Preconditioners for saddle point problems arising in computational fluid dynamics. (English) Zbl 1168.76348

Elman, Howard C.

Appl. Numer. Math. 43, No. 1-2, 75-89 (2002).

Summary: Discretization and linearization of the incompressible Navier–Stokes equations leads to linear algebraic systems in which the coefficient matrix has the form of a saddle point problem \[ \left( \begin{matrix} F & B^T\\ B & O \end{matrix} \right) \left( \begin{matrix} u\\ p \end{matrix} \right) = \left( \begin{matrix} f\\ g \end{matrix} \right). \] In this paper, we describe the development of efficient and general iterative solution algorithms for this class of problems. We review the case where (1) arises from the steady-state Stokes equations and show that solution methods such as the Uzawa algorithm lead naturally to a focus on the Schur complement operator \(BF^{-1}B^T\) together with efficient strategies of applying the action of \(F^{-1}\) to a vector. We then discuss the advantages of explicitly working with the coupled form of the block system (1). Using this point of view, we describe some new algorithms derived by developing efficient methods for the Schur complement systems arising from the Navier-Stokes equations, and we demonstrate their effectiveness for solving both steady-state and evolutionary problems.

MSC:

76M25	Other numerical methods (fluid mechanics) (MSC2010)
65F35	Numerical computation of matrix norms, conditioning, scaling

Software:

Wesseling

References:

[1]	Arrow, K.; Hurwicz, L.; Uzawa, H., Studies in nonlinear programming, (1958), Stanford University Press Stanford, CA
[2]	Bey, J.; Wittum, G., Downwind numbering: robust multigrid for convection – diffusion problems, Appl. numer. math., 23, 177-192, (1997) · Zbl 0879.65088
[3]	Bramble, J.H.; Pasciak, J.E., Iterative techniques for time dependent Stokes problems, (), 201-216 · Zbl 1030.76506
[4]	Bramble, J.H.; Pasciak, J.E.; Vassilev, A.T., Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. numer. anal., 34, 1072-1092, (1997) · Zbl 0873.65031
[5]	Brandt, A.; Dinar, N., Multigrid solutions to elliptic flow problems, (), 53-147
[6]	Cahouet, J.; Chabard, J.-P., Some fast 3D finite element solvers for the generalized Stokes problem, Internat. J. numer. methods fluids, 8, 869-895, (1988) · Zbl 0665.76038
[7]	Dean, E.; Glowinski, R., On some finite element methods for the numerical simulation of incompressible viscous flow, (), 17-65 · Zbl 1189.76336
[8]	Elman, H.; Silvester, D., Fast nonsymmetric iterations and preconditioning for navier – stokes equations, SIAM J. sci. comput., 17, 33-46, (1996) · Zbl 0843.65080
[9]	Elman, H.C., Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems, Bit, 29, 890-915, (1989) · Zbl 0694.65011
[10]	Elman, H.C., Multigrid and Krylov subspace methods for the discrete Stokes equations, Internat. J. numer. methods fluids, 227, 755-770, (1996) · Zbl 0865.76078
[11]	Elman, H.C.; Golub, G.H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. numer. anal., 30, 1645-1661, (1994) · Zbl 0815.65041
[12]	H.C. Elman, V.E. Howles, J. Shadid, R. Tuminaro, 2001, in preparation
[13]	Elman, H.C.; Silvester, D.J.; Wathen, A.J., Performance and analysis of saddle point preconditioners for the discrete steady-state navier – stokes equations, Numer. math., 90, 641-664, (2002)
[14]	Freund, R.; Nachtigal, N.M., QMR: A quasi-minimal residual method for non-Hermitian linear systems, Numer. math., 60, 315-339, (1991) · Zbl 0754.65034
[15]	Girault, V.; Raviart, P.A., Finite element approximation of the navier – stokes equations, (1986), Springer-Verlag New York · Zbl 0396.65070
[16]	Gresho, P.M.; Sani, R.L., Incompressible flow and the finite element method, (1998), Wiley New York · Zbl 0941.76002
[17]	Harlow, F.H.; Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. fluids, 8, 2182-2189, (1965) · Zbl 1180.76043
[18]	D. Kay, D. Loghin, A Green’s function preconditioner for the steady-state Navier-Stokes equations, Technical Report 99/06, Oxford University Computing Laboratory, Oxford, 1999
[19]	Klawonn, A.; Starke, G., Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis, Numer. math., 81, 577-594, (1999) · Zbl 0922.65021
[20]	D. Loghin, Analysis of preconditioned Picard iterations for the Navier-Stokes equations, Technical Report 01/10, Oxford University Computing Laboratory, Oxford, 2001
[21]	Nicolaides, R.A., Analysis and convergence of the MAC scheme I, SIAM J. numer. anal., 29, 1579-1591, (1992) · Zbl 0764.76051
[22]	Paige, C.C.; Saunders, M.A., Solution of sparse indefinite systems of linear equations, SIAM J. numer. anal., 12, 617-629, (1975) · Zbl 0319.65025
[23]	Pitkäranta, J.; Saarinen, T., A multigrid version of a simple finite element method for the Stokes problem, Math. comp., 45, 1-14, (1985) · Zbl 0584.65080
[24]	A. Reusken, Convergence analysis of a multigrid method for convection – diffusion equations, Technical Report IGPM Report 190, RWTH Aachen, 2000 · Zbl 0996.65110
[25]	Rusten, T.; Winther, R., A preconditioned iterative method for saddle point problems, SIAM J. matrix anal. appl., 13, 887-904, (1992) · Zbl 0760.65033
[26]	Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing Boston, MA · Zbl 1002.65042
[27]	Saad, Y.; Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[28]	Silvester, D.; Elman, H.; Kay, D.; Wathen, A., Efficient preconditioning of the linearized navier – stokes equations for incompressible flow, J. comput. appl. math., 128, 261-279, (2001) · Zbl 0983.76051
[29]	Silvester, D.; Wathen, A., Fast iterative solution of stabilized Stokes systems. part II: using block preconditioners, SIAM J. numer. anal., 31, 1352-1367, (1994) · Zbl 0810.76044
[30]	Silvester, D.J.; Wathen, A.J., Fast and robust solvers for time-discretised incompressible navier – stokes equations, () · Zbl 0851.76042
[31]	Simo, J.C.; Armero, F., Unconditional stability and long-term behavior of transient algorithms for the incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 111, 111-154, (1994) · Zbl 0846.76075
[32]	Sleijpen, G.L.G.; Fokkema, D.R., BICGSTAB(L) for linear equations involving unsymmetric matrices with complex spectrum, Electron. trans. numer. anal., 6, 162-181, (1997)
[33]	Smith, A.; Silvester, D., Implicit algorithms and their linearisation for the transient incompressible navier – stokes equations, IMA J. numer. anal., 17, 527-545, (1997) · Zbl 0892.76060
[34]	Turek, S., Efficient solvers for incompressible flow problems, (1999), Springer-Verlag Berlin
[35]	Vanka, S.P., Block-implicit multigrid solution of navier – stokes in primitive variables, J. comput. phys., 65, 138-158, (1986) · Zbl 0606.76035
[36]	Varga, R.S., Matrix iterative analysis, (1962), Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602
[37]	Verfürth, R., A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem, IMA J. numer. anal., 4, 441-455, (1984) · Zbl 0563.76028
[38]	Verfürth, R., A multilevel algorithm for mixed problems, SIAM J. numer. anal., 21, 264-271, (1984) · Zbl 0534.65065
[39]	Wathen, A.; Silvester, D., Fast iterative solution of stabilized Stokes systems. part I: using simple diagonal preconditioners, SIAM J. numer. anal., 30, 630-649, (1993) · Zbl 0776.76024
[40]	Wathen, A.J., Realistic eigenvalue bounds for the Galerkin mass matrix, IMA J. numer. anal., 7, 449-457, (1987) · Zbl 0648.65076
[41]	B.D. Welfert, Convergence of inexact Uzawa algorithms for saddle point problems, Technical Report, Mathematics Department, University of Arizona, Tempe, AZ, 1993
[42]	Wesseling, P., An introduction to multigrid methods, (1992), Wiley New York · Zbl 0760.65092
[43]	Wittum, G., Multi-grid methods for the Stokes and navier – stokes equations, Numer. math., 54, 543-564, (1989) · Zbl 0645.76031