underline{x},\underline{y} \in \mathbb{R}^{dN_{v}+N_{p} }.}$$
If C = 0, then (9.1) is a linear saddle point problem.


Linear Coupled System Linear Saddle Point Problem Incompressible Flow Incomplete Factorization Iterative Solver 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Amestoy PR, Duff IS, L’Excellent J-Y, Koster J (2001) A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J Matrix Anal Appl 23:15–41 (electronic)Google Scholar
  2. Amestoy PR, Guermouche A, L’Excellent J-Y, Pralet S (2006) Hybrid scheduling for the parallel solution of linear systems. Parallel Comput 32:136–156MathSciNetCrossRefGoogle Scholar
  3. Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Dalcin L, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zampini S, Zhang H, Zhang H (2016) PETSc Web page.
  4. Benzi M, Olshanskii MA (2006) An augmented Lagrangian-based approach to the Oseen problem. SIAM J Sci Comput 28:2095–2113MathSciNetCrossRefzbMATHGoogle Scholar
  5. Benzi M, Wang Z (2011) Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier-Stokes equations. SIAM J Sci Comput 33:2761–2784MathSciNetCrossRefzbMATHGoogle Scholar
  6. Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137MathSciNetCrossRefzbMATHGoogle Scholar
  7. Briggs WL, Henson VE, McCormick SF (2000) A multigrid tutorial, 2nd edn. Society for industrial and applied mathematics (SIAM), Philadelphia, PA, pp xii+193Google Scholar
  8. Dahl O, Wille S (1992) An ILU preconditioner with coupled node fill-in for iterative solution of the mixed finite element formulation of the 2D and 3D Navier-Stokes equations. Int J Numer Methods Fluids 15:525–544CrossRefzbMATHGoogle Scholar
  9. Davis TA (2004) Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans Math Softw 30:196–199MathSciNetCrossRefzbMATHGoogle Scholar
  10. Elman HC, Tuminaro RS (2009) Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations. Electron Trans Numer Anal 35:257–280MathSciNetzbMATHGoogle Scholar
  11. Elman H, Howle VE, Shadid J, Shuttleworth R, Tuminaro R (2006) Block preconditioners based on approximate commutators. SIAM J Sci Comput 27:1651–1668MathSciNetCrossRefzbMATHGoogle Scholar
  12. Elman H, Howle VE, Shadid J, Silvester D, Tuminaro R (2008a) Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations. SIAM J Sci Comput 30:290–311MathSciNetCrossRefzbMATHGoogle Scholar
  13. Elman H, Howle VE, Shadid J, Shuttleworth R, Tuminaro R (2008b) A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations. J Comput Phys 227:1790–1808MathSciNetCrossRefzbMATHGoogle Scholar
  14. Elman HC, Silvester DJ, Wathen AJ (2014) Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Numerical mathematics and scientific computation, 2nd edn. Oxford University Press, Oxford, pp xiv+479Google Scholar
  15. Hackbusch W (1985) Multigrid methods and applications. Springer series in computational mathematics, vol 4. Springer, Berlin, pp xiv+377Google Scholar
  16. John V (2002) Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations. Int J Numer Methods Fluids 40:775–798MathSciNetCrossRefzbMATHGoogle Scholar
  17. John V (2006) On the efficiency of linearization schemes and coupled multigrid methods in the simulation of a 3D flow around a cylinder. Int J Numer Methods Fluids 50:845–862MathSciNetCrossRefzbMATHGoogle Scholar
  18. John V, Matthies G (2001) Higher-order finite element discretizations in a benchmark problem for incompressible flows. Int J Numer Methods Fluids 37:885–903CrossRefzbMATHGoogle Scholar
  19. John V, Tobiska L (2000) Numerical performance of smoothers in coupled multigrid methods for the parallel solution of the incompressible Navier-Stokes equations. Int J Numer Methods Fluids 33:453–473CrossRefzbMATHGoogle Scholar
  20. John V, Knobloch P, Matthies G, Tobiska L (2002) Non-nested multi-level solvers for finite element discretisations of mixed problems. Computing 68:313–341MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kay D, Loghin D, Wathen A (2002) A preconditioner for the steady-state Navier-Stokes equations. SIAM J Sci Comput 24:237–256MathSciNetCrossRefzbMATHGoogle Scholar
  22. Konshin IN, Olshanskii MA, Vassilevski YV (2015) ILU preconditioners for nonsymmetric saddle-point matrices with application to the incompressible Navier-Stokes equations. SIAM J Sci Comput 37:A2171–A2197MathSciNetCrossRefzbMATHGoogle Scholar
  23. Olshanskii MA, Tyrtyshnikov EE (2014) Iterative methods for linear systems. Theory and applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, pp xvi+247Google Scholar
  24. Olshanskii MA, Vassilevski YV (2007) Pressure Schur complement preconditioners for the discrete Oseen problem. SIAM J Sci Comput 29:2686–2704MathSciNetCrossRefzbMATHGoogle Scholar
  25. Patankar SV (1980) Numerical heat transfer and fluid flow. Series in computational methods in mechanics and thermal sciences. Hemisphere Publishing Corporation, Washington, New York, London/McGraw-Hill Book Company, New York, pp XIII+197Google Scholar
  26. Saad Y (1993) A flexible inner-outer preconditioned GMRES algorithm. SIAM J Sci Comput 14:461–469MathSciNetCrossRefzbMATHGoogle Scholar
  27. Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA, pp xviii+528Google Scholar
  28. Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869MathSciNetCrossRefzbMATHGoogle Scholar
  29. Schenk O, Bollhöfer M, Römer RA (2008) On large-scale diagonalization techniques for the anderson model of localization SIAM Rev 50:91–112Google Scholar
  30. Schieweck F (2000) A general transfer operator for arbitrary finite element spaces. Preprint 00-25. Fakultät für Mathematik, Otto-von-Guericke-Universität MagdeburgGoogle Scholar
  31. Schönknecht N (2015) On solvers for saddle point problems arising in finite element discretizations of incompressible flow problems. Master thesis, Freie Universität BerlinGoogle Scholar
  32. Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J Sci Stat Comput 10:36–52MathSciNetCrossRefzbMATHGoogle Scholar
  33. Trilinos (2016) Accessed 22 June 2016
  34. Trottenberg U, Oosterlee CW, Schüller A (2001) Multigrid. Academic Press, Inc., San Diego, CA, pp xvi+631. With contributions by A. Brandt, P. Oswald and K. StübenGoogle Scholar
  35. ur Rehman M, Vuik C, Segal G (2008) A comparison of preconditioners for incompressible Navier-Stokes solvers. Int J Numer Methods Fluids 57:1731–1751Google Scholar
  36. van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Statist Comput 13:631–644MathSciNetCrossRefzbMATHGoogle Scholar
  37. Vanka SP (1986) Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J Comput Phys 65:138–158MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Volker John
    • 1
    • 2
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

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