Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 489295 6 pageshttpdxdoiorg1011552013489295
Research ArticleA Parameterized Splitting Preconditioner for GeneralizedSaddle Point Problems
Wei-Hua Luo12 and Ting-Zhu Huang1
1 School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu Sichuan 611731 China2 Key Laboratory of Numerical Simulation of Sichuan Province University Neijiang Normal University Neijiang Sichuan 641112 China
Correspondence should be addressed to Ting-Zhu Huang tingzhuhuang126com
Received 4 January 2013 Revised 31 March 2013 Accepted 1 April 2013
Academic Editor P N Shivakumar
Copyright copy 2013 W-H Luo and T-Z Huang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
By using Sherman-Morrison-Woodbury formula we introduce a preconditioner based on parameterized splitting idea forgeneralized saddle point problems which may be singular and nonsymmetric By analyzing the eigenvalues of the preconditionedmatrix we find that when 120572 is big enough it has an eigenvalue at 1 with multiplicity at least n and the remaining eigenvalues are alllocated in a unit circle centered at 1 Particularly when the preconditioner is used in general saddle point problems it guaranteeseigenvalue at 1 with the same multiplicity and the remaining eigenvalues will tend to 1 as the parameter 120572 rarr 0 Consequently thiscan lead to a good convergence when some GMRES iterative methods are used in Krylov subspace Numerical results of Stokesproblems and Oseen problems are presented to illustrate the behavior of the preconditioner
1 Introduction
In some scientific and engineering applications such as finiteelement methods for solving partial differential equations [12] and computational fluid dynamics [3 4] we often considersolutions of the generalized saddle point problems of the form
(119860 119861119879
minus119861 119862
)(
119909
119910) = (
119891
119892) (1)
where 119860 isin R119899times119899 119861 isin R119898times119899 (119898 le 119899) and 119862 isin R119898times119898
are positive semidefinite 119909 119891 isin R119899 and 119910 119892 isin R119898 When119862 = 0 (1) is a general saddle point problem which is also aresearching object for many authors
It is well known that when the matrices 119860 119861 and 119862
are large and sparse the iterative methods are more efficientand attractive than direct methods assuming that (1) has agood preconditioner In recent years a lot of preconditioningtechniques have arisen for solving linear system for exampleSaad [5] and Chen [6] have comprehensively surveyed someclassical preconditioning techniques including ILU pre-conditioner triangular preconditioner SPAI preconditionermultilevel recursive Schur complements preconditioner and
sparse wavelet preconditioner Particularly many precondi-tioning methods for saddle problems have been presentedrecently such as dimensional splitting (DS) [7] relaxeddimensional factorization (RDF) [8] splitting preconditioner[9] and Hermitian and skew-Hermitian splitting precondi-tioner [10]
Among these results Cao et al [9] have used splitting ideato give a preconditioner for saddle point problems where thematrix 119860 is symmetric and positive definite and 119861 is of fullrow rank According to his preconditioner the eigenvaluesof the preconditioned matrix would tend to 1 when theparameter 119905 rarr infin Consequently just as we have seen fromthose examples of [9] preconditioner has guaranteed a goodconvergence when some iterative methods were used
In this paper being motivated by [9] we use the splittingidea to present a preconditioner for the system (1) where 119860may be nonsymmetric and singular (when rank(119861) lt 119898) Wefind that when the parameter is big enough the precondi-tioned matrix has the eigenvalue at 1 with multiplicity at least119899 and the remaining eigenvalues are all located in a unit circlecentered at 1 Particularly when the precondidtioner is usedin some general saddle point problems (namely 119862 = 0) we
2 Journal of Applied Mathematics
see that the multiplicity of the eigenvalue at 1 is also at least 119899but the remaining eigenvalues will tend to 1 as the parameter120572 rarr 0
The remainder of the paper is organized as followsIn Section 2 we present our preconditioner based on thesplitting idea and analyze the bound of eigenvalues of thepreconditioned matrix In Section 3 we use some numericalexamples to show the behavior of the new preconditionerFinally we draw some conclusions and outline our futurework in Section 4
2 A Parameterized Splitting Preconditioner
Now we consider using splitting idea with a variable param-eter to present a preconditioner for the system (1)
Firstly it is evident that when 120572 = 0 the system (1) isequivalent to
(
119860 119861119879
minus119861
120572
119862
120572
)(
119909
119910) = (
119891
119892
120572
) (2)
Let
119872 = (
119860 119861119879
minus119861
120572
119868
) 119873 = (
0 0
0 119868 minus
119862
120572
)
119883 = (
119909
119910) 119865 = (
119891
119892
120572
)
(3)
Then the coefficient matrix of (2) can be expressed by119872minus119873Multiplying both sides of system (2) from the left with matrix119872minus1 we have
(119868 minus119872minus1119873)119883 = 119872
minus1119865 (4)
Hence we obtain a preconditioned linear system from (1)using the idea of splitting and the corresponding precondi-tioner is
119867 = (
119860 119861119879
minus119861
120572
119868
)
minus1
(
119868 0
0
119868
120572
) (5)
Nowwe analyze the eigenvalues of the preconditioned system(4)
Theorem 1 The preconditioned matrix 119868 minus 119872minus1119873 has an
eigenvalue at 1 with multiplicity at least 119899 The remainingeigenvalues 120582 satisfy
120582 =
1199041+ 1199042
1199041+ 120572
(6)
where 1199041= 120596119879119861119860minus1119861119879120596 1199042= 120596119879119862120596 and 120596 isin C119898 satisfies
(
119861119860minus1119861119879
120572
+
119862
120572
)120596 = 120582(119868 +
119861119860minus1119861119879
120572
)120596 120596 = 1 (7)
Proof Because
119872minus1=(
(119860 +
119861119879119861
120572
)
minus1
minus119860minus1119861119879(119868 +
119861119860minus1119861119879
120572
)
minus1
119861
120572
(119860 +
119861119879119861
120572
)
minus1
(119868 +
119861119860minus1119861119879
120572
)
minus1)
(8)
we can easily get
119868 minus119872minus1119873 =(
119868 119860minus1119861119879(119868 +
119861119860minus1119861119879
120572
)
minus1
(119868 minus
119862
120572
)
0 (119868 +
119861119860minus1119861119879
120572
)
minus1
(
119861119860minus1119861119879
120572
+
119862
120572
)
)
(9)
which implies the preconditioned matrix 119868 minus 119872minus1119873 has aneigenvalue at 1 with multiplicity at least 119899
For the remaining eigenvalues let
(119868 +
119861119860minus1119861119879
120572
)
minus1
(
119861119860minus1119861119879
120572
+
119862
120572
)120596 = 120582120596 (10)
with 120596 = 1 then we have
(
119861119860minus1119861119879
120572
+
119862
120572
)120596 = 120582(119868 +
119861119860minus1119861119879
120572
)120596 (11)
By multiplying both sides of this equality from the left with120596119879 we can get
120582 =
1199041+ 1199042
1199041+ 120572
(12)
This completes the proof of Theorem 1
Remark 2 FromTheorem 1 we can get that when parameter120572 is big enough the modulus of nonnil eigenvalues 120582 will belocated in interval (0 1)
Remark 3 InTheorem 1 if the matrix 119862 = 0 then for nonnileigenvalues we have
lim120572rarr0
120582 = 1 (13)
Figures 1 2 and 3 are the eigenvalues plots of the pre-conditioned matrices obtained with our preconditioner Aswe can see in the following numerical experiments thisgood phenomenon is useful for accelerating convergence ofiterative methods in Krylov subspace
Additionally for the purpose of practically executing ourpreconditioner
119867 =(
(119860 +
119861119879119861
120572
)
minus1
minus119860minus1119861119879
120572
(119868 +
119861119860minus1119861119879
120572
)
minus1
119861
120572
(119860 +
119861119879119861
120572
)
minus1
1
120572
(119868 +
119861119860minus1119861119879
120572
)
minus1)
(14)
Journal of Applied Mathematics 3
0 02 04 06 08 12 141
0
1
2
3
4
minus1
minus2
minus3
times10minus4
= 01 120572 = 00001
(a)
0
002
004
006
008
0 02 04 06 08 12 141minus008
minus006
minus004
minus002
= 01 120572 = norm(119862 fro)20
(b)
Figure 1 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 01 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
0
0 02 04 06 08 121
02
04
06
08
1
minus02
minus04
minus06
minus08
minus1
minus02
times10minus3
= 001 120572 = 00001
(a)
0 02 04 06 08 12 141minus02
minus015
minus01
minus005
0
005
01
015
02 = 001 120572 = norm(119862 fro)20
(b)
Figure 2 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 001 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
0
2
4
6
8
0 02 04 06 08 12 141
times10minus4
minus8
minus6
minus4
minus2
= 0001 120572 = 00001
(a)
0
002
004
006
008
minus008
minus006
minus004
minus002
0 02 04 06 08 121minus02
= 0001 120572 = norm(119862 fro)20
(b)
Figure 3 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 0001 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
Grid 120572 its LU time its time Total time16 times 16 00001 4 00457 00267 0072432 times 32 00001 6 03912 00813 0472564 times 64 00001 9 54472 05519 59991128 times 128 00001 14 914698 57732 972430
Table 2 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 00001 3 00479 00244 0072332 times 32 00001 3 06312 00696 0700964 times 64 00001 4 132959 03811 136770128 times 128 00001 6 1305463 37727 1343190
Table 3 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00442 00236 0067832 times 32 00001 3 04160 00557 0471764 times 64 00001 3 73645 02623 76268128 times 128 00001 4 1694009 31709 1725718
Table 4 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00481 00206 0068732 times 32 00001 3 04661 00585 0524664 times 64 00001 3 66240 02728 68969128 times 128 00001 4 1773130 29814 1802944
Grid 120572 its LU time its time Total time16 times 16 10000 5 00471 00272 0074332 times 32 10000 7 03914 00906 0482064 times 64 10000 9 56107 04145 60252128 times 128 10000 14 927154 48908 976062
Table 6 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 10000 4 00458 00223 0068132 times 32 10000 4 05748 00670 0641864 times 64 10000 5 122642 04179 126821128 times 128 10000 7 1282758 16275 1299033
Table 7 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00458 00224 0068232 times 32 10000 3 04309 00451 0476064 times 64 10000 4 76537 01712 78249128 times 128 10000 5 1751587 34554 1786141
Table 8 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00507 00212 0071932 times 32 10000 3 04735 00449 0518464 times 64 10000 4 66482 01645 68127128 times 128 10000 4 1720516 21216 1741732
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro) 21 00240 00473 0071332 times 32 norm(119862 fro) 20 00890 01497 0238764 times 64 norm(119862 fro) 20 08387 06191 14578128 times 128 norm(119862 fro) 20 68917 30866 99783
Table 10 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 10 00250 00302 0055232 times 32 norm(119862 fro)20 10 00816 00832 0164864 times 64 norm(119862 fro)20 12 08466 03648 12114128 times 128 norm(119862 fro)20 14 69019 20398 89417
Table 11 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 7 00238 00286 0052432 times 32 norm(119862 fro)20 7 00850 00552 0140264 times 64 norm(119862 fro)20 11 08400 03177 11577128 times 128 norm(119862 fro)20 16 69537 22306 91844
Table 12 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 5 00245 00250 0049532 times 32 norm(119862 fro)20 5 00905 00587 0149264 times 64 norm(119862 fro)20 8 12916 03200 16116128 times 128 norm(119862 fro)20 15 104399 27468 131867
Journal of Applied Mathematics 5
Table 13 Size and number of non-nil elements of the coefficient matrix generalized by using Q2-Q1 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 6178 81 times 578 2318 3690432 times 32 2178 times 2178 28418 289 times 2178 10460 19322064 times 64 8450 times 8450 122206 1089 times 8450 44314 875966128 times 128 33282 times 33282 506376 4225 times 33282 184316 3741110
we should efficiently deal with the computation of (119868 +(119861119860minus1119861119879120572))minus1 This can been tackled by the well-known
Sherman-Morrison-Woodbury formula
(1198601+ 11988311198602119883119879
2)
minus1
= 119860minus1
1minus 119860minus1
11198831(119860minus1
2+ 119883119879
2119860minus1
11198831)
minus1
119883119879
2119860minus1
1
(15)
where 1198601isin R119899times119899 and 119860
2isin R1199031times1199031 are invertible matrices
1198831isin R119899times1199031 and 119883
2isin R119899times1199031 are any matrices and 119899 119903
1are
any positive integersFrom (15) we immediately get
(119868 +
119861119860minus1119861119879
120572
)
minus1
= 119868 minus 119861(120572119860 + 119861119879119861)
minus1
119861119879 (16)
In the following numerical examples we will always use (16)to compute (119868 + (119861119860minus1119861119879120572))minus1 in (14)
3 Numerical Examples
In this sectionwe give numerical experiments to illustrate thebehavior of our preconditioner The numerical experimentsare done by using MATLAB 71 The linear systems areobtained by using finite element methods in the Stokes prob-lems and steady Oseen problems and they are respectivelythe cases of
(1) 119862 = 0 which is caused by using Q2-Q1 FEM(2) 119862 = 0 which is caused by using Q1-P0 FEM
Furthermore we compare our preconditionerwith that of[9] in the case of general saddle point problems (namely 119862 =0) For the general saddle point problem [9] has presentedthe preconditioner
= (
119860 + 119905119861119879119861 0
minus2119861
119868
119905
)
minus1
(17)
with 119905 as a parameter and has proved that when 119860 issymmetric positive definite the preconditioned matrix hasan eigenvalue 1 with multiplicity at 119899 and the remainingeigenvalues satisfy
120582 =
1199051205902
119894
1 + 1199051205902
119894
(18)
lim119905rarrinfin
120582 = 1 (19)
where 120590119894 119894 = 1 2 119898 are119898 positive singular values of the
matrix 119861119860minus12All these systems can be generalized by
using IFISS software package [11] (this is a freepackage that can be downloaded from the sitehttpwwwmathsmanchesteracuksimdjsifiss) We userestarted GMRES(20) as the Krylov subspace method andwe always take a zero initial guess The stopping criterion is
1003817100381710038171003817119903119896
10038171003817100381710038172
10038171003817100381710038171199030
10038171003817100381710038172
le 10minus6 (20)
where 119903119896is the residual vector at the 119896th iteration
In the whole course of computation we always replace(119868 + (119861119860
minus1119861119879120572))
minus1 in (14) with (16) and use the 119871119880 factor-ization of 119860 + (119861
119879119861120572) to tackle (119860 + (119861
119879119861120572))
minus1V whereV is a corresponding vector in the iteration Concretely let119860 + (119861
119879119861120572) = 119871119880 then we complete the matrix-vector
product (119860 + (119861119879119861120572))
minus1V by 119880 119871 V in MATLAB termIn the following tables the denotation norm (119862 fro) meansthe Frobenius form of the matrix 119862 The total time is the sumof LU time and iterative time and the LU time is the time tocompute LU factorization of 119860 + (119861119879119861120572)
Case 1 (for our preconditioner) 119862 = 0 (using Q2-Q1 FEM inStokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 1 2 3 4 )
Case 1rsquo (for preconditioner of [9]) 119862 = 0 (using Q2-Q1 FEMin Stokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 5 6 7 8)
Case 2 119862 = 0 (using Q1-P0 FEM in Stokes problems andsteady Oseen problems with different viscosity coefficientsThe results are in Tables 9 10 11 12)
From Tables 1 2 3 4 5 6 7 and 8 we can see that theseresults are in agreement with the theoretical analyses (13) and(19) respectively Additionally comparing with the results inTables 9 10 11 and 12 we find that although the iterationsused in Case 1 (either for the preconditioner of [9] or ourpreconditioner) are less than those in Case 2 the time spentby Case 1 ismuchmore than that of Case 2This is because thedensity of the coefficient matrix generalized by Q2-Q1 FEMis much larger than that generalized by Q1-P0 FEMThis canbe partly illustrated by Tables 13 and 14 and the others can beillustrated similarly
6 Journal of Applied Mathematics
Table 14 Size and number of non-nil elements of the coefficient matrix generalized by using Q1-P0 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 3826 256 times 578 1800 707632 times 32 2178 times 2178 16818 1024 times 2178 7688 3187464 times 64 8450 times 8450 70450 4096 times 8450 31752 136582128 times 128 33282 times 33282 288306 16384 times 33282 129032 567192
4 Conclusions
In this paper we have introduced a splitting preconditionerfor solving generalized saddle point systems Theoreticalanalysis showed the modulus of eigenvalues of the precon-ditioned matrix would be located in interval (0 1) when theparameter is big enough Particularly when the submatrix119862 = 0 the eigenvalues will tend to 1 as the parameter 120572 rarr 0These performances are tested by some examples and theresults are in agreement with the theoretical analysis
There are still some future works to be done how to prop-erly choose a parameter 120572 so that the preconditioned matrixhas better propertiesHow to further precondition submatrix(119868 + (119861119860
minus1119861119879120572))
minus1
((119861119860minus1119861119879120572) + (119862120572)) to improve our
preconditioner
Acknowledgments
The authors would express their great thankfulness to thereferees and the editor Professor P N Shivakumar for theirhelpful suggestions for revising this paperThe authors wouldlike to thank H C Elman A Ramage and D J Silvester fortheir free IFISS software package This research is supportedby Chinese Universities Specialized Research Fund for theDoctoral Program (20110185110020) Sichuan Province Sci ampTech Research Project (2012GZX0080) and the Fundamen-tal Research Funds for the Central Universities
References
[1] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991
[2] Z Chen Q Du and J Zou ldquoFinite element methods withmatching and nonmatching meshes for Maxwell equationswith discontinuous coefficientsrdquo SIAM Journal on NumericalAnalysis vol 37 no 5 pp 1542ndash1570 2000
[3] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Numerical Mathematics and Scientific Com-putation Oxford University Press New York NY USA 2005
[4] C Cuvelier A Segal and A A van Steenhoven Finite ElementMethods and Navier-Stokes Equations vol 22 of Mathematicsand its Applications D Reidel Dordrecht The Netherlands1986
[5] Y Saad Iterative Methods for Sparse Linear Systems Society forIndustrial and Applied Mathematics Philadelphia Pa USA2nd edition 2003
[6] K Chen Matrix Preconditioning Techniques and Applicationsvol 19 ofCambridgeMonographs onApplied andComputational
Mathematics Cambridge University Press Cambridge UK2005
[7] M Benzi and X-P Guo ldquoA dimensional split preconditionerfor Stokes and linearized Navier-Stokes equationsrdquo AppliedNumerical Mathematics vol 61 no 1 pp 66ndash76 2011
[8] M Benzi M Ng Q Niu and Z Wang ldquoA relaxed dimensionalfactorization preconditioner for the incompressible Navier-Stokes equationsrdquo Journal of Computational Physics vol 230no 16 pp 6185ndash6202 2011
[9] YCaoM-Q Jiang andY-L Zheng ldquoA splitting preconditionerfor saddle point problemsrdquo Numerical Linear Algebra withApplications vol 18 no 5 pp 875ndash895 2011
[10] V Simoncini and M Benzi ldquoSpectral properties of the Her-mitian and skew-Hermitian splitting preconditioner for saddlepoint problemsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 26 no 2 pp 377ndash389 2004
[11] H C Elman A Ramage and D J Silvester ldquoAlgorithm 886IFISS a Matlab toolbox for modelling incompressible flowrdquoACM Transactions on Mathematical Software vol 33 no 2article 14 2007
see that the multiplicity of the eigenvalue at 1 is also at least 119899but the remaining eigenvalues will tend to 1 as the parameter120572 rarr 0
The remainder of the paper is organized as followsIn Section 2 we present our preconditioner based on thesplitting idea and analyze the bound of eigenvalues of thepreconditioned matrix In Section 3 we use some numericalexamples to show the behavior of the new preconditionerFinally we draw some conclusions and outline our futurework in Section 4
2 A Parameterized Splitting Preconditioner
Now we consider using splitting idea with a variable param-eter to present a preconditioner for the system (1)
Firstly it is evident that when 120572 = 0 the system (1) isequivalent to
(
119860 119861119879
minus119861
120572
119862
120572
)(
119909
119910) = (
119891
119892
120572
) (2)
Let
119872 = (
119860 119861119879
minus119861
120572
119868
) 119873 = (
0 0
0 119868 minus
119862
120572
)
119883 = (
119909
119910) 119865 = (
119891
119892
120572
)
(3)
Then the coefficient matrix of (2) can be expressed by119872minus119873Multiplying both sides of system (2) from the left with matrix119872minus1 we have
(119868 minus119872minus1119873)119883 = 119872
minus1119865 (4)
Hence we obtain a preconditioned linear system from (1)using the idea of splitting and the corresponding precondi-tioner is
119867 = (
119860 119861119879
minus119861
120572
119868
)
minus1
(
119868 0
0
119868
120572
) (5)
Nowwe analyze the eigenvalues of the preconditioned system(4)
Theorem 1 The preconditioned matrix 119868 minus 119872minus1119873 has an
eigenvalue at 1 with multiplicity at least 119899 The remainingeigenvalues 120582 satisfy
120582 =
1199041+ 1199042
1199041+ 120572
(6)
where 1199041= 120596119879119861119860minus1119861119879120596 1199042= 120596119879119862120596 and 120596 isin C119898 satisfies
(
119861119860minus1119861119879
120572
+
119862
120572
)120596 = 120582(119868 +
119861119860minus1119861119879
120572
)120596 120596 = 1 (7)
Proof Because
119872minus1=(
(119860 +
119861119879119861
120572
)
minus1
minus119860minus1119861119879(119868 +
119861119860minus1119861119879
120572
)
minus1
119861
120572
(119860 +
119861119879119861
120572
)
minus1
(119868 +
119861119860minus1119861119879
120572
)
minus1)
(8)
we can easily get
119868 minus119872minus1119873 =(
119868 119860minus1119861119879(119868 +
119861119860minus1119861119879
120572
)
minus1
(119868 minus
119862
120572
)
0 (119868 +
119861119860minus1119861119879
120572
)
minus1
(
119861119860minus1119861119879
120572
+
119862
120572
)
)
(9)
which implies the preconditioned matrix 119868 minus 119872minus1119873 has aneigenvalue at 1 with multiplicity at least 119899
For the remaining eigenvalues let
(119868 +
119861119860minus1119861119879
120572
)
minus1
(
119861119860minus1119861119879
120572
+
119862
120572
)120596 = 120582120596 (10)
with 120596 = 1 then we have
(
119861119860minus1119861119879
120572
+
119862
120572
)120596 = 120582(119868 +
119861119860minus1119861119879
120572
)120596 (11)
By multiplying both sides of this equality from the left with120596119879 we can get
120582 =
1199041+ 1199042
1199041+ 120572
(12)
This completes the proof of Theorem 1
Remark 2 FromTheorem 1 we can get that when parameter120572 is big enough the modulus of nonnil eigenvalues 120582 will belocated in interval (0 1)
Remark 3 InTheorem 1 if the matrix 119862 = 0 then for nonnileigenvalues we have
lim120572rarr0
120582 = 1 (13)
Figures 1 2 and 3 are the eigenvalues plots of the pre-conditioned matrices obtained with our preconditioner Aswe can see in the following numerical experiments thisgood phenomenon is useful for accelerating convergence ofiterative methods in Krylov subspace
Additionally for the purpose of practically executing ourpreconditioner
119867 =(
(119860 +
119861119879119861
120572
)
minus1
minus119860minus1119861119879
120572
(119868 +
119861119860minus1119861119879
120572
)
minus1
119861
120572
(119860 +
119861119879119861
120572
)
minus1
1
120572
(119868 +
119861119860minus1119861119879
120572
)
minus1)
(14)
Journal of Applied Mathematics 3
0 02 04 06 08 12 141
0
1
2
3
4
minus1
minus2
minus3
times10minus4
= 01 120572 = 00001
(a)
0
002
004
006
008
0 02 04 06 08 12 141minus008
minus006
minus004
minus002
= 01 120572 = norm(119862 fro)20
(b)
Figure 1 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 01 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
0
0 02 04 06 08 121
02
04
06
08
1
minus02
minus04
minus06
minus08
minus1
minus02
times10minus3
= 001 120572 = 00001
(a)
0 02 04 06 08 12 141minus02
minus015
minus01
minus005
0
005
01
015
02 = 001 120572 = norm(119862 fro)20
(b)
Figure 2 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 001 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
0
2
4
6
8
0 02 04 06 08 12 141
times10minus4
minus8
minus6
minus4
minus2
= 0001 120572 = 00001
(a)
0
002
004
006
008
minus008
minus006
minus004
minus002
0 02 04 06 08 121minus02
= 0001 120572 = norm(119862 fro)20
(b)
Figure 3 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 0001 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
Grid 120572 its LU time its time Total time16 times 16 00001 4 00457 00267 0072432 times 32 00001 6 03912 00813 0472564 times 64 00001 9 54472 05519 59991128 times 128 00001 14 914698 57732 972430
Table 2 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 00001 3 00479 00244 0072332 times 32 00001 3 06312 00696 0700964 times 64 00001 4 132959 03811 136770128 times 128 00001 6 1305463 37727 1343190
Table 3 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00442 00236 0067832 times 32 00001 3 04160 00557 0471764 times 64 00001 3 73645 02623 76268128 times 128 00001 4 1694009 31709 1725718
Table 4 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00481 00206 0068732 times 32 00001 3 04661 00585 0524664 times 64 00001 3 66240 02728 68969128 times 128 00001 4 1773130 29814 1802944
Grid 120572 its LU time its time Total time16 times 16 10000 5 00471 00272 0074332 times 32 10000 7 03914 00906 0482064 times 64 10000 9 56107 04145 60252128 times 128 10000 14 927154 48908 976062
Table 6 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 10000 4 00458 00223 0068132 times 32 10000 4 05748 00670 0641864 times 64 10000 5 122642 04179 126821128 times 128 10000 7 1282758 16275 1299033
Table 7 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00458 00224 0068232 times 32 10000 3 04309 00451 0476064 times 64 10000 4 76537 01712 78249128 times 128 10000 5 1751587 34554 1786141
Table 8 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00507 00212 0071932 times 32 10000 3 04735 00449 0518464 times 64 10000 4 66482 01645 68127128 times 128 10000 4 1720516 21216 1741732
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro) 21 00240 00473 0071332 times 32 norm(119862 fro) 20 00890 01497 0238764 times 64 norm(119862 fro) 20 08387 06191 14578128 times 128 norm(119862 fro) 20 68917 30866 99783
Table 10 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 10 00250 00302 0055232 times 32 norm(119862 fro)20 10 00816 00832 0164864 times 64 norm(119862 fro)20 12 08466 03648 12114128 times 128 norm(119862 fro)20 14 69019 20398 89417
Table 11 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 7 00238 00286 0052432 times 32 norm(119862 fro)20 7 00850 00552 0140264 times 64 norm(119862 fro)20 11 08400 03177 11577128 times 128 norm(119862 fro)20 16 69537 22306 91844
Table 12 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 5 00245 00250 0049532 times 32 norm(119862 fro)20 5 00905 00587 0149264 times 64 norm(119862 fro)20 8 12916 03200 16116128 times 128 norm(119862 fro)20 15 104399 27468 131867
Journal of Applied Mathematics 5
Table 13 Size and number of non-nil elements of the coefficient matrix generalized by using Q2-Q1 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 6178 81 times 578 2318 3690432 times 32 2178 times 2178 28418 289 times 2178 10460 19322064 times 64 8450 times 8450 122206 1089 times 8450 44314 875966128 times 128 33282 times 33282 506376 4225 times 33282 184316 3741110
we should efficiently deal with the computation of (119868 +(119861119860minus1119861119879120572))minus1 This can been tackled by the well-known
Sherman-Morrison-Woodbury formula
(1198601+ 11988311198602119883119879
2)
minus1
= 119860minus1
1minus 119860minus1
11198831(119860minus1
2+ 119883119879
2119860minus1
11198831)
minus1
119883119879
2119860minus1
1
(15)
where 1198601isin R119899times119899 and 119860
2isin R1199031times1199031 are invertible matrices
1198831isin R119899times1199031 and 119883
2isin R119899times1199031 are any matrices and 119899 119903
1are
any positive integersFrom (15) we immediately get
(119868 +
119861119860minus1119861119879
120572
)
minus1
= 119868 minus 119861(120572119860 + 119861119879119861)
minus1
119861119879 (16)
In the following numerical examples we will always use (16)to compute (119868 + (119861119860minus1119861119879120572))minus1 in (14)
3 Numerical Examples
In this sectionwe give numerical experiments to illustrate thebehavior of our preconditioner The numerical experimentsare done by using MATLAB 71 The linear systems areobtained by using finite element methods in the Stokes prob-lems and steady Oseen problems and they are respectivelythe cases of
(1) 119862 = 0 which is caused by using Q2-Q1 FEM(2) 119862 = 0 which is caused by using Q1-P0 FEM
Furthermore we compare our preconditionerwith that of[9] in the case of general saddle point problems (namely 119862 =0) For the general saddle point problem [9] has presentedthe preconditioner
= (
119860 + 119905119861119879119861 0
minus2119861
119868
119905
)
minus1
(17)
with 119905 as a parameter and has proved that when 119860 issymmetric positive definite the preconditioned matrix hasan eigenvalue 1 with multiplicity at 119899 and the remainingeigenvalues satisfy
120582 =
1199051205902
119894
1 + 1199051205902
119894
(18)
lim119905rarrinfin
120582 = 1 (19)
where 120590119894 119894 = 1 2 119898 are119898 positive singular values of the
matrix 119861119860minus12All these systems can be generalized by
using IFISS software package [11] (this is a freepackage that can be downloaded from the sitehttpwwwmathsmanchesteracuksimdjsifiss) We userestarted GMRES(20) as the Krylov subspace method andwe always take a zero initial guess The stopping criterion is
1003817100381710038171003817119903119896
10038171003817100381710038172
10038171003817100381710038171199030
10038171003817100381710038172
le 10minus6 (20)
where 119903119896is the residual vector at the 119896th iteration
In the whole course of computation we always replace(119868 + (119861119860
minus1119861119879120572))
minus1 in (14) with (16) and use the 119871119880 factor-ization of 119860 + (119861
119879119861120572) to tackle (119860 + (119861
119879119861120572))
minus1V whereV is a corresponding vector in the iteration Concretely let119860 + (119861
119879119861120572) = 119871119880 then we complete the matrix-vector
product (119860 + (119861119879119861120572))
minus1V by 119880 119871 V in MATLAB termIn the following tables the denotation norm (119862 fro) meansthe Frobenius form of the matrix 119862 The total time is the sumof LU time and iterative time and the LU time is the time tocompute LU factorization of 119860 + (119861119879119861120572)
Case 1 (for our preconditioner) 119862 = 0 (using Q2-Q1 FEM inStokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 1 2 3 4 )
Case 1rsquo (for preconditioner of [9]) 119862 = 0 (using Q2-Q1 FEMin Stokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 5 6 7 8)
Case 2 119862 = 0 (using Q1-P0 FEM in Stokes problems andsteady Oseen problems with different viscosity coefficientsThe results are in Tables 9 10 11 12)
From Tables 1 2 3 4 5 6 7 and 8 we can see that theseresults are in agreement with the theoretical analyses (13) and(19) respectively Additionally comparing with the results inTables 9 10 11 and 12 we find that although the iterationsused in Case 1 (either for the preconditioner of [9] or ourpreconditioner) are less than those in Case 2 the time spentby Case 1 ismuchmore than that of Case 2This is because thedensity of the coefficient matrix generalized by Q2-Q1 FEMis much larger than that generalized by Q1-P0 FEMThis canbe partly illustrated by Tables 13 and 14 and the others can beillustrated similarly
6 Journal of Applied Mathematics
Table 14 Size and number of non-nil elements of the coefficient matrix generalized by using Q1-P0 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 3826 256 times 578 1800 707632 times 32 2178 times 2178 16818 1024 times 2178 7688 3187464 times 64 8450 times 8450 70450 4096 times 8450 31752 136582128 times 128 33282 times 33282 288306 16384 times 33282 129032 567192
4 Conclusions
In this paper we have introduced a splitting preconditionerfor solving generalized saddle point systems Theoreticalanalysis showed the modulus of eigenvalues of the precon-ditioned matrix would be located in interval (0 1) when theparameter is big enough Particularly when the submatrix119862 = 0 the eigenvalues will tend to 1 as the parameter 120572 rarr 0These performances are tested by some examples and theresults are in agreement with the theoretical analysis
There are still some future works to be done how to prop-erly choose a parameter 120572 so that the preconditioned matrixhas better propertiesHow to further precondition submatrix(119868 + (119861119860
minus1119861119879120572))
minus1
((119861119860minus1119861119879120572) + (119862120572)) to improve our
preconditioner
Acknowledgments
The authors would express their great thankfulness to thereferees and the editor Professor P N Shivakumar for theirhelpful suggestions for revising this paperThe authors wouldlike to thank H C Elman A Ramage and D J Silvester fortheir free IFISS software package This research is supportedby Chinese Universities Specialized Research Fund for theDoctoral Program (20110185110020) Sichuan Province Sci ampTech Research Project (2012GZX0080) and the Fundamen-tal Research Funds for the Central Universities
References
[1] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991
[2] Z Chen Q Du and J Zou ldquoFinite element methods withmatching and nonmatching meshes for Maxwell equationswith discontinuous coefficientsrdquo SIAM Journal on NumericalAnalysis vol 37 no 5 pp 1542ndash1570 2000
[3] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Numerical Mathematics and Scientific Com-putation Oxford University Press New York NY USA 2005
[4] C Cuvelier A Segal and A A van Steenhoven Finite ElementMethods and Navier-Stokes Equations vol 22 of Mathematicsand its Applications D Reidel Dordrecht The Netherlands1986
[5] Y Saad Iterative Methods for Sparse Linear Systems Society forIndustrial and Applied Mathematics Philadelphia Pa USA2nd edition 2003
[6] K Chen Matrix Preconditioning Techniques and Applicationsvol 19 ofCambridgeMonographs onApplied andComputational
Mathematics Cambridge University Press Cambridge UK2005
[7] M Benzi and X-P Guo ldquoA dimensional split preconditionerfor Stokes and linearized Navier-Stokes equationsrdquo AppliedNumerical Mathematics vol 61 no 1 pp 66ndash76 2011
[8] M Benzi M Ng Q Niu and Z Wang ldquoA relaxed dimensionalfactorization preconditioner for the incompressible Navier-Stokes equationsrdquo Journal of Computational Physics vol 230no 16 pp 6185ndash6202 2011
[9] YCaoM-Q Jiang andY-L Zheng ldquoA splitting preconditionerfor saddle point problemsrdquo Numerical Linear Algebra withApplications vol 18 no 5 pp 875ndash895 2011
[10] V Simoncini and M Benzi ldquoSpectral properties of the Her-mitian and skew-Hermitian splitting preconditioner for saddlepoint problemsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 26 no 2 pp 377ndash389 2004
[11] H C Elman A Ramage and D J Silvester ldquoAlgorithm 886IFISS a Matlab toolbox for modelling incompressible flowrdquoACM Transactions on Mathematical Software vol 33 no 2article 14 2007
Figure 1 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 01 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
0
0 02 04 06 08 121
02
04
06
08
1
minus02
minus04
minus06
minus08
minus1
minus02
times10minus3
= 001 120572 = 00001
(a)
0 02 04 06 08 12 141minus02
minus015
minus01
minus005
0
005
01
015
02 = 001 120572 = norm(119862 fro)20
(b)
Figure 2 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 001 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
0
2
4
6
8
0 02 04 06 08 12 141
times10minus4
minus8
minus6
minus4
minus2
= 0001 120572 = 00001
(a)
0
002
004
006
008
minus008
minus006
minus004
minus002
0 02 04 06 08 121minus02
= 0001 120572 = norm(119862 fro)20
(b)
Figure 3 Spectrum of the preconditioned steady Oseen matrix with viscosity coefficient V = 0001 32 times 32 grid (a) Q2-Q1 FEM 119862 = 0 (b)Q1-P0 FEM 119862 = 0
Grid 120572 its LU time its time Total time16 times 16 00001 4 00457 00267 0072432 times 32 00001 6 03912 00813 0472564 times 64 00001 9 54472 05519 59991128 times 128 00001 14 914698 57732 972430
Table 2 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 00001 3 00479 00244 0072332 times 32 00001 3 06312 00696 0700964 times 64 00001 4 132959 03811 136770128 times 128 00001 6 1305463 37727 1343190
Table 3 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00442 00236 0067832 times 32 00001 3 04160 00557 0471764 times 64 00001 3 73645 02623 76268128 times 128 00001 4 1694009 31709 1725718
Table 4 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00481 00206 0068732 times 32 00001 3 04661 00585 0524664 times 64 00001 3 66240 02728 68969128 times 128 00001 4 1773130 29814 1802944
Grid 120572 its LU time its time Total time16 times 16 10000 5 00471 00272 0074332 times 32 10000 7 03914 00906 0482064 times 64 10000 9 56107 04145 60252128 times 128 10000 14 927154 48908 976062
Table 6 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 10000 4 00458 00223 0068132 times 32 10000 4 05748 00670 0641864 times 64 10000 5 122642 04179 126821128 times 128 10000 7 1282758 16275 1299033
Table 7 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00458 00224 0068232 times 32 10000 3 04309 00451 0476064 times 64 10000 4 76537 01712 78249128 times 128 10000 5 1751587 34554 1786141
Table 8 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00507 00212 0071932 times 32 10000 3 04735 00449 0518464 times 64 10000 4 66482 01645 68127128 times 128 10000 4 1720516 21216 1741732
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro) 21 00240 00473 0071332 times 32 norm(119862 fro) 20 00890 01497 0238764 times 64 norm(119862 fro) 20 08387 06191 14578128 times 128 norm(119862 fro) 20 68917 30866 99783
Table 10 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 10 00250 00302 0055232 times 32 norm(119862 fro)20 10 00816 00832 0164864 times 64 norm(119862 fro)20 12 08466 03648 12114128 times 128 norm(119862 fro)20 14 69019 20398 89417
Table 11 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 7 00238 00286 0052432 times 32 norm(119862 fro)20 7 00850 00552 0140264 times 64 norm(119862 fro)20 11 08400 03177 11577128 times 128 norm(119862 fro)20 16 69537 22306 91844
Table 12 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 5 00245 00250 0049532 times 32 norm(119862 fro)20 5 00905 00587 0149264 times 64 norm(119862 fro)20 8 12916 03200 16116128 times 128 norm(119862 fro)20 15 104399 27468 131867
Journal of Applied Mathematics 5
Table 13 Size and number of non-nil elements of the coefficient matrix generalized by using Q2-Q1 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 6178 81 times 578 2318 3690432 times 32 2178 times 2178 28418 289 times 2178 10460 19322064 times 64 8450 times 8450 122206 1089 times 8450 44314 875966128 times 128 33282 times 33282 506376 4225 times 33282 184316 3741110
we should efficiently deal with the computation of (119868 +(119861119860minus1119861119879120572))minus1 This can been tackled by the well-known
Sherman-Morrison-Woodbury formula
(1198601+ 11988311198602119883119879
2)
minus1
= 119860minus1
1minus 119860minus1
11198831(119860minus1
2+ 119883119879
2119860minus1
11198831)
minus1
119883119879
2119860minus1
1
(15)
where 1198601isin R119899times119899 and 119860
2isin R1199031times1199031 are invertible matrices
1198831isin R119899times1199031 and 119883
2isin R119899times1199031 are any matrices and 119899 119903
1are
any positive integersFrom (15) we immediately get
(119868 +
119861119860minus1119861119879
120572
)
minus1
= 119868 minus 119861(120572119860 + 119861119879119861)
minus1
119861119879 (16)
In the following numerical examples we will always use (16)to compute (119868 + (119861119860minus1119861119879120572))minus1 in (14)
3 Numerical Examples
In this sectionwe give numerical experiments to illustrate thebehavior of our preconditioner The numerical experimentsare done by using MATLAB 71 The linear systems areobtained by using finite element methods in the Stokes prob-lems and steady Oseen problems and they are respectivelythe cases of
(1) 119862 = 0 which is caused by using Q2-Q1 FEM(2) 119862 = 0 which is caused by using Q1-P0 FEM
Furthermore we compare our preconditionerwith that of[9] in the case of general saddle point problems (namely 119862 =0) For the general saddle point problem [9] has presentedthe preconditioner
= (
119860 + 119905119861119879119861 0
minus2119861
119868
119905
)
minus1
(17)
with 119905 as a parameter and has proved that when 119860 issymmetric positive definite the preconditioned matrix hasan eigenvalue 1 with multiplicity at 119899 and the remainingeigenvalues satisfy
120582 =
1199051205902
119894
1 + 1199051205902
119894
(18)
lim119905rarrinfin
120582 = 1 (19)
where 120590119894 119894 = 1 2 119898 are119898 positive singular values of the
matrix 119861119860minus12All these systems can be generalized by
using IFISS software package [11] (this is a freepackage that can be downloaded from the sitehttpwwwmathsmanchesteracuksimdjsifiss) We userestarted GMRES(20) as the Krylov subspace method andwe always take a zero initial guess The stopping criterion is
1003817100381710038171003817119903119896
10038171003817100381710038172
10038171003817100381710038171199030
10038171003817100381710038172
le 10minus6 (20)
where 119903119896is the residual vector at the 119896th iteration
In the whole course of computation we always replace(119868 + (119861119860
minus1119861119879120572))
minus1 in (14) with (16) and use the 119871119880 factor-ization of 119860 + (119861
119879119861120572) to tackle (119860 + (119861
119879119861120572))
minus1V whereV is a corresponding vector in the iteration Concretely let119860 + (119861
119879119861120572) = 119871119880 then we complete the matrix-vector
product (119860 + (119861119879119861120572))
minus1V by 119880 119871 V in MATLAB termIn the following tables the denotation norm (119862 fro) meansthe Frobenius form of the matrix 119862 The total time is the sumof LU time and iterative time and the LU time is the time tocompute LU factorization of 119860 + (119861119879119861120572)
Case 1 (for our preconditioner) 119862 = 0 (using Q2-Q1 FEM inStokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 1 2 3 4 )
Case 1rsquo (for preconditioner of [9]) 119862 = 0 (using Q2-Q1 FEMin Stokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 5 6 7 8)
Case 2 119862 = 0 (using Q1-P0 FEM in Stokes problems andsteady Oseen problems with different viscosity coefficientsThe results are in Tables 9 10 11 12)
From Tables 1 2 3 4 5 6 7 and 8 we can see that theseresults are in agreement with the theoretical analyses (13) and(19) respectively Additionally comparing with the results inTables 9 10 11 and 12 we find that although the iterationsused in Case 1 (either for the preconditioner of [9] or ourpreconditioner) are less than those in Case 2 the time spentby Case 1 ismuchmore than that of Case 2This is because thedensity of the coefficient matrix generalized by Q2-Q1 FEMis much larger than that generalized by Q1-P0 FEMThis canbe partly illustrated by Tables 13 and 14 and the others can beillustrated similarly
6 Journal of Applied Mathematics
Table 14 Size and number of non-nil elements of the coefficient matrix generalized by using Q1-P0 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 3826 256 times 578 1800 707632 times 32 2178 times 2178 16818 1024 times 2178 7688 3187464 times 64 8450 times 8450 70450 4096 times 8450 31752 136582128 times 128 33282 times 33282 288306 16384 times 33282 129032 567192
4 Conclusions
In this paper we have introduced a splitting preconditionerfor solving generalized saddle point systems Theoreticalanalysis showed the modulus of eigenvalues of the precon-ditioned matrix would be located in interval (0 1) when theparameter is big enough Particularly when the submatrix119862 = 0 the eigenvalues will tend to 1 as the parameter 120572 rarr 0These performances are tested by some examples and theresults are in agreement with the theoretical analysis
There are still some future works to be done how to prop-erly choose a parameter 120572 so that the preconditioned matrixhas better propertiesHow to further precondition submatrix(119868 + (119861119860
minus1119861119879120572))
minus1
((119861119860minus1119861119879120572) + (119862120572)) to improve our
preconditioner
Acknowledgments
The authors would express their great thankfulness to thereferees and the editor Professor P N Shivakumar for theirhelpful suggestions for revising this paperThe authors wouldlike to thank H C Elman A Ramage and D J Silvester fortheir free IFISS software package This research is supportedby Chinese Universities Specialized Research Fund for theDoctoral Program (20110185110020) Sichuan Province Sci ampTech Research Project (2012GZX0080) and the Fundamen-tal Research Funds for the Central Universities
References
[1] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991
[2] Z Chen Q Du and J Zou ldquoFinite element methods withmatching and nonmatching meshes for Maxwell equationswith discontinuous coefficientsrdquo SIAM Journal on NumericalAnalysis vol 37 no 5 pp 1542ndash1570 2000
[3] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Numerical Mathematics and Scientific Com-putation Oxford University Press New York NY USA 2005
[4] C Cuvelier A Segal and A A van Steenhoven Finite ElementMethods and Navier-Stokes Equations vol 22 of Mathematicsand its Applications D Reidel Dordrecht The Netherlands1986
[5] Y Saad Iterative Methods for Sparse Linear Systems Society forIndustrial and Applied Mathematics Philadelphia Pa USA2nd edition 2003
[6] K Chen Matrix Preconditioning Techniques and Applicationsvol 19 ofCambridgeMonographs onApplied andComputational
Mathematics Cambridge University Press Cambridge UK2005
[7] M Benzi and X-P Guo ldquoA dimensional split preconditionerfor Stokes and linearized Navier-Stokes equationsrdquo AppliedNumerical Mathematics vol 61 no 1 pp 66ndash76 2011
[8] M Benzi M Ng Q Niu and Z Wang ldquoA relaxed dimensionalfactorization preconditioner for the incompressible Navier-Stokes equationsrdquo Journal of Computational Physics vol 230no 16 pp 6185ndash6202 2011
[9] YCaoM-Q Jiang andY-L Zheng ldquoA splitting preconditionerfor saddle point problemsrdquo Numerical Linear Algebra withApplications vol 18 no 5 pp 875ndash895 2011
[10] V Simoncini and M Benzi ldquoSpectral properties of the Her-mitian and skew-Hermitian splitting preconditioner for saddlepoint problemsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 26 no 2 pp 377ndash389 2004
[11] H C Elman A Ramage and D J Silvester ldquoAlgorithm 886IFISS a Matlab toolbox for modelling incompressible flowrdquoACM Transactions on Mathematical Software vol 33 no 2article 14 2007
Grid 120572 its LU time its time Total time16 times 16 00001 4 00457 00267 0072432 times 32 00001 6 03912 00813 0472564 times 64 00001 9 54472 05519 59991128 times 128 00001 14 914698 57732 972430
Table 2 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 00001 3 00479 00244 0072332 times 32 00001 3 06312 00696 0700964 times 64 00001 4 132959 03811 136770128 times 128 00001 6 1305463 37727 1343190
Table 3 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00442 00236 0067832 times 32 00001 3 04160 00557 0471764 times 64 00001 3 73645 02623 76268128 times 128 00001 4 1694009 31709 1725718
Table 4 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 00001 2 00481 00206 0068732 times 32 00001 3 04661 00585 0524664 times 64 00001 3 66240 02728 68969128 times 128 00001 4 1773130 29814 1802944
Grid 120572 its LU time its time Total time16 times 16 10000 5 00471 00272 0074332 times 32 10000 7 03914 00906 0482064 times 64 10000 9 56107 04145 60252128 times 128 10000 14 927154 48908 976062
Table 6 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 10000 4 00458 00223 0068132 times 32 10000 4 05748 00670 0641864 times 64 10000 5 122642 04179 126821128 times 128 10000 7 1282758 16275 1299033
Table 7 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00458 00224 0068232 times 32 10000 3 04309 00451 0476064 times 64 10000 4 76537 01712 78249128 times 128 10000 5 1751587 34554 1786141
Table 8 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 10000 3 00507 00212 0071932 times 32 10000 3 04735 00449 0518464 times 64 10000 4 66482 01645 68127128 times 128 10000 4 1720516 21216 1741732
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro) 21 00240 00473 0071332 times 32 norm(119862 fro) 20 00890 01497 0238764 times 64 norm(119862 fro) 20 08387 06191 14578128 times 128 norm(119862 fro) 20 68917 30866 99783
Table 10 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 01
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 10 00250 00302 0055232 times 32 norm(119862 fro)20 10 00816 00832 0164864 times 64 norm(119862 fro)20 12 08466 03648 12114128 times 128 norm(119862 fro)20 14 69019 20398 89417
Table 11 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 7 00238 00286 0052432 times 32 norm(119862 fro)20 7 00850 00552 0140264 times 64 norm(119862 fro)20 11 08400 03177 11577128 times 128 norm(119862 fro)20 16 69537 22306 91844
Table 12 Preconditioned GMRES(20) on steady Oseen problemswith different grid sizes (uniform grids) viscosity V = 0001
Grid 120572 its LU time its time Total time16 times 16 norm(119862 fro)20 5 00245 00250 0049532 times 32 norm(119862 fro)20 5 00905 00587 0149264 times 64 norm(119862 fro)20 8 12916 03200 16116128 times 128 norm(119862 fro)20 15 104399 27468 131867
Journal of Applied Mathematics 5
Table 13 Size and number of non-nil elements of the coefficient matrix generalized by using Q2-Q1 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 6178 81 times 578 2318 3690432 times 32 2178 times 2178 28418 289 times 2178 10460 19322064 times 64 8450 times 8450 122206 1089 times 8450 44314 875966128 times 128 33282 times 33282 506376 4225 times 33282 184316 3741110
we should efficiently deal with the computation of (119868 +(119861119860minus1119861119879120572))minus1 This can been tackled by the well-known
Sherman-Morrison-Woodbury formula
(1198601+ 11988311198602119883119879
2)
minus1
= 119860minus1
1minus 119860minus1
11198831(119860minus1
2+ 119883119879
2119860minus1
11198831)
minus1
119883119879
2119860minus1
1
(15)
where 1198601isin R119899times119899 and 119860
2isin R1199031times1199031 are invertible matrices
1198831isin R119899times1199031 and 119883
2isin R119899times1199031 are any matrices and 119899 119903
1are
any positive integersFrom (15) we immediately get
(119868 +
119861119860minus1119861119879
120572
)
minus1
= 119868 minus 119861(120572119860 + 119861119879119861)
minus1
119861119879 (16)
In the following numerical examples we will always use (16)to compute (119868 + (119861119860minus1119861119879120572))minus1 in (14)
3 Numerical Examples
In this sectionwe give numerical experiments to illustrate thebehavior of our preconditioner The numerical experimentsare done by using MATLAB 71 The linear systems areobtained by using finite element methods in the Stokes prob-lems and steady Oseen problems and they are respectivelythe cases of
(1) 119862 = 0 which is caused by using Q2-Q1 FEM(2) 119862 = 0 which is caused by using Q1-P0 FEM
Furthermore we compare our preconditionerwith that of[9] in the case of general saddle point problems (namely 119862 =0) For the general saddle point problem [9] has presentedthe preconditioner
= (
119860 + 119905119861119879119861 0
minus2119861
119868
119905
)
minus1
(17)
with 119905 as a parameter and has proved that when 119860 issymmetric positive definite the preconditioned matrix hasan eigenvalue 1 with multiplicity at 119899 and the remainingeigenvalues satisfy
120582 =
1199051205902
119894
1 + 1199051205902
119894
(18)
lim119905rarrinfin
120582 = 1 (19)
where 120590119894 119894 = 1 2 119898 are119898 positive singular values of the
matrix 119861119860minus12All these systems can be generalized by
using IFISS software package [11] (this is a freepackage that can be downloaded from the sitehttpwwwmathsmanchesteracuksimdjsifiss) We userestarted GMRES(20) as the Krylov subspace method andwe always take a zero initial guess The stopping criterion is
1003817100381710038171003817119903119896
10038171003817100381710038172
10038171003817100381710038171199030
10038171003817100381710038172
le 10minus6 (20)
where 119903119896is the residual vector at the 119896th iteration
In the whole course of computation we always replace(119868 + (119861119860
minus1119861119879120572))
minus1 in (14) with (16) and use the 119871119880 factor-ization of 119860 + (119861
119879119861120572) to tackle (119860 + (119861
119879119861120572))
minus1V whereV is a corresponding vector in the iteration Concretely let119860 + (119861
119879119861120572) = 119871119880 then we complete the matrix-vector
product (119860 + (119861119879119861120572))
minus1V by 119880 119871 V in MATLAB termIn the following tables the denotation norm (119862 fro) meansthe Frobenius form of the matrix 119862 The total time is the sumof LU time and iterative time and the LU time is the time tocompute LU factorization of 119860 + (119861119879119861120572)
Case 1 (for our preconditioner) 119862 = 0 (using Q2-Q1 FEM inStokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 1 2 3 4 )
Case 1rsquo (for preconditioner of [9]) 119862 = 0 (using Q2-Q1 FEMin Stokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 5 6 7 8)
Case 2 119862 = 0 (using Q1-P0 FEM in Stokes problems andsteady Oseen problems with different viscosity coefficientsThe results are in Tables 9 10 11 12)
From Tables 1 2 3 4 5 6 7 and 8 we can see that theseresults are in agreement with the theoretical analyses (13) and(19) respectively Additionally comparing with the results inTables 9 10 11 and 12 we find that although the iterationsused in Case 1 (either for the preconditioner of [9] or ourpreconditioner) are less than those in Case 2 the time spentby Case 1 ismuchmore than that of Case 2This is because thedensity of the coefficient matrix generalized by Q2-Q1 FEMis much larger than that generalized by Q1-P0 FEMThis canbe partly illustrated by Tables 13 and 14 and the others can beillustrated similarly
6 Journal of Applied Mathematics
Table 14 Size and number of non-nil elements of the coefficient matrix generalized by using Q1-P0 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 3826 256 times 578 1800 707632 times 32 2178 times 2178 16818 1024 times 2178 7688 3187464 times 64 8450 times 8450 70450 4096 times 8450 31752 136582128 times 128 33282 times 33282 288306 16384 times 33282 129032 567192
4 Conclusions
In this paper we have introduced a splitting preconditionerfor solving generalized saddle point systems Theoreticalanalysis showed the modulus of eigenvalues of the precon-ditioned matrix would be located in interval (0 1) when theparameter is big enough Particularly when the submatrix119862 = 0 the eigenvalues will tend to 1 as the parameter 120572 rarr 0These performances are tested by some examples and theresults are in agreement with the theoretical analysis
There are still some future works to be done how to prop-erly choose a parameter 120572 so that the preconditioned matrixhas better propertiesHow to further precondition submatrix(119868 + (119861119860
minus1119861119879120572))
minus1
((119861119860minus1119861119879120572) + (119862120572)) to improve our
preconditioner
Acknowledgments
The authors would express their great thankfulness to thereferees and the editor Professor P N Shivakumar for theirhelpful suggestions for revising this paperThe authors wouldlike to thank H C Elman A Ramage and D J Silvester fortheir free IFISS software package This research is supportedby Chinese Universities Specialized Research Fund for theDoctoral Program (20110185110020) Sichuan Province Sci ampTech Research Project (2012GZX0080) and the Fundamen-tal Research Funds for the Central Universities
References
[1] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991
[2] Z Chen Q Du and J Zou ldquoFinite element methods withmatching and nonmatching meshes for Maxwell equationswith discontinuous coefficientsrdquo SIAM Journal on NumericalAnalysis vol 37 no 5 pp 1542ndash1570 2000
[3] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Numerical Mathematics and Scientific Com-putation Oxford University Press New York NY USA 2005
[4] C Cuvelier A Segal and A A van Steenhoven Finite ElementMethods and Navier-Stokes Equations vol 22 of Mathematicsand its Applications D Reidel Dordrecht The Netherlands1986
[5] Y Saad Iterative Methods for Sparse Linear Systems Society forIndustrial and Applied Mathematics Philadelphia Pa USA2nd edition 2003
[6] K Chen Matrix Preconditioning Techniques and Applicationsvol 19 ofCambridgeMonographs onApplied andComputational
Mathematics Cambridge University Press Cambridge UK2005
[7] M Benzi and X-P Guo ldquoA dimensional split preconditionerfor Stokes and linearized Navier-Stokes equationsrdquo AppliedNumerical Mathematics vol 61 no 1 pp 66ndash76 2011
[8] M Benzi M Ng Q Niu and Z Wang ldquoA relaxed dimensionalfactorization preconditioner for the incompressible Navier-Stokes equationsrdquo Journal of Computational Physics vol 230no 16 pp 6185ndash6202 2011
[9] YCaoM-Q Jiang andY-L Zheng ldquoA splitting preconditionerfor saddle point problemsrdquo Numerical Linear Algebra withApplications vol 18 no 5 pp 875ndash895 2011
[10] V Simoncini and M Benzi ldquoSpectral properties of the Her-mitian and skew-Hermitian splitting preconditioner for saddlepoint problemsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 26 no 2 pp 377ndash389 2004
[11] H C Elman A Ramage and D J Silvester ldquoAlgorithm 886IFISS a Matlab toolbox for modelling incompressible flowrdquoACM Transactions on Mathematical Software vol 33 no 2article 14 2007
Table 13 Size and number of non-nil elements of the coefficient matrix generalized by using Q2-Q1 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 6178 81 times 578 2318 3690432 times 32 2178 times 2178 28418 289 times 2178 10460 19322064 times 64 8450 times 8450 122206 1089 times 8450 44314 875966128 times 128 33282 times 33282 506376 4225 times 33282 184316 3741110
we should efficiently deal with the computation of (119868 +(119861119860minus1119861119879120572))minus1 This can been tackled by the well-known
Sherman-Morrison-Woodbury formula
(1198601+ 11988311198602119883119879
2)
minus1
= 119860minus1
1minus 119860minus1
11198831(119860minus1
2+ 119883119879
2119860minus1
11198831)
minus1
119883119879
2119860minus1
1
(15)
where 1198601isin R119899times119899 and 119860
2isin R1199031times1199031 are invertible matrices
1198831isin R119899times1199031 and 119883
2isin R119899times1199031 are any matrices and 119899 119903
1are
any positive integersFrom (15) we immediately get
(119868 +
119861119860minus1119861119879
120572
)
minus1
= 119868 minus 119861(120572119860 + 119861119879119861)
minus1
119861119879 (16)
In the following numerical examples we will always use (16)to compute (119868 + (119861119860minus1119861119879120572))minus1 in (14)
3 Numerical Examples
In this sectionwe give numerical experiments to illustrate thebehavior of our preconditioner The numerical experimentsare done by using MATLAB 71 The linear systems areobtained by using finite element methods in the Stokes prob-lems and steady Oseen problems and they are respectivelythe cases of
(1) 119862 = 0 which is caused by using Q2-Q1 FEM(2) 119862 = 0 which is caused by using Q1-P0 FEM
Furthermore we compare our preconditionerwith that of[9] in the case of general saddle point problems (namely 119862 =0) For the general saddle point problem [9] has presentedthe preconditioner
= (
119860 + 119905119861119879119861 0
minus2119861
119868
119905
)
minus1
(17)
with 119905 as a parameter and has proved that when 119860 issymmetric positive definite the preconditioned matrix hasan eigenvalue 1 with multiplicity at 119899 and the remainingeigenvalues satisfy
120582 =
1199051205902
119894
1 + 1199051205902
119894
(18)
lim119905rarrinfin
120582 = 1 (19)
where 120590119894 119894 = 1 2 119898 are119898 positive singular values of the
matrix 119861119860minus12All these systems can be generalized by
using IFISS software package [11] (this is a freepackage that can be downloaded from the sitehttpwwwmathsmanchesteracuksimdjsifiss) We userestarted GMRES(20) as the Krylov subspace method andwe always take a zero initial guess The stopping criterion is
1003817100381710038171003817119903119896
10038171003817100381710038172
10038171003817100381710038171199030
10038171003817100381710038172
le 10minus6 (20)
where 119903119896is the residual vector at the 119896th iteration
In the whole course of computation we always replace(119868 + (119861119860
minus1119861119879120572))
minus1 in (14) with (16) and use the 119871119880 factor-ization of 119860 + (119861
119879119861120572) to tackle (119860 + (119861
119879119861120572))
minus1V whereV is a corresponding vector in the iteration Concretely let119860 + (119861
119879119861120572) = 119871119880 then we complete the matrix-vector
product (119860 + (119861119879119861120572))
minus1V by 119880 119871 V in MATLAB termIn the following tables the denotation norm (119862 fro) meansthe Frobenius form of the matrix 119862 The total time is the sumof LU time and iterative time and the LU time is the time tocompute LU factorization of 119860 + (119861119879119861120572)
Case 1 (for our preconditioner) 119862 = 0 (using Q2-Q1 FEM inStokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 1 2 3 4 )
Case 1rsquo (for preconditioner of [9]) 119862 = 0 (using Q2-Q1 FEMin Stokes problems and steady Oseen problems with differentviscosity coefficients The results are in Tables 5 6 7 8)
Case 2 119862 = 0 (using Q1-P0 FEM in Stokes problems andsteady Oseen problems with different viscosity coefficientsThe results are in Tables 9 10 11 12)
From Tables 1 2 3 4 5 6 7 and 8 we can see that theseresults are in agreement with the theoretical analyses (13) and(19) respectively Additionally comparing with the results inTables 9 10 11 and 12 we find that although the iterationsused in Case 1 (either for the preconditioner of [9] or ourpreconditioner) are less than those in Case 2 the time spentby Case 1 ismuchmore than that of Case 2This is because thedensity of the coefficient matrix generalized by Q2-Q1 FEMis much larger than that generalized by Q1-P0 FEMThis canbe partly illustrated by Tables 13 and 14 and the others can beillustrated similarly
6 Journal of Applied Mathematics
Table 14 Size and number of non-nil elements of the coefficient matrix generalized by using Q1-P0 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 3826 256 times 578 1800 707632 times 32 2178 times 2178 16818 1024 times 2178 7688 3187464 times 64 8450 times 8450 70450 4096 times 8450 31752 136582128 times 128 33282 times 33282 288306 16384 times 33282 129032 567192
4 Conclusions
In this paper we have introduced a splitting preconditionerfor solving generalized saddle point systems Theoreticalanalysis showed the modulus of eigenvalues of the precon-ditioned matrix would be located in interval (0 1) when theparameter is big enough Particularly when the submatrix119862 = 0 the eigenvalues will tend to 1 as the parameter 120572 rarr 0These performances are tested by some examples and theresults are in agreement with the theoretical analysis
There are still some future works to be done how to prop-erly choose a parameter 120572 so that the preconditioned matrixhas better propertiesHow to further precondition submatrix(119868 + (119861119860
minus1119861119879120572))
minus1
((119861119860minus1119861119879120572) + (119862120572)) to improve our
preconditioner
Acknowledgments
The authors would express their great thankfulness to thereferees and the editor Professor P N Shivakumar for theirhelpful suggestions for revising this paperThe authors wouldlike to thank H C Elman A Ramage and D J Silvester fortheir free IFISS software package This research is supportedby Chinese Universities Specialized Research Fund for theDoctoral Program (20110185110020) Sichuan Province Sci ampTech Research Project (2012GZX0080) and the Fundamen-tal Research Funds for the Central Universities
References
[1] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991
[2] Z Chen Q Du and J Zou ldquoFinite element methods withmatching and nonmatching meshes for Maxwell equationswith discontinuous coefficientsrdquo SIAM Journal on NumericalAnalysis vol 37 no 5 pp 1542ndash1570 2000
[3] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Numerical Mathematics and Scientific Com-putation Oxford University Press New York NY USA 2005
[4] C Cuvelier A Segal and A A van Steenhoven Finite ElementMethods and Navier-Stokes Equations vol 22 of Mathematicsand its Applications D Reidel Dordrecht The Netherlands1986
[5] Y Saad Iterative Methods for Sparse Linear Systems Society forIndustrial and Applied Mathematics Philadelphia Pa USA2nd edition 2003
[6] K Chen Matrix Preconditioning Techniques and Applicationsvol 19 ofCambridgeMonographs onApplied andComputational
Mathematics Cambridge University Press Cambridge UK2005
[7] M Benzi and X-P Guo ldquoA dimensional split preconditionerfor Stokes and linearized Navier-Stokes equationsrdquo AppliedNumerical Mathematics vol 61 no 1 pp 66ndash76 2011
[8] M Benzi M Ng Q Niu and Z Wang ldquoA relaxed dimensionalfactorization preconditioner for the incompressible Navier-Stokes equationsrdquo Journal of Computational Physics vol 230no 16 pp 6185ndash6202 2011
[9] YCaoM-Q Jiang andY-L Zheng ldquoA splitting preconditionerfor saddle point problemsrdquo Numerical Linear Algebra withApplications vol 18 no 5 pp 875ndash895 2011
[10] V Simoncini and M Benzi ldquoSpectral properties of the Her-mitian and skew-Hermitian splitting preconditioner for saddlepoint problemsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 26 no 2 pp 377ndash389 2004
[11] H C Elman A Ramage and D J Silvester ldquoAlgorithm 886IFISS a Matlab toolbox for modelling incompressible flowrdquoACM Transactions on Mathematical Software vol 33 no 2article 14 2007
Table 14 Size and number of non-nil elements of the coefficient matrix generalized by using Q1-P0 FEM in steady Stokes problems
Grid dim(119860) nnz(119860) dim(119861) nnz(119861) nnz(119860 + (1120572)119861119879119861)16 times 16 578 times 578 3826 256 times 578 1800 707632 times 32 2178 times 2178 16818 1024 times 2178 7688 3187464 times 64 8450 times 8450 70450 4096 times 8450 31752 136582128 times 128 33282 times 33282 288306 16384 times 33282 129032 567192
4 Conclusions
In this paper we have introduced a splitting preconditionerfor solving generalized saddle point systems Theoreticalanalysis showed the modulus of eigenvalues of the precon-ditioned matrix would be located in interval (0 1) when theparameter is big enough Particularly when the submatrix119862 = 0 the eigenvalues will tend to 1 as the parameter 120572 rarr 0These performances are tested by some examples and theresults are in agreement with the theoretical analysis
There are still some future works to be done how to prop-erly choose a parameter 120572 so that the preconditioned matrixhas better propertiesHow to further precondition submatrix(119868 + (119861119860
minus1119861119879120572))
minus1
((119861119860minus1119861119879120572) + (119862120572)) to improve our
preconditioner
Acknowledgments
The authors would express their great thankfulness to thereferees and the editor Professor P N Shivakumar for theirhelpful suggestions for revising this paperThe authors wouldlike to thank H C Elman A Ramage and D J Silvester fortheir free IFISS software package This research is supportedby Chinese Universities Specialized Research Fund for theDoctoral Program (20110185110020) Sichuan Province Sci ampTech Research Project (2012GZX0080) and the Fundamen-tal Research Funds for the Central Universities
References
[1] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991
[2] Z Chen Q Du and J Zou ldquoFinite element methods withmatching and nonmatching meshes for Maxwell equationswith discontinuous coefficientsrdquo SIAM Journal on NumericalAnalysis vol 37 no 5 pp 1542ndash1570 2000
[3] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Numerical Mathematics and Scientific Com-putation Oxford University Press New York NY USA 2005
[4] C Cuvelier A Segal and A A van Steenhoven Finite ElementMethods and Navier-Stokes Equations vol 22 of Mathematicsand its Applications D Reidel Dordrecht The Netherlands1986
[5] Y Saad Iterative Methods for Sparse Linear Systems Society forIndustrial and Applied Mathematics Philadelphia Pa USA2nd edition 2003
[6] K Chen Matrix Preconditioning Techniques and Applicationsvol 19 ofCambridgeMonographs onApplied andComputational
Mathematics Cambridge University Press Cambridge UK2005
[7] M Benzi and X-P Guo ldquoA dimensional split preconditionerfor Stokes and linearized Navier-Stokes equationsrdquo AppliedNumerical Mathematics vol 61 no 1 pp 66ndash76 2011
[8] M Benzi M Ng Q Niu and Z Wang ldquoA relaxed dimensionalfactorization preconditioner for the incompressible Navier-Stokes equationsrdquo Journal of Computational Physics vol 230no 16 pp 6185ndash6202 2011
[9] YCaoM-Q Jiang andY-L Zheng ldquoA splitting preconditionerfor saddle point problemsrdquo Numerical Linear Algebra withApplications vol 18 no 5 pp 875ndash895 2011
[10] V Simoncini and M Benzi ldquoSpectral properties of the Her-mitian and skew-Hermitian splitting preconditioner for saddlepoint problemsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 26 no 2 pp 377ndash389 2004
[11] H C Elman A Ramage and D J Silvester ldquoAlgorithm 886IFISS a Matlab toolbox for modelling incompressible flowrdquoACM Transactions on Mathematical Software vol 33 no 2article 14 2007
![Augmented Lagrangian Preconditioner for Linear Stability ...+days/2017/pdf/C-JM.pdf · Introduction How to precondition this ? o SIMPLE [Patankar 1980] o Stokes Preconditioner [Tuckerman,](https://static.documents.pub/doc/80x56/5fbfc5c476c329002220b1f7/augmented-lagrangian-preconditioner-for-linear-stability-days2017pdfc-jmpdf.jpg)
