Schur complement preconditioners for the Navier-Stokes equations
by
Daniel Loghin
Oxford University Computing Laboratory

Schur complement preconditioners for the Navier-Stokes equations

Daniel Loghin
Oxford University Computing Laboratory

This work is concerned with the efficient numerical solution of the Navier-Stokes equations. In particular, we are interested in suitably preconditioned solution methods of Krylov subspace type applied to linear systems arising from mixed finite element discretizations of the Navier-Stokes equations. This paper analyses a class of preconditioners introduced in [] for the Oseen equations arising from a Picard linearisation and investigates their performance for the solution of linear systems involving the Jacobian matrix resulting from a Newton method of linearisation.

Consider the steady-state incompressible Navier-Stokes equations together with suitable boundary conditions in an open bounded domain \Omega subset R^d

- \epsilon\Deltau + ( u ·Ñ) u + Ñp = f ,    Ñ·u = 0,

(1)

where (u)_i = u_i are the components of velocity, p is the pressure and \epsilon is the viscosity parameter.

Writing (1) as N(w)w=[f\tilde]=(f, 0) with w=(u, p) a solution is sought via an iteration of the form

K(wn)(wn+1-wn)= ~ f   -N(wn)wn=g

(2)

We note here that when \epsilon is not too small, a computationally cheap choice for K is given by K=N. This leads to a Picard (or fixed point) iteration. However, for \epsilon << 1 this choice leads to slow convergence or even divergence. For this class of problems the more expensive Newton iteration given by the choice K(w)=D_w(N(w)w) becomes mandatory. In either case a mixed finite element discretization of (2) leads to a system of linear equations of the form

K\boldsymbol\deltaw=K æ ç è \boldsymbol\deltau \boldsymbol\deltap ö ÷ ø = æ ç è F Bt B C ö ÷ ø æ ç è \boldsymbol\deltau \boldsymbol\deltap ö ÷ ø =g

(3)

for which a useful preconditioner is known to be []

P= æ ç è F Bt 0 S ö ÷ ø

(4)

where S=C-BF^-1B^t is the Schur complement of F in K. This choice requires the action of the inverses of F and S which in general is not trivial to compute. We note here that a pressure solution method (which ammounts to eliminating the velocity from (3) and solving for the pressure) also requires the inverses of F and S. The aim of this work is to provide a useful preconditioner for S.

In the case of the Picard iteration, F is a discrete convection-diffusion operator whose efficient inverse can be achieved via a multigrid scheme or an incomplete factorisation preconditioning strategy. A useful approximation X for S was introduced in [] and is given by

X=ApFp-1Mp

where A_p, F_p, M_p are the projections of the Laplacian, convection-diffusion and identity operators onto the pressure finite element space. For this choice of preconditioner for S analytic results showing mesh independence of the condition number and the eigenvalues are presented in []; we review them briefly in this presentation. We also demonstrate here that the number of GMRES iterations grows like (|| u ||/\epsilon)^1/2.

In the case of the Newton iteration a similar preconditioning strategy is available. Despite the complicated nature of both F and S, the preconditioners used for the Picard iteration lead to convergence independent of the mesh in this case also. We also consider a choice of preconditioners for F. In each case the dependence on the viscosity parameter \epsilon is investigated with numerical results presented for a variety of standard test problems and finite element discretizations.

References

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D. Kay and D. Loghin. A Green's function preconditioner for the steady-state Navier-Stokes equations. Technical Report 99/06, Oxford University Computing Laboratory, 1999.

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D. Loghin. Schur complement preconditioners for the Navier-Stokes equations. submitted to Num. Mathematik, 2001.

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M. F. Murphy, G. H. Golub, and A. J. Wathen. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comp., 21:1969-1972, 2000.

Date received: January 15, 2001