Mathematical Computation December 2013, Volume 2, Issue 4, PP.105-108
A Parameterized Preconditioner for Incompressible Navier-Stokes Equations Weihua Luo 1, 2, #, Tingzhu Huang 1 1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P. R. China 2. Key Lab of Numer. Simulation of Sichuan Province Univ., Neijiang Normal University, Neijiang, Sichuan, 641112, PR, China #Email: huaweiluo2012@163.com
Abstract In this paper, a preconditioner was introduced based on the parameterized splitting thought for saddle point problems. It was found out that the preconditioned matrix has an eigenvalue at 1 with multiplicity at least n1 n2 , and the remaining eigenvalues are close to 1 when the parameter α is small enough. Numerical examples of some Navier-Stokes problems were presented to illustrate the behavior of the preconditioner. Keywords: Saddle Point Problem; Navier-Stokes Equations; Sherman-Morrison-Woodbury Formula
1 INTRODUCTION In the study of well-known Navier-Stokes equations in computational fluid dynamics (the details can be found in [1], [3]), we usually obtain the linear system of the form (called saddle point problem)
HX b , A1 0 B1 x f with H 0 A2 B2 , X= y , b g , where A1 n1 n1 , A2 n2 n2 , B1 mn1 , B2 mn2 , z h B B 0 2 1
(1)
x, f n1 , y, g n2 , z, h m , and A1 , A2 are positive definite.
There have been a lot of preconditioning techniques for efficiently solving saddle point problems, including ILU factorization, wavelet Schur preconditioning, relaxed dimensional factorization (RDF), which can be respectively found in [4,6,7]. A comprehensive survey of existing preconditioning approaches for saddle point problems has been done by M. Benzi in [5]. Among these techniques, RDF preconditioning has been shown to be very effective, which is because it can be guaranteed that the preconditioned matrix has an eigenvalue at 1 with multiplicity at least n1 n2 , and the remaining eigenvalues are all located between 0 and 1. In this paper, using parameterized splitting thought, a preconditioner for Eqs. (1) was presented. Besides keeping the eigenvalue at 1 with the same multiplicity in RDF, it was shown that the remaining eigenvalues are more close to 1 than those of RDF preconditioned matrix. Furthermore, for saving CPU time, the Sherman-Morrison-Woodbury formula was utilized to approximately deal with the inverse of sub-matrix.
2 A NEW PRECONDITIONER BASED ON PARAMETERIZED THOUGHT In this section, a new preconditioner was proposed using parameterized splitting thought, and the eigenvalue of the preconditioned system was analysed. Define K1 B1 A11B1T , K2 B2 A21B2T , and let be a real number. Using the parameterized splitting thought, the preconditioner of the form was presented
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A11 I 1 H A2 K1 K 2 1 B1 A11 (I )
I
I B2 A2
1
(2)
For the preconditioner (2), we have Theorem 1 The preconditioned matrix T HH has an eigenvalue at 1 with multiplicity at least n1 n2 , and the remaining eigenvalues satisfy
s1 s2 , s1 s2
where s1 T K1, s2 T K2 , and the unit vector
(
K1
K2
m
(3)
satisfies
) ( I
K1
K2
) .
Proof Firstly, we can easily get
I A11 B1T 1 T HH I A2 B2 , K K 2 1 K1 K 2 (I 1 ) hence, it can be immediately obtained that T has an eigenvalue at 1 with multiplicity at least n1 n2 . For
the
remaining
have
K1 K2
s1 s2
(I
s1 s2
eigenvalues
K1 K2
,
assuming
that
(I
K1 K 2
)1
K1 K 2
,
then
we
) , by multiplying both sides of this equality from left with T , we obtain
, and this completes the proof of theorem 1.
Remark 1 From Eqs.(3) it can be easily gotten that lim 1 . 0
In order to efficiently compute the matrix ( I
K1 K 2
)1 , the well-known Sherman-Morrison-Woodbury formula
can be utilized: (S X1GX 2T )1 S 1 S 1 X1 (G 1 X 2T S 1 X1 )1 X 2T S 1 , nn
where S integers.
r1 r1
, G
nr1
are invertible matrices, X1
nr1
, X 2
(4)
are any matrices, and n, r1 are any positive
From Eqs. (4) we immediately get
(I
K1 K 2
)1 I B( A BT B)1 BT
(5)
with B ( B1 , B2 ), A diag ( A1 , A2 ) . In the following numerical examples, when applying preconditioner (2), we will always use Eqs. (5).
3 NUMERICAL EXAMPLES In this section, some numerical experiments were carried out to illustrate the behaviour of above preconditioner. In these experiments, the linear systems which are generated by the IFISS software package done by H.C. Elman et al.[2], come from discretion of the cavity problem using Q2-Q1 and Q2-P1 finite element. Restarted GMRES (20) was used as the Krylov subspace method, and we always take a zero initial guess. Let rk be the residual vector at the - 106 www.ivypub.org/MC
k th iteration, the iteration stops when
|| rk ||2 106 . In the following table 1-4, the viscosity parameter is denoted by || r0 ||2
v , and the iterations is denoted by its. We always take 106 . TABLE 1 RESULTS ON STEADY OSEEN PROBLEMS (Q2-Q1FEM, UNIFORM GRIDS)
Grid 16 32 32 32
16 32 32 32
v=0.1
v=0.01
v=0.001
its
its
its
3 3 3 4
3 3 3 3
3 3 3 3
TABLE 2 RESULTS ON STEADY OSEEN PROBLEMS (Q2-Q1FEM,, STRETCHED GRIDS)
Grid 16 32 32 32
16 32 32 32
v=0.1
v=0.01
v=0.001
its
its
its
2 3 4 4
2 2 3 4
2 2 2 3
TABLE 3 RESULTS ON STEADY OSEEN PROBLEMS (Q2-P1FEM, UNIFORM GRIDS)
Grid 16 32 32 32
16 32 32 32
v=0.1
v=0.01
v=0.001
its
its
its
3 3 3 3
3 3 3 3
3 3 3 3
TABLE 4 RESULTS ON STEADY OSEEN PROBLEMS (Q2-P1FEM,, STRETCHED GRIDS)
Grid 16 32 32 32
16 32 32 32
v=0.1
v=0.01
v=0.001
its
its
its
2 2 3 3
2 2 2 3
2 2 2 3
4 CONCLUSIONS A preconditioner was introduced based on the parameterized splitting thought for solving saddle point systems. Theoretical analysis showed good clustering properties of the spectra of the preconditioned matrix when the parameter is small enough. The numerical examples has illustrated that this preconditioner can greatly promote the convergence of iterative methods in the Krylov subspace.
ACKNOWLEDGMENT This research was supported by NSFC (61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci.\& Tech. Research Project (12ZC1802) and the Fundamental Research Funds for the Central Universities.
REFERENCES [1]
C. Cuvelier, A. Segal and A. van Steenhoven, Finite Element Methods and the Navier-Stokes Equations, D. Reidel Publishing Co, 1986.
[2]
H. C. Elman, A. Ramage, D. J. Silvester, Algorithm 886: IFISS, A Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw., vol. 33 no. 14, 2007. DOI 10.1145/1236463.1236469. - 107 www.ivypub.org/MC
[3]
H. C. Elman, D.J. Silvester, A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford Series in Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2005.
[4]
Ke Chen, Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge, UK, 2005.
[5]
M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numerica, 14(2005): 1-137.
[6]
M. Benzi, M.K. Ng, Q. Niu and Z. Wang, A relaxed dimensional factorization preconditioner for the incompressible NavierStokes equations, J. Comput. Phys., 230 (2011): 6185-6202.
[7]
Y. Saad, Iterative Methods for Sparse Linear Systems, seconded edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003.
AUTHORS 1Weihua
Luo was born in Hunan province
two papers in Applied Mathematics and Computation, Journal of
in 1982. PhD in School of Mathematical
Applied Mathematics. Nowadays, he is being interested in
Sciences, University of Electronic Science
preconditioning techniques for large sparse linear systems.
and Technology of China, Chengdu. The author’s major field is numerical algebra and scientific computation.
2Tingzhu
Huang was born in Sichuan province in 1964.
Doctoral supervisor in School of Mathematical Sciences, University of Electronic Science and Technology of China,
He once worked as a teacher in Neijiang normal university in Sichuan province. So far, he has published
Chengdu. His major field is numerical algebra and scientific computation.
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