Research Article Smoothing Analysis of Distributive...
Transcript of Research Article Smoothing Analysis of Distributive...
Research ArticleSmoothing Analysis of Distributive Red-Black Jacobi Relaxationfor Solving 2D Stokes Flow by Multigrid Method
Xingwen Zhu12 and Lixiang Zhang1
1Department of Engineering Mechanics Kunming University of Science and TechnologyKunming Yunnan 650500 China2School of Mathematics and Computer Dali University Dali Yunnan 671003 China
Correspondence should be addressed to Lixiang Zhang zlxzcc126com
Received 15 September 2014 Revised 7 March 2015 Accepted 8 March 2015
Academic Editor Vassilios C Loukopoulos
Copyright copy 2015 X Zhu and L ZhangThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Smoothing analysis process of distributive red-black Jacobi relaxation in multigrid method for solving 2D Stokes flow is mainlyinvestigated on the nonstaggered grid by using local Fourier analysis (LFA) For multigrid relaxation the nonstaggered discretizingscheme of Stokes flow is generally stabilized by adding an artificial pressure term Therefore an important problem is how todetermine the zone of parameter in adding artificial pressure term in order to make stabilization of the algorithm for multigridrelaxation To end that a distributive red-black Jacobi relaxation technique for the 2D Stokes flow is established According to the2h-harmonics invariant subspaces in LFA the Fourier representation of the distributive red-black Jacobi relaxation for discretizingStokes flow is given by the form of square matrix whose eigenvalues are meanwhile analytically computed Based on optimal one-stage relaxation a mathematical relation of the parameter in artificial pressure term between the optimal relaxation parameterand related smoothing factor is well yielded The analysis results show that the numerical schemes for solving 2D Stokes flow bymultigridmethod on the distributive red-black Jacobi relaxation have a specified convergence parameter zone of the added artificialpressure term
1 Introduction
Multigrid methods [1ndash7] are generally considered as oneof the fastest numerical methods which have an optimallycomputational complexity for solving partial differentialequations (PDEs) especially for 3D steady incompressibleNewtonian flow governed by Navier-Stokes equations
In multigrid methods smoothing relaxations play animportant role Several multigrid relaxation methods weredeveloped for solving PDEs which are roughly classifiedinto two categories collective and decoupled relaxations [8]The collective relaxations are considered as a straightforwardgeneralization of the scalar case [2] The early decoupledrelaxation is on a distributive Gauss-Seidel relaxation [9]Gradually it is generalized to an incomplete LU factor-ization relaxation [10] Recently Stokes system with dis-tributive Gauss-Seidel relaxation based on the least squares
commutator has been researched [11]Much of the relaxationsfor Stokes system is seen in [12 13]
Formultigridmethods LFA is a very useful tool to designefficient algorithms and to predict convergence factors forsolving PDEs with high order accuracy [1ndash7] Distributiverelaxation for poroelasticity equations is optimized by LFA[14] Using LFA textbook efficiency multigrid solver forcompressible Navier-Stokes equations is designed [15] All-at-once multigrid approach for optimality systems with LFAis discussed in detail and an analytical expression of theconvergence factors is given by using symbolic computation[16ndash18]
The smoothing analysis of the distributive relaxationsfor solving 2D Stokes flow is investigated with LFA As weknow the discretizing Stokes flow in computational domainis not stable by means of standard central differencing onnonstaggered grid Thus in order to overcome the stability
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 572198 7 pageshttpdxdoiorg1011552015572198
2 Mathematical Problems in Engineering
problem an artificial pressure term is generally added bythe method in [1 2] The optimal one-stage relaxationparameter and related smoothing factor of the distributiverelaxation with the red-black Jacobi point relaxation needto be developed In deriving an explicit formulation of thesmoothing factor for the multigrid method the symbolicoperation process is carried out by using the MATLAB andMathematica software especially by the cylindrical algebraicdecomposition (CAD) function in the Mathematica build-incommand [19]
2 Discretizing Stokes Flow and LFA
21 Discrete Stokes Flow 3D steady incompressible Newto-nian flow governed by Navier-Stokes equations is given as
minusΔ
997888
119906 +
997888
nabla119901 =
997888
119891 (119909 119910 119911) isin Ω
997888
nabla sdot
997888
119906 = 0 (119909 119910 119911) isin Ω
997888
119906 =
997888
119892 (119909 119910 119911) isin 120597Ω
(1)
where 997888119906 = (119906(119909 119910 119911) V(119909 119910 119911) 119908(119909 119910 119911)) is the veloc-ity field 119901 = 119901(119909 119910 119911) represents the pressure
997888
119891 =
(119891
1(119909 119910 119911) 119891
2(119909 119910 119911) 119891
3(119909 119910 119911)) is the external force field
(119909 119910 119911) isin Ω sube R3 and 120597Ω is the Dirichlet boundary of thecomputing domain From (1) 2D Stokes operator is writtenas
119871 = (
minusΔ 0 120597
119909
0 minusΔ 120597
119910
120597
119909120597
1199100
) (2)
on nonstaggered grid
119866
ℎ=
997888
119909 = (119909 119910) = (119896
1ℎ 119896
2ℎ) | (119896
1 119896
2) isin Z2 (3)
Discretizing Stokes operator (2) by means of standard centraldifferencing is given as
119871
1015840
ℎ= (
minusΔ
ℎ0 120597
ℎ
119909
0 minusΔ
ℎ120597
ℎ
119910
120597
ℎ
119909120597
ℎ
1199100
) (4)
where ℎ denotes the uniform mesh size and minusΔℎ 120597ℎ119909 and 120597ℎ
119910
are the second-order difference operator with the followingdiscrete stencils
minusΔ
ℎ=
1
ℎ
2
[
[
[
minus1
minus1 4 minus1
minus1
]
]
]
ℎ
120597
ℎ
119909=
1
2ℎ
[minus1 0 1
]
ℎ 120597
ℎ
119910=
1
2ℎ
[
[
[
1
0
minus1
]
]
]
ℎ
(5)
From [1] the above nonstaggered schemes (4) are not stableStabilization may be achieved by adding an artificial ellipticpressure term minus119888ℎ
2Δ
ℎto the continuity equation in (2) [1 2
6] With discrete operator in (5) and parameter 119888 the discreteStokes operator is changed to
119871
ℎ= (
minusΔ
ℎ0 120597
ℎ
119909
0 minusΔ
ℎ120597
ℎ
119910
120597
ℎ
119909120597
ℎ
119910minus119888ℎ
2Δ
ℎ
) (6)
22 Elements of LFA in Multigrid In LFA a current approx-imation and its corresponding error and residual are rep-resented by a linear combination of certain exponentialfunctions for example Fourier modes which form a unitarybasis in space on a bounded infinite grid functions [1ndash7]
From [1 2] on nonstaggered grid (3) a unitary basis ofthe Fourier modes is defined by
120593
ℎ(
997888
120579
997888
119909) = exp(119894997888
120579 sdot
997888
119909
ℎ
)
(7)
in which997888
120579 = (120579
1 120579
2) isin Θ = (minus120587 120587]
2 is called Fourierfrequency 997888119909 isin 119866
ℎ and complex unit 119894 = radic
minus1 Thus aFourier space is yielded as
119865 (
997888
120579) = span 120593ℎ(
997888
120579
997888
119909) |
997888
120579 isin Θ (8)
From [1ndash7] applying (3) and (7) for 2D scalar discreteoperator119863
ℎwith discrete stencil
119863
ℎ= [119897997888119896]
ℎ (9)
where 119897997888119896isin R and
997888
119896 isin 119869 sub Z2 containing (0 0) the Fouriermode of (9) is defined by
119863
ℎ(
997888
120579) = sum
997888119896isin119869
119897997888119896exp (119894
997888
120579 sdot
997888
119896) (10)
with997888
120579 sdot
997888
119896 = 120579
1119896
1+ 120579
2119896
2 subjected to
119863
ℎ120593
ℎ(
997888
120579
997888
119909) =
119863
ℎ(
997888
120579)120593
ℎ(
997888
120579
997888
119909) (11)
A main idea of LFA is to analyze relaxation properties inmultigrid for (6) by evaluating their effects on the Fouriercomponents From [2 14 16] if standard coarsening in 2D is
selected each low frequency997888
120579 =
997888
120579
00
isin Θ
2ℎ
low = (minus1205872 1205872]2
is coupled with three high frequencies 997888
120579
11
997888
120579
10
997888
120579
01
isin
Θ
2ℎ
high in the transition from119866
ℎto1198662ℎ whereΘ2ℎhigh = ΘΘ
2ℎ
lowand
997888
120579
997888120572
=
997888
120579 minus (120572
1sign (120579
1) 120572
2sign (120579
2)) 120587
(12)
where 997888120572 isin Λ = 00 11 10 01 and 997888120572 = (1205721 120572
2) are denoted
by (1205721 120572
2) = 120572
1120572
2 In this paper standard coarsening is to
Mathematical Problems in Engineering 3
be assumed Then the Fourier space (8) is subdivided into2ℎ-harmonics subspaces
119865
2ℎ(
997888
120579) = span120593ℎ(
997888
120579
00
997888
119909) 120593
ℎ(
997888
120579
11
997888
119909)
120593
ℎ(
997888
120579
10
997888
119909) 120593
ℎ(
997888
120579
01
997888
119909)
(13)
3 Smoothing Process
31 Distributive Relaxation of System (6) From [1 2 7] adistributive operator for the discrete system (6) is constructedas
119862
ℎ= (
119868
ℎ0 minus120597
ℎ
119909
0 119868
ℎminus120597
ℎ
119910
0 0 minusΔ
ℎ
) (14)
where 119868ℎis the unit operator with discrete stencil [1]
ℎ From
(14) the discrete system (6) is transformed as
119871
ℎ119862
ℎ= (
minusΔ
ℎ0 0
0 minusΔ
ℎ0
120597
ℎ
119909120597
ℎ
119910119888ℎ
2Δ
2
ℎminus Δ
2ℎ
) (15)
where the discrete stencils of Δ2ℎand minusΔ
2ℎare
Δ
2
ℎ=
1
ℎ
4
[
[
[
[
[
[
[
[
[
1
2 minus8 2
1 minus8 20 minus8 1
2 minus8 2
1
]
]
]
]
]
]
]
]
]ℎ
minusΔ
2ℎ=
1
4ℎ
2
[
[
[
[
[
[
[
[
[
0 0 minus1 0 0
0 0 0 0 0
minus1 0 4 0 minus1
0 0 0 0 0
0 0 minus1 0 0
]
]
]
]
]
]
]
]
]ℎ
=
1
4ℎ
2
[
[
[
minus1
minus1 4 minus1
minus1
]
]
]
2ℎ
(16)
From (9)ndash(11) the Fourier modes of the scalar discreteoperators of (16) are
Δ
2
ℎ(
997888
120579) = (minus
Δ
ℎ(
997888
120579))
2
minus
Δ
2ℎ(
997888
120579) = minus [
120597
ℎ
119909(
997888
120579)]
2
minus [
120597
ℎ
119910(
997888
120579)]
2
(17)
where
minus
Δ
ℎ(
997888
120579) =
1
ℎ
2(4 minus exp (minus119894120579
1) minus exp (119894120579
1)
minus exp (minus1198941205792) minus exp (119894120579
2))
=
1
ℎ
2(4 minus 2 cos 120579
1minus 2 cos 120579
2)
(18)
120597
ℎ
119909(
997888
120579) =
1
2ℎ
(exp (1198941205791) minus exp (minus119894120579
1)) =
1
ℎ
119894 sin 1205791
(19)
120597
ℎ
119910(
997888
120579) =
1
2ℎ
(exp (1198941205792) minus exp (minus119894120579
2)) =
1
ℎ
119894 sin 1205792
(20)
32 Optimal One-Stage Relaxation For the discrete scalaroperator of (15) standard coarsening and an ideal coarse gridcorrection operator [2] are applied as
119876
2ℎ
ℎ
1003816
1003816
1003816
1003816
10038161198652ℎ(997888120579 )=
_119876
2ℎ
ℎ= diag (0 1 1 1) isin C
4times4
(21)
where_119876
2ℎ
ℎis the Fourier representation of the operator
119876
2ℎ
ℎwith subspace (13) which suppresses the low frequency
error components andmakes the high frequency componentsunchangedThen from [2] the smoothing factor for discreteoperator (9) is defined by
120588 (119899119863
ℎ) = sup997888120579isinΘlow
(120588(
_119876
2ℎ
ℎ(
_119878 ℎ(
997888
120579 120596))
119899
))
1119899
(22)
It implies that the asymptotic error reduction of the highfrequency error components is given by n sweeps of the relax-ation method where
_119878 ℎ(
997888
120579 120596) is the Fourier representationof the relaxation operator 119878
ℎ(120596) on subspace (13) and 120596 is the
relaxation parameterFrom [2 14] a good smoothing factor is obtained by using
one-stage parameter 120596 in the relaxation operator 119878ℎ(120596) the
optimal smoothing factor and related smoothing parameterare given by
120596opt =2
2 minus 119878max minus 119878min
120588opt =119878max minus 119878min
2 minus 119878max minus 119878min
(23)
where 119878max and 119878min are the max and min eigenvalues of the
product matrix_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579 1) with the relaxation parameter120596 = 1 for 120579 isin Θ2ℎlow From [2 19] the smoothing factor of(6) with the distributive relaxation (14) is determined by thediagonal blocks of the transformed system (15) which is givenby
120588 (119899 119871
ℎ) = max 120588 (119899 minusΔ
ℎ) 120588 (119899 119888ℎ
2Δ
2
ℎminus Δ
2ℎ) (24)
33 Optimal Smoothing for Stokes Flow The red-black Jacobipoint relaxation 119878119877119861
ℎis applied to (15) to discuss the optimal
4 Mathematical Problems in Engineering
smoothing problems for Stokes flow From [1 2 14] the oper-ator 119878119877119861ℎ
makes the 2ℎ-harmonics subspace (13) invariant thatis
119878
119877119861
ℎ
1003816
1003816
1003816
1003816
10038161198652ℎ(997888120579 )=
_119878
119877119861
ℎ(
997888
120579) isin C4times4
(25)
where_119878
119877119861
ℎ(
997888
120579 ) is the Fourier representation of 119878119877119861ℎ(120596) with
relaxation parameter 120596 = 1 and is given as
_119878
119877119861
ℎ(
997888
120579 1)
=
_119878
119877119861
ℎ(
997888
120579)
=
_119878
119861
ℎ(
997888
120579)
_119878
119877
ℎ(
997888
120579)
=
1
2
(
119860
00+ 1 minus119860
11+ 1 0 0
minus119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 minus119860
01+ 1
0 0 minus119860
10+ 1 119860
01+ 1
)
sdot
1
2
(
119860
00+ 1 119860
11minus 1 0 0
119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 119860
01minus 1
0 0 119860
10minus 1 119860
01+ 1
)
(26)
in which
119860997888120572= 1 minus
119863
ℎ(
997888
120579
997888120572
)
119863
0
ℎ(
997888
120579
997888120572
)
(27)
denotes the Fourier mode of the point Jacobi relaxation for
the discrete operator (9) on subspace (13) and 1198630ℎ(
997888
120579
997888120572
) is theFourier mode of the discrete operator with the stencil [119897
(00)]
ℎ
in (9) For the sake of convenient discussion in the followingtwo variables are introduced as
119904
1= sin2
120579
0
1
2
= sin2 12057912
119904
2= sin2
120579
0
2
2
= sin2 12057922
(28)
Thus997888
120579 = (120579
1 120579
2) isin Θ
2ℎ
low = (minus1205872 1205872]2 is transformed to
997888
119904 = (119904
1 119904
2) isin 119878low = [0 12]
2
Theorem 1 For the Poisson operator minusΔℎ the optimal one-
stage relaxation parameter and related smoothing factor of thered-black Jacobi point relaxation are stated as
120596
119900119901119905=
16
15
120588
119900119901119905=
1
5
(29)
Proof For the red-black Jacobi point relaxation for thePoisson operator 119863
ℎ= minusΔ
ℎ substituting (12) (18) and (28)
into (26) and (27) and from (5) the product of (21) and (25)is written as
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579)
=
1
2
(
0 0 0 0
(119904
1+ 119904
2) (1 minus 119904
1minus 119904
2) (119904
1+ 119904
2) (119904
1+ 119904
2minus 1) 0 0
0 0 (119904
1minus 119904
2) (119904
1minus 119904
2+ 1) (119904
2minus 119904
1) (119904
1minus 119904
2+ 1)
0 0 (119904
1minus 119904
2) (119904
2minus 119904
1+ 1) (119904
2minus 119904
1) (119904
2minus 119904
1+ 1)
)
(30)
Thus eigenvalues of (30) are obtained as
120582
1= 0 120582
2= (119904
1minus 119904
2)
2
120582
3= 0 120582
4=
(119904
1+ 119904
2) (119904
1+ 119904
2minus 1)
2
(31)
From (31) the max andmin eigenvalues of (30) are yielded as
119878max = max(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = max(11990411199042)isin[012]
2
120582
2=
1
4
119878min = min(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = min(11990411199042)isin[012]
2
120582
4= minus
1
8
(32)
From (23) and (32) (29) is obtained Theorem 1 holds
Next 120588(119899 119888ℎ2Δ2ℎminus Δ
2ℎ) for the red-black Jacobi point
relaxation need to be computed Meanwhile the smoothingfactor of distributive relaxation (15) is given as follows
Theorem 2 For the discrete operator 119888ℎ2Δ2ℎminus Δ
2ℎwith 119888 gt
0 the optimal one-stage relaxation parameter and related
Mathematical Problems in Engineering 5
smoothing factor of the red-black Jacobi point relaxation aregiven by
120596
119900119901119905=
1 + 20119888
1 + 16119888
0 lt 119888 le
1
32
2 (1 + 20119888)
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
120588
119900119901119905=
1
1 + 16119888
0 lt 119888 le
1
32
1 + 24119888 + 1104119888
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
(33)
Proof For the discrete operator
119863
ℎ= 119888ℎ
2Δ
2
ℎminus Δ
2ℎ
(34)
from (17)ndash(20) the Fourier mode of (34) is given by
119863
ℎ(
997888
120579) =
1
ℎ
2[4119888 (2 minus cos 120579
1minus cos 120579
2)
2
+ sin21205791+ sin2120579
2]
(35)
Thus when the red-black point relaxation is applied to (34)from (16) substituting (12) (28) and (35) into (26) and (27)the product of (21) and (25) is
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579) =
1
4
diag (11987711 119877
22)
(36)
where both 11987711
and 11987722
are 2 times 2 square matrices whoseexpressions are below
119877
11= (
0 0
0 1
) sdot (
119860
119861
00+ 1 minus119860
119861
11+ 1
minus119860
119861
00minus 1 119860
119861
11+ 1
) sdot (
119860
119877
00+ 1 119860
119877
11minus 1
119860
119877
00minus 1 119860
119877
11+ 1
)
=
4
(1 + 20119888)
2
sdot (
0 0
1 + 4 [minus1 minus 36119888 + 16119888 (119904
1+ 119904
2)]
sdot [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (119904
1+ 119904
2)
2
]
64119888 (minus1 + 119904
1+ 119904
2) [
119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2]
)
119877
22= (
1 0
0 1
) sdot (
119860
119861
10+ 1 minus119860
119861
01+ 1
minus119860
119861
10+ 1 119860
119861
01+ 1
) sdot (
119860
119877
10+ 1 119860
119877
01minus 1
119860
119877
10minus 1 119860
119877
01+ 1
)
=
4
(1 + 20119888)
2
sdot
(
(
(
(
(
(
(
(
(
(
(
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
+384119888
2(119904
1minus 119904
2) + (192119888
2+ 4) (119904
2
1+ 119904
2
2)
minus (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
2
+128119888119904
2
2+ 32119888119904
1119904
2(2119904
2minus 2119904
1minus 1) (1 minus 12119888)
]
]
]
]
]
]
minus64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
+ 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
minus 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
minus384119888
2(119904
1minus 119904
2) + (4 + 192119888
2) (119904
2
1+ 119904
2
2)
+ (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
1
+128119888119904
2
1+ 32119888119904
1119904
2(2119904
1minus 2119904
2+ 1) (1 minus 12119888)
]
]
]
]
]
]
)
)
)
)
)
)
)
)
)
)
)
(37)
Thus the eigenvalues of matrix (36) are obtained as
120582
1= 0
120582
2=
64119888
(1 + 20119888)
2(minus1 + 119904
1+ 119904
2) [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2
]
(38)
120582
34=
1
(1 + 20119888)
2
[
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (4 + 80119888) (119904
1+ 119904
2)
+ (4 + 64119888 + 192119888
2) (119904
2
1+ 119904
2
2) + (32119888 minus 384119888
2) 119904
1119904
2
plusmn32119888 (119904
1minus 119904
2)radic
1 + 80119888
2+ 24119888 + (minus64119888
2+ 64119888 + 4) (119904
2
1+ 119904
2
2)
minus (80119888 + 4) (119904
1+ 119904
2) + (128119888
2+ 32119888) 119904
1119904
2
]
]
]
]
]
]
]
(39)
6 Mathematical Problems in Engineering
By using the MATLAB and Mathematica software withcylindrical algebraic decomposition function [19] for 997888119904 =
(119904
1 119904
2) isin (0 12)
2 there is no extreme value for (39) when0 lt 119888 le 132 one of extreme values of (38) is obtained as
119904
lowast
1=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
119904
lowast
2=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
(40)
Thus for 997888119904 isin 119878low = [0 12]
2 besides (40) the possibleextreme values of the eigenvalues of matrix (36) are placedon the boundary of 119878low From minus1 le 120582
119896le 1 with 119896 = 1 4
then 0 lt 119888 le 112 Noting that (40) exists with 0 lt 119888 le 132From (38)ndash(40) when 0 lt 119888 le 112 for 997888119904 isin 119878low the maxand min eigenvalues of (36) are yielded as
119878max = 12058234 (0 0) =1 + 4119888
1 + 20119888
119878min =
120582
34(
1
2
1
2
) =
4119888 minus 1
1 + 20119888
0 lt 119888 le
1
32
120582
2(0 0) = minus (
32119888
20119888 + 1
)
21
32
lt 119888 le
1
12
(41)
Substituting (41) into (23) (33) is obtainedTheorem 2 holds
From (33) 12 le 120588opt(119888ℎ2Δ
2
ℎminusΔ
2ℎ) lt 1holdswith 0 lt 119888 le
112 Therefore fromTheorems 1 and 2 when 0 lt 119888 le 112the smoothing factor of (6) with the distributive relaxation(14) is as
1
2
le 120588opt (119871ℎ)
= max 120588opt (minusΔ ℎ) 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ)
= 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ) lt 1
(42)
4 Conclusions
The smoothing analysis process of the distributive red-black Jacobi point relaxation for solving 2D Stokes flow isanalytically presented Applying (28) the Fouriermodes withthe trigonometric functions for the discrete operator andrelaxation are mapped to rational functions So it is possibleto apply the cylindrical algebraic decomposition function intheMathematica software to realize complex smoothing anal-ysis and the computation process is simplifiedThe analyticalexpressions of the smoothing factor for the distributive red-black Jacobi point relaxation are obtained which is an upperbound for the smoothing rates and is independent of themesh size with the parameter 119888 Obviously it is valuable tounderstand numerical experiments in multigrid method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors were supported by the National Natural ScienceFoundation of China (NSFC) (Grant no 51279071) and theDoctoral Foundation of Ministry of Education of China(Grant no 20135314130002)
References
[1] U Trottenberg C W Oosterlee and A Schuller MultigridAcademic Press San Diego Calif USA 2001
[2] R Wienands and W Joppich Practical Fourier Analysis forMultigrid Methods Chapman amp Hall CRC Press 2005
[3] W L Briggs V E Henson and S McCormick A MultigridTutorial Society for Industrial and Applied Mathematics 2ndedition 2000
[4] W Hackbusch Multigrid Methods and Applications SpringerBerlin Germany 1985
[5] P Wesseling An Introduction to Multigrid Methods JohnWileyamp Sons Chichester UK 1992
[6] K Stuben and U Trottenberg ldquoMultigrid methods fundamen-tal algorithms model problem analysis and applicationsrdquo inMultigridMethods W Hackbusch andU Trottenberg Eds vol960 of Lectwe Notes in Mathematics pp 1ndash176 Springer BerlinGermany 1982
[7] A Brandt and O E Livne 1984 Guide to Multigrid Develop-ment in Multigrid Methods Society for Industrial and AppliedMathematics 2011 httpwwwwisdomweizmannacilsimachiclassicspdf
[8] C W Oosterlee and F J G Lorenz ldquoMultigrid methods for thestokes systemrdquoComputing in Science and Engineering vol 8 no6 Article ID 1717313 pp 34ndash43 2006
[9] A Brandt and N Dinar Multigrid Solutions to Elliptic LlowProblems Institute for Computer Applications in Science andEngineering NASA Langley Research Center 1979
[10] G Wittum ldquoMulti-grid methods for stokes and navier-stokesequationsrdquoNumerische Mathematik vol 54 no 5 pp 543ndash5631989
[11] M Wang and L Chen ldquoMultigrid methods for the Stokesequations using distributive Gauss-Seidel relaxations based onthe least squares commutatorrdquo Journal of Scientific Computingvol 56 no 2 pp 409ndash431 2013
[12] M ur Rehman T Geenen C Vuik G Segal and S PMacLachlan ldquoOn iterative methods for the incompressibleStokes problemrdquo International Journal for Numerical Methodsin Fluids vol 65 no 10 pp 1180ndash1200 2011
[13] C Bacuta P S Vassilevski and S Zhang ldquoA new approachfor solving Stokes systems arising from a distributive relaxationmethodrdquo Numerical Methods for Partial Differential Equationsvol 27 no 4 pp 898ndash914 2011
[14] R Wienands F J Gaspar F J Lisbona and C W OosterleeldquoAn efficient multigrid solver based on distributive smoothingfor poroelasticity equationsrdquo Computing vol 73 no 2 pp 99ndash119 2004
[15] W Liao B Diskin Y Peng and L-S Luo ldquoTextbook-efficiencymultigrid solver for three-dimensional unsteady compressibleNavier-Stokes equationsrdquo Journal of Computational Physics vol227 no 15 pp 7160ndash7177 2008
[16] V Pillwein and S Takacs ldquoA local Fourier convergence analysisof a multigrid method using symbolic computationrdquo Journal ofSymbolic Computation vol 63 pp 1ndash20 2014
Mathematical Problems in Engineering 7
[17] S Takacs All-at-once multigrid methods for optimality systemsarising from optimal control problems [PhD thesis] JohannesKepler University Linz Doctoral Program ComputationalMathematics 2012
[18] V Pillwein and S Takacs ldquoSmoothing analysis of an all-at-oncemultigrid approach for optimal control problems using symboliccomputationrdquo inNumerical and Symbolic Scientific ComputingProgress and Prospects U Langer and P Paule Eds SpringerWien Austria 2011
[19] M Kauers ldquoHow to use cylindrical algebraic decompositionrdquoSeminaire Lotharingien de Combinatoire vol 65 article B65app 1ndash16 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
problem an artificial pressure term is generally added bythe method in [1 2] The optimal one-stage relaxationparameter and related smoothing factor of the distributiverelaxation with the red-black Jacobi point relaxation needto be developed In deriving an explicit formulation of thesmoothing factor for the multigrid method the symbolicoperation process is carried out by using the MATLAB andMathematica software especially by the cylindrical algebraicdecomposition (CAD) function in the Mathematica build-incommand [19]
2 Discretizing Stokes Flow and LFA
21 Discrete Stokes Flow 3D steady incompressible Newto-nian flow governed by Navier-Stokes equations is given as
minusΔ
997888
119906 +
997888
nabla119901 =
997888
119891 (119909 119910 119911) isin Ω
997888
nabla sdot
997888
119906 = 0 (119909 119910 119911) isin Ω
997888
119906 =
997888
119892 (119909 119910 119911) isin 120597Ω
(1)
where 997888119906 = (119906(119909 119910 119911) V(119909 119910 119911) 119908(119909 119910 119911)) is the veloc-ity field 119901 = 119901(119909 119910 119911) represents the pressure
997888
119891 =
(119891
1(119909 119910 119911) 119891
2(119909 119910 119911) 119891
3(119909 119910 119911)) is the external force field
(119909 119910 119911) isin Ω sube R3 and 120597Ω is the Dirichlet boundary of thecomputing domain From (1) 2D Stokes operator is writtenas
119871 = (
minusΔ 0 120597
119909
0 minusΔ 120597
119910
120597
119909120597
1199100
) (2)
on nonstaggered grid
119866
ℎ=
997888
119909 = (119909 119910) = (119896
1ℎ 119896
2ℎ) | (119896
1 119896
2) isin Z2 (3)
Discretizing Stokes operator (2) by means of standard centraldifferencing is given as
119871
1015840
ℎ= (
minusΔ
ℎ0 120597
ℎ
119909
0 minusΔ
ℎ120597
ℎ
119910
120597
ℎ
119909120597
ℎ
1199100
) (4)
where ℎ denotes the uniform mesh size and minusΔℎ 120597ℎ119909 and 120597ℎ
119910
are the second-order difference operator with the followingdiscrete stencils
minusΔ
ℎ=
1
ℎ
2
[
[
[
minus1
minus1 4 minus1
minus1
]
]
]
ℎ
120597
ℎ
119909=
1
2ℎ
[minus1 0 1
]
ℎ 120597
ℎ
119910=
1
2ℎ
[
[
[
1
0
minus1
]
]
]
ℎ
(5)
From [1] the above nonstaggered schemes (4) are not stableStabilization may be achieved by adding an artificial ellipticpressure term minus119888ℎ
2Δ
ℎto the continuity equation in (2) [1 2
6] With discrete operator in (5) and parameter 119888 the discreteStokes operator is changed to
119871
ℎ= (
minusΔ
ℎ0 120597
ℎ
119909
0 minusΔ
ℎ120597
ℎ
119910
120597
ℎ
119909120597
ℎ
119910minus119888ℎ
2Δ
ℎ
) (6)
22 Elements of LFA in Multigrid In LFA a current approx-imation and its corresponding error and residual are rep-resented by a linear combination of certain exponentialfunctions for example Fourier modes which form a unitarybasis in space on a bounded infinite grid functions [1ndash7]
From [1 2] on nonstaggered grid (3) a unitary basis ofthe Fourier modes is defined by
120593
ℎ(
997888
120579
997888
119909) = exp(119894997888
120579 sdot
997888
119909
ℎ
)
(7)
in which997888
120579 = (120579
1 120579
2) isin Θ = (minus120587 120587]
2 is called Fourierfrequency 997888119909 isin 119866
ℎ and complex unit 119894 = radic
minus1 Thus aFourier space is yielded as
119865 (
997888
120579) = span 120593ℎ(
997888
120579
997888
119909) |
997888
120579 isin Θ (8)
From [1ndash7] applying (3) and (7) for 2D scalar discreteoperator119863
ℎwith discrete stencil
119863
ℎ= [119897997888119896]
ℎ (9)
where 119897997888119896isin R and
997888
119896 isin 119869 sub Z2 containing (0 0) the Fouriermode of (9) is defined by
119863
ℎ(
997888
120579) = sum
997888119896isin119869
119897997888119896exp (119894
997888
120579 sdot
997888
119896) (10)
with997888
120579 sdot
997888
119896 = 120579
1119896
1+ 120579
2119896
2 subjected to
119863
ℎ120593
ℎ(
997888
120579
997888
119909) =
119863
ℎ(
997888
120579)120593
ℎ(
997888
120579
997888
119909) (11)
A main idea of LFA is to analyze relaxation properties inmultigrid for (6) by evaluating their effects on the Fouriercomponents From [2 14 16] if standard coarsening in 2D is
selected each low frequency997888
120579 =
997888
120579
00
isin Θ
2ℎ
low = (minus1205872 1205872]2
is coupled with three high frequencies 997888
120579
11
997888
120579
10
997888
120579
01
isin
Θ
2ℎ
high in the transition from119866
ℎto1198662ℎ whereΘ2ℎhigh = ΘΘ
2ℎ
lowand
997888
120579
997888120572
=
997888
120579 minus (120572
1sign (120579
1) 120572
2sign (120579
2)) 120587
(12)
where 997888120572 isin Λ = 00 11 10 01 and 997888120572 = (1205721 120572
2) are denoted
by (1205721 120572
2) = 120572
1120572
2 In this paper standard coarsening is to
Mathematical Problems in Engineering 3
be assumed Then the Fourier space (8) is subdivided into2ℎ-harmonics subspaces
119865
2ℎ(
997888
120579) = span120593ℎ(
997888
120579
00
997888
119909) 120593
ℎ(
997888
120579
11
997888
119909)
120593
ℎ(
997888
120579
10
997888
119909) 120593
ℎ(
997888
120579
01
997888
119909)
(13)
3 Smoothing Process
31 Distributive Relaxation of System (6) From [1 2 7] adistributive operator for the discrete system (6) is constructedas
119862
ℎ= (
119868
ℎ0 minus120597
ℎ
119909
0 119868
ℎminus120597
ℎ
119910
0 0 minusΔ
ℎ
) (14)
where 119868ℎis the unit operator with discrete stencil [1]
ℎ From
(14) the discrete system (6) is transformed as
119871
ℎ119862
ℎ= (
minusΔ
ℎ0 0
0 minusΔ
ℎ0
120597
ℎ
119909120597
ℎ
119910119888ℎ
2Δ
2
ℎminus Δ
2ℎ
) (15)
where the discrete stencils of Δ2ℎand minusΔ
2ℎare
Δ
2
ℎ=
1
ℎ
4
[
[
[
[
[
[
[
[
[
1
2 minus8 2
1 minus8 20 minus8 1
2 minus8 2
1
]
]
]
]
]
]
]
]
]ℎ
minusΔ
2ℎ=
1
4ℎ
2
[
[
[
[
[
[
[
[
[
0 0 minus1 0 0
0 0 0 0 0
minus1 0 4 0 minus1
0 0 0 0 0
0 0 minus1 0 0
]
]
]
]
]
]
]
]
]ℎ
=
1
4ℎ
2
[
[
[
minus1
minus1 4 minus1
minus1
]
]
]
2ℎ
(16)
From (9)ndash(11) the Fourier modes of the scalar discreteoperators of (16) are
Δ
2
ℎ(
997888
120579) = (minus
Δ
ℎ(
997888
120579))
2
minus
Δ
2ℎ(
997888
120579) = minus [
120597
ℎ
119909(
997888
120579)]
2
minus [
120597
ℎ
119910(
997888
120579)]
2
(17)
where
minus
Δ
ℎ(
997888
120579) =
1
ℎ
2(4 minus exp (minus119894120579
1) minus exp (119894120579
1)
minus exp (minus1198941205792) minus exp (119894120579
2))
=
1
ℎ
2(4 minus 2 cos 120579
1minus 2 cos 120579
2)
(18)
120597
ℎ
119909(
997888
120579) =
1
2ℎ
(exp (1198941205791) minus exp (minus119894120579
1)) =
1
ℎ
119894 sin 1205791
(19)
120597
ℎ
119910(
997888
120579) =
1
2ℎ
(exp (1198941205792) minus exp (minus119894120579
2)) =
1
ℎ
119894 sin 1205792
(20)
32 Optimal One-Stage Relaxation For the discrete scalaroperator of (15) standard coarsening and an ideal coarse gridcorrection operator [2] are applied as
119876
2ℎ
ℎ
1003816
1003816
1003816
1003816
10038161198652ℎ(997888120579 )=
_119876
2ℎ
ℎ= diag (0 1 1 1) isin C
4times4
(21)
where_119876
2ℎ
ℎis the Fourier representation of the operator
119876
2ℎ
ℎwith subspace (13) which suppresses the low frequency
error components andmakes the high frequency componentsunchangedThen from [2] the smoothing factor for discreteoperator (9) is defined by
120588 (119899119863
ℎ) = sup997888120579isinΘlow
(120588(
_119876
2ℎ
ℎ(
_119878 ℎ(
997888
120579 120596))
119899
))
1119899
(22)
It implies that the asymptotic error reduction of the highfrequency error components is given by n sweeps of the relax-ation method where
_119878 ℎ(
997888
120579 120596) is the Fourier representationof the relaxation operator 119878
ℎ(120596) on subspace (13) and 120596 is the
relaxation parameterFrom [2 14] a good smoothing factor is obtained by using
one-stage parameter 120596 in the relaxation operator 119878ℎ(120596) the
optimal smoothing factor and related smoothing parameterare given by
120596opt =2
2 minus 119878max minus 119878min
120588opt =119878max minus 119878min
2 minus 119878max minus 119878min
(23)
where 119878max and 119878min are the max and min eigenvalues of the
product matrix_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579 1) with the relaxation parameter120596 = 1 for 120579 isin Θ2ℎlow From [2 19] the smoothing factor of(6) with the distributive relaxation (14) is determined by thediagonal blocks of the transformed system (15) which is givenby
120588 (119899 119871
ℎ) = max 120588 (119899 minusΔ
ℎ) 120588 (119899 119888ℎ
2Δ
2
ℎminus Δ
2ℎ) (24)
33 Optimal Smoothing for Stokes Flow The red-black Jacobipoint relaxation 119878119877119861
ℎis applied to (15) to discuss the optimal
4 Mathematical Problems in Engineering
smoothing problems for Stokes flow From [1 2 14] the oper-ator 119878119877119861ℎ
makes the 2ℎ-harmonics subspace (13) invariant thatis
119878
119877119861
ℎ
1003816
1003816
1003816
1003816
10038161198652ℎ(997888120579 )=
_119878
119877119861
ℎ(
997888
120579) isin C4times4
(25)
where_119878
119877119861
ℎ(
997888
120579 ) is the Fourier representation of 119878119877119861ℎ(120596) with
relaxation parameter 120596 = 1 and is given as
_119878
119877119861
ℎ(
997888
120579 1)
=
_119878
119877119861
ℎ(
997888
120579)
=
_119878
119861
ℎ(
997888
120579)
_119878
119877
ℎ(
997888
120579)
=
1
2
(
119860
00+ 1 minus119860
11+ 1 0 0
minus119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 minus119860
01+ 1
0 0 minus119860
10+ 1 119860
01+ 1
)
sdot
1
2
(
119860
00+ 1 119860
11minus 1 0 0
119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 119860
01minus 1
0 0 119860
10minus 1 119860
01+ 1
)
(26)
in which
119860997888120572= 1 minus
119863
ℎ(
997888
120579
997888120572
)
119863
0
ℎ(
997888
120579
997888120572
)
(27)
denotes the Fourier mode of the point Jacobi relaxation for
the discrete operator (9) on subspace (13) and 1198630ℎ(
997888
120579
997888120572
) is theFourier mode of the discrete operator with the stencil [119897
(00)]
ℎ
in (9) For the sake of convenient discussion in the followingtwo variables are introduced as
119904
1= sin2
120579
0
1
2
= sin2 12057912
119904
2= sin2
120579
0
2
2
= sin2 12057922
(28)
Thus997888
120579 = (120579
1 120579
2) isin Θ
2ℎ
low = (minus1205872 1205872]2 is transformed to
997888
119904 = (119904
1 119904
2) isin 119878low = [0 12]
2
Theorem 1 For the Poisson operator minusΔℎ the optimal one-
stage relaxation parameter and related smoothing factor of thered-black Jacobi point relaxation are stated as
120596
119900119901119905=
16
15
120588
119900119901119905=
1
5
(29)
Proof For the red-black Jacobi point relaxation for thePoisson operator 119863
ℎ= minusΔ
ℎ substituting (12) (18) and (28)
into (26) and (27) and from (5) the product of (21) and (25)is written as
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579)
=
1
2
(
0 0 0 0
(119904
1+ 119904
2) (1 minus 119904
1minus 119904
2) (119904
1+ 119904
2) (119904
1+ 119904
2minus 1) 0 0
0 0 (119904
1minus 119904
2) (119904
1minus 119904
2+ 1) (119904
2minus 119904
1) (119904
1minus 119904
2+ 1)
0 0 (119904
1minus 119904
2) (119904
2minus 119904
1+ 1) (119904
2minus 119904
1) (119904
2minus 119904
1+ 1)
)
(30)
Thus eigenvalues of (30) are obtained as
120582
1= 0 120582
2= (119904
1minus 119904
2)
2
120582
3= 0 120582
4=
(119904
1+ 119904
2) (119904
1+ 119904
2minus 1)
2
(31)
From (31) the max andmin eigenvalues of (30) are yielded as
119878max = max(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = max(11990411199042)isin[012]
2
120582
2=
1
4
119878min = min(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = min(11990411199042)isin[012]
2
120582
4= minus
1
8
(32)
From (23) and (32) (29) is obtained Theorem 1 holds
Next 120588(119899 119888ℎ2Δ2ℎminus Δ
2ℎ) for the red-black Jacobi point
relaxation need to be computed Meanwhile the smoothingfactor of distributive relaxation (15) is given as follows
Theorem 2 For the discrete operator 119888ℎ2Δ2ℎminus Δ
2ℎwith 119888 gt
0 the optimal one-stage relaxation parameter and related
Mathematical Problems in Engineering 5
smoothing factor of the red-black Jacobi point relaxation aregiven by
120596
119900119901119905=
1 + 20119888
1 + 16119888
0 lt 119888 le
1
32
2 (1 + 20119888)
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
120588
119900119901119905=
1
1 + 16119888
0 lt 119888 le
1
32
1 + 24119888 + 1104119888
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
(33)
Proof For the discrete operator
119863
ℎ= 119888ℎ
2Δ
2
ℎminus Δ
2ℎ
(34)
from (17)ndash(20) the Fourier mode of (34) is given by
119863
ℎ(
997888
120579) =
1
ℎ
2[4119888 (2 minus cos 120579
1minus cos 120579
2)
2
+ sin21205791+ sin2120579
2]
(35)
Thus when the red-black point relaxation is applied to (34)from (16) substituting (12) (28) and (35) into (26) and (27)the product of (21) and (25) is
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579) =
1
4
diag (11987711 119877
22)
(36)
where both 11987711
and 11987722
are 2 times 2 square matrices whoseexpressions are below
119877
11= (
0 0
0 1
) sdot (
119860
119861
00+ 1 minus119860
119861
11+ 1
minus119860
119861
00minus 1 119860
119861
11+ 1
) sdot (
119860
119877
00+ 1 119860
119877
11minus 1
119860
119877
00minus 1 119860
119877
11+ 1
)
=
4
(1 + 20119888)
2
sdot (
0 0
1 + 4 [minus1 minus 36119888 + 16119888 (119904
1+ 119904
2)]
sdot [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (119904
1+ 119904
2)
2
]
64119888 (minus1 + 119904
1+ 119904
2) [
119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2]
)
119877
22= (
1 0
0 1
) sdot (
119860
119861
10+ 1 minus119860
119861
01+ 1
minus119860
119861
10+ 1 119860
119861
01+ 1
) sdot (
119860
119877
10+ 1 119860
119877
01minus 1
119860
119877
10minus 1 119860
119877
01+ 1
)
=
4
(1 + 20119888)
2
sdot
(
(
(
(
(
(
(
(
(
(
(
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
+384119888
2(119904
1minus 119904
2) + (192119888
2+ 4) (119904
2
1+ 119904
2
2)
minus (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
2
+128119888119904
2
2+ 32119888119904
1119904
2(2119904
2minus 2119904
1minus 1) (1 minus 12119888)
]
]
]
]
]
]
minus64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
+ 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
minus 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
minus384119888
2(119904
1minus 119904
2) + (4 + 192119888
2) (119904
2
1+ 119904
2
2)
+ (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
1
+128119888119904
2
1+ 32119888119904
1119904
2(2119904
1minus 2119904
2+ 1) (1 minus 12119888)
]
]
]
]
]
]
)
)
)
)
)
)
)
)
)
)
)
(37)
Thus the eigenvalues of matrix (36) are obtained as
120582
1= 0
120582
2=
64119888
(1 + 20119888)
2(minus1 + 119904
1+ 119904
2) [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2
]
(38)
120582
34=
1
(1 + 20119888)
2
[
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (4 + 80119888) (119904
1+ 119904
2)
+ (4 + 64119888 + 192119888
2) (119904
2
1+ 119904
2
2) + (32119888 minus 384119888
2) 119904
1119904
2
plusmn32119888 (119904
1minus 119904
2)radic
1 + 80119888
2+ 24119888 + (minus64119888
2+ 64119888 + 4) (119904
2
1+ 119904
2
2)
minus (80119888 + 4) (119904
1+ 119904
2) + (128119888
2+ 32119888) 119904
1119904
2
]
]
]
]
]
]
]
(39)
6 Mathematical Problems in Engineering
By using the MATLAB and Mathematica software withcylindrical algebraic decomposition function [19] for 997888119904 =
(119904
1 119904
2) isin (0 12)
2 there is no extreme value for (39) when0 lt 119888 le 132 one of extreme values of (38) is obtained as
119904
lowast
1=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
119904
lowast
2=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
(40)
Thus for 997888119904 isin 119878low = [0 12]
2 besides (40) the possibleextreme values of the eigenvalues of matrix (36) are placedon the boundary of 119878low From minus1 le 120582
119896le 1 with 119896 = 1 4
then 0 lt 119888 le 112 Noting that (40) exists with 0 lt 119888 le 132From (38)ndash(40) when 0 lt 119888 le 112 for 997888119904 isin 119878low the maxand min eigenvalues of (36) are yielded as
119878max = 12058234 (0 0) =1 + 4119888
1 + 20119888
119878min =
120582
34(
1
2
1
2
) =
4119888 minus 1
1 + 20119888
0 lt 119888 le
1
32
120582
2(0 0) = minus (
32119888
20119888 + 1
)
21
32
lt 119888 le
1
12
(41)
Substituting (41) into (23) (33) is obtainedTheorem 2 holds
From (33) 12 le 120588opt(119888ℎ2Δ
2
ℎminusΔ
2ℎ) lt 1holdswith 0 lt 119888 le
112 Therefore fromTheorems 1 and 2 when 0 lt 119888 le 112the smoothing factor of (6) with the distributive relaxation(14) is as
1
2
le 120588opt (119871ℎ)
= max 120588opt (minusΔ ℎ) 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ)
= 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ) lt 1
(42)
4 Conclusions
The smoothing analysis process of the distributive red-black Jacobi point relaxation for solving 2D Stokes flow isanalytically presented Applying (28) the Fouriermodes withthe trigonometric functions for the discrete operator andrelaxation are mapped to rational functions So it is possibleto apply the cylindrical algebraic decomposition function intheMathematica software to realize complex smoothing anal-ysis and the computation process is simplifiedThe analyticalexpressions of the smoothing factor for the distributive red-black Jacobi point relaxation are obtained which is an upperbound for the smoothing rates and is independent of themesh size with the parameter 119888 Obviously it is valuable tounderstand numerical experiments in multigrid method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors were supported by the National Natural ScienceFoundation of China (NSFC) (Grant no 51279071) and theDoctoral Foundation of Ministry of Education of China(Grant no 20135314130002)
References
[1] U Trottenberg C W Oosterlee and A Schuller MultigridAcademic Press San Diego Calif USA 2001
[2] R Wienands and W Joppich Practical Fourier Analysis forMultigrid Methods Chapman amp Hall CRC Press 2005
[3] W L Briggs V E Henson and S McCormick A MultigridTutorial Society for Industrial and Applied Mathematics 2ndedition 2000
[4] W Hackbusch Multigrid Methods and Applications SpringerBerlin Germany 1985
[5] P Wesseling An Introduction to Multigrid Methods JohnWileyamp Sons Chichester UK 1992
[6] K Stuben and U Trottenberg ldquoMultigrid methods fundamen-tal algorithms model problem analysis and applicationsrdquo inMultigridMethods W Hackbusch andU Trottenberg Eds vol960 of Lectwe Notes in Mathematics pp 1ndash176 Springer BerlinGermany 1982
[7] A Brandt and O E Livne 1984 Guide to Multigrid Develop-ment in Multigrid Methods Society for Industrial and AppliedMathematics 2011 httpwwwwisdomweizmannacilsimachiclassicspdf
[8] C W Oosterlee and F J G Lorenz ldquoMultigrid methods for thestokes systemrdquoComputing in Science and Engineering vol 8 no6 Article ID 1717313 pp 34ndash43 2006
[9] A Brandt and N Dinar Multigrid Solutions to Elliptic LlowProblems Institute for Computer Applications in Science andEngineering NASA Langley Research Center 1979
[10] G Wittum ldquoMulti-grid methods for stokes and navier-stokesequationsrdquoNumerische Mathematik vol 54 no 5 pp 543ndash5631989
[11] M Wang and L Chen ldquoMultigrid methods for the Stokesequations using distributive Gauss-Seidel relaxations based onthe least squares commutatorrdquo Journal of Scientific Computingvol 56 no 2 pp 409ndash431 2013
[12] M ur Rehman T Geenen C Vuik G Segal and S PMacLachlan ldquoOn iterative methods for the incompressibleStokes problemrdquo International Journal for Numerical Methodsin Fluids vol 65 no 10 pp 1180ndash1200 2011
[13] C Bacuta P S Vassilevski and S Zhang ldquoA new approachfor solving Stokes systems arising from a distributive relaxationmethodrdquo Numerical Methods for Partial Differential Equationsvol 27 no 4 pp 898ndash914 2011
[14] R Wienands F J Gaspar F J Lisbona and C W OosterleeldquoAn efficient multigrid solver based on distributive smoothingfor poroelasticity equationsrdquo Computing vol 73 no 2 pp 99ndash119 2004
[15] W Liao B Diskin Y Peng and L-S Luo ldquoTextbook-efficiencymultigrid solver for three-dimensional unsteady compressibleNavier-Stokes equationsrdquo Journal of Computational Physics vol227 no 15 pp 7160ndash7177 2008
[16] V Pillwein and S Takacs ldquoA local Fourier convergence analysisof a multigrid method using symbolic computationrdquo Journal ofSymbolic Computation vol 63 pp 1ndash20 2014
Mathematical Problems in Engineering 7
[17] S Takacs All-at-once multigrid methods for optimality systemsarising from optimal control problems [PhD thesis] JohannesKepler University Linz Doctoral Program ComputationalMathematics 2012
[18] V Pillwein and S Takacs ldquoSmoothing analysis of an all-at-oncemultigrid approach for optimal control problems using symboliccomputationrdquo inNumerical and Symbolic Scientific ComputingProgress and Prospects U Langer and P Paule Eds SpringerWien Austria 2011
[19] M Kauers ldquoHow to use cylindrical algebraic decompositionrdquoSeminaire Lotharingien de Combinatoire vol 65 article B65app 1ndash16 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
be assumed Then the Fourier space (8) is subdivided into2ℎ-harmonics subspaces
119865
2ℎ(
997888
120579) = span120593ℎ(
997888
120579
00
997888
119909) 120593
ℎ(
997888
120579
11
997888
119909)
120593
ℎ(
997888
120579
10
997888
119909) 120593
ℎ(
997888
120579
01
997888
119909)
(13)
3 Smoothing Process
31 Distributive Relaxation of System (6) From [1 2 7] adistributive operator for the discrete system (6) is constructedas
119862
ℎ= (
119868
ℎ0 minus120597
ℎ
119909
0 119868
ℎminus120597
ℎ
119910
0 0 minusΔ
ℎ
) (14)
where 119868ℎis the unit operator with discrete stencil [1]
ℎ From
(14) the discrete system (6) is transformed as
119871
ℎ119862
ℎ= (
minusΔ
ℎ0 0
0 minusΔ
ℎ0
120597
ℎ
119909120597
ℎ
119910119888ℎ
2Δ
2
ℎminus Δ
2ℎ
) (15)
where the discrete stencils of Δ2ℎand minusΔ
2ℎare
Δ
2
ℎ=
1
ℎ
4
[
[
[
[
[
[
[
[
[
1
2 minus8 2
1 minus8 20 minus8 1
2 minus8 2
1
]
]
]
]
]
]
]
]
]ℎ
minusΔ
2ℎ=
1
4ℎ
2
[
[
[
[
[
[
[
[
[
0 0 minus1 0 0
0 0 0 0 0
minus1 0 4 0 minus1
0 0 0 0 0
0 0 minus1 0 0
]
]
]
]
]
]
]
]
]ℎ
=
1
4ℎ
2
[
[
[
minus1
minus1 4 minus1
minus1
]
]
]
2ℎ
(16)
From (9)ndash(11) the Fourier modes of the scalar discreteoperators of (16) are
Δ
2
ℎ(
997888
120579) = (minus
Δ
ℎ(
997888
120579))
2
minus
Δ
2ℎ(
997888
120579) = minus [
120597
ℎ
119909(
997888
120579)]
2
minus [
120597
ℎ
119910(
997888
120579)]
2
(17)
where
minus
Δ
ℎ(
997888
120579) =
1
ℎ
2(4 minus exp (minus119894120579
1) minus exp (119894120579
1)
minus exp (minus1198941205792) minus exp (119894120579
2))
=
1
ℎ
2(4 minus 2 cos 120579
1minus 2 cos 120579
2)
(18)
120597
ℎ
119909(
997888
120579) =
1
2ℎ
(exp (1198941205791) minus exp (minus119894120579
1)) =
1
ℎ
119894 sin 1205791
(19)
120597
ℎ
119910(
997888
120579) =
1
2ℎ
(exp (1198941205792) minus exp (minus119894120579
2)) =
1
ℎ
119894 sin 1205792
(20)
32 Optimal One-Stage Relaxation For the discrete scalaroperator of (15) standard coarsening and an ideal coarse gridcorrection operator [2] are applied as
119876
2ℎ
ℎ
1003816
1003816
1003816
1003816
10038161198652ℎ(997888120579 )=
_119876
2ℎ
ℎ= diag (0 1 1 1) isin C
4times4
(21)
where_119876
2ℎ
ℎis the Fourier representation of the operator
119876
2ℎ
ℎwith subspace (13) which suppresses the low frequency
error components andmakes the high frequency componentsunchangedThen from [2] the smoothing factor for discreteoperator (9) is defined by
120588 (119899119863
ℎ) = sup997888120579isinΘlow
(120588(
_119876
2ℎ
ℎ(
_119878 ℎ(
997888
120579 120596))
119899
))
1119899
(22)
It implies that the asymptotic error reduction of the highfrequency error components is given by n sweeps of the relax-ation method where
_119878 ℎ(
997888
120579 120596) is the Fourier representationof the relaxation operator 119878
ℎ(120596) on subspace (13) and 120596 is the
relaxation parameterFrom [2 14] a good smoothing factor is obtained by using
one-stage parameter 120596 in the relaxation operator 119878ℎ(120596) the
optimal smoothing factor and related smoothing parameterare given by
120596opt =2
2 minus 119878max minus 119878min
120588opt =119878max minus 119878min
2 minus 119878max minus 119878min
(23)
where 119878max and 119878min are the max and min eigenvalues of the
product matrix_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579 1) with the relaxation parameter120596 = 1 for 120579 isin Θ2ℎlow From [2 19] the smoothing factor of(6) with the distributive relaxation (14) is determined by thediagonal blocks of the transformed system (15) which is givenby
120588 (119899 119871
ℎ) = max 120588 (119899 minusΔ
ℎ) 120588 (119899 119888ℎ
2Δ
2
ℎminus Δ
2ℎ) (24)
33 Optimal Smoothing for Stokes Flow The red-black Jacobipoint relaxation 119878119877119861
ℎis applied to (15) to discuss the optimal
4 Mathematical Problems in Engineering
smoothing problems for Stokes flow From [1 2 14] the oper-ator 119878119877119861ℎ
makes the 2ℎ-harmonics subspace (13) invariant thatis
119878
119877119861
ℎ
1003816
1003816
1003816
1003816
10038161198652ℎ(997888120579 )=
_119878
119877119861
ℎ(
997888
120579) isin C4times4
(25)
where_119878
119877119861
ℎ(
997888
120579 ) is the Fourier representation of 119878119877119861ℎ(120596) with
relaxation parameter 120596 = 1 and is given as
_119878
119877119861
ℎ(
997888
120579 1)
=
_119878
119877119861
ℎ(
997888
120579)
=
_119878
119861
ℎ(
997888
120579)
_119878
119877
ℎ(
997888
120579)
=
1
2
(
119860
00+ 1 minus119860
11+ 1 0 0
minus119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 minus119860
01+ 1
0 0 minus119860
10+ 1 119860
01+ 1
)
sdot
1
2
(
119860
00+ 1 119860
11minus 1 0 0
119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 119860
01minus 1
0 0 119860
10minus 1 119860
01+ 1
)
(26)
in which
119860997888120572= 1 minus
119863
ℎ(
997888
120579
997888120572
)
119863
0
ℎ(
997888
120579
997888120572
)
(27)
denotes the Fourier mode of the point Jacobi relaxation for
the discrete operator (9) on subspace (13) and 1198630ℎ(
997888
120579
997888120572
) is theFourier mode of the discrete operator with the stencil [119897
(00)]
ℎ
in (9) For the sake of convenient discussion in the followingtwo variables are introduced as
119904
1= sin2
120579
0
1
2
= sin2 12057912
119904
2= sin2
120579
0
2
2
= sin2 12057922
(28)
Thus997888
120579 = (120579
1 120579
2) isin Θ
2ℎ
low = (minus1205872 1205872]2 is transformed to
997888
119904 = (119904
1 119904
2) isin 119878low = [0 12]
2
Theorem 1 For the Poisson operator minusΔℎ the optimal one-
stage relaxation parameter and related smoothing factor of thered-black Jacobi point relaxation are stated as
120596
119900119901119905=
16
15
120588
119900119901119905=
1
5
(29)
Proof For the red-black Jacobi point relaxation for thePoisson operator 119863
ℎ= minusΔ
ℎ substituting (12) (18) and (28)
into (26) and (27) and from (5) the product of (21) and (25)is written as
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579)
=
1
2
(
0 0 0 0
(119904
1+ 119904
2) (1 minus 119904
1minus 119904
2) (119904
1+ 119904
2) (119904
1+ 119904
2minus 1) 0 0
0 0 (119904
1minus 119904
2) (119904
1minus 119904
2+ 1) (119904
2minus 119904
1) (119904
1minus 119904
2+ 1)
0 0 (119904
1minus 119904
2) (119904
2minus 119904
1+ 1) (119904
2minus 119904
1) (119904
2minus 119904
1+ 1)
)
(30)
Thus eigenvalues of (30) are obtained as
120582
1= 0 120582
2= (119904
1minus 119904
2)
2
120582
3= 0 120582
4=
(119904
1+ 119904
2) (119904
1+ 119904
2minus 1)
2
(31)
From (31) the max andmin eigenvalues of (30) are yielded as
119878max = max(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = max(11990411199042)isin[012]
2
120582
2=
1
4
119878min = min(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = min(11990411199042)isin[012]
2
120582
4= minus
1
8
(32)
From (23) and (32) (29) is obtained Theorem 1 holds
Next 120588(119899 119888ℎ2Δ2ℎminus Δ
2ℎ) for the red-black Jacobi point
relaxation need to be computed Meanwhile the smoothingfactor of distributive relaxation (15) is given as follows
Theorem 2 For the discrete operator 119888ℎ2Δ2ℎminus Δ
2ℎwith 119888 gt
0 the optimal one-stage relaxation parameter and related
Mathematical Problems in Engineering 5
smoothing factor of the red-black Jacobi point relaxation aregiven by
120596
119900119901119905=
1 + 20119888
1 + 16119888
0 lt 119888 le
1
32
2 (1 + 20119888)
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
120588
119900119901119905=
1
1 + 16119888
0 lt 119888 le
1
32
1 + 24119888 + 1104119888
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
(33)
Proof For the discrete operator
119863
ℎ= 119888ℎ
2Δ
2
ℎminus Δ
2ℎ
(34)
from (17)ndash(20) the Fourier mode of (34) is given by
119863
ℎ(
997888
120579) =
1
ℎ
2[4119888 (2 minus cos 120579
1minus cos 120579
2)
2
+ sin21205791+ sin2120579
2]
(35)
Thus when the red-black point relaxation is applied to (34)from (16) substituting (12) (28) and (35) into (26) and (27)the product of (21) and (25) is
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579) =
1
4
diag (11987711 119877
22)
(36)
where both 11987711
and 11987722
are 2 times 2 square matrices whoseexpressions are below
119877
11= (
0 0
0 1
) sdot (
119860
119861
00+ 1 minus119860
119861
11+ 1
minus119860
119861
00minus 1 119860
119861
11+ 1
) sdot (
119860
119877
00+ 1 119860
119877
11minus 1
119860
119877
00minus 1 119860
119877
11+ 1
)
=
4
(1 + 20119888)
2
sdot (
0 0
1 + 4 [minus1 minus 36119888 + 16119888 (119904
1+ 119904
2)]
sdot [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (119904
1+ 119904
2)
2
]
64119888 (minus1 + 119904
1+ 119904
2) [
119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2]
)
119877
22= (
1 0
0 1
) sdot (
119860
119861
10+ 1 minus119860
119861
01+ 1
minus119860
119861
10+ 1 119860
119861
01+ 1
) sdot (
119860
119877
10+ 1 119860
119877
01minus 1
119860
119877
10minus 1 119860
119877
01+ 1
)
=
4
(1 + 20119888)
2
sdot
(
(
(
(
(
(
(
(
(
(
(
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
+384119888
2(119904
1minus 119904
2) + (192119888
2+ 4) (119904
2
1+ 119904
2
2)
minus (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
2
+128119888119904
2
2+ 32119888119904
1119904
2(2119904
2minus 2119904
1minus 1) (1 minus 12119888)
]
]
]
]
]
]
minus64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
+ 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
minus 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
minus384119888
2(119904
1minus 119904
2) + (4 + 192119888
2) (119904
2
1+ 119904
2
2)
+ (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
1
+128119888119904
2
1+ 32119888119904
1119904
2(2119904
1minus 2119904
2+ 1) (1 minus 12119888)
]
]
]
]
]
]
)
)
)
)
)
)
)
)
)
)
)
(37)
Thus the eigenvalues of matrix (36) are obtained as
120582
1= 0
120582
2=
64119888
(1 + 20119888)
2(minus1 + 119904
1+ 119904
2) [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2
]
(38)
120582
34=
1
(1 + 20119888)
2
[
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (4 + 80119888) (119904
1+ 119904
2)
+ (4 + 64119888 + 192119888
2) (119904
2
1+ 119904
2
2) + (32119888 minus 384119888
2) 119904
1119904
2
plusmn32119888 (119904
1minus 119904
2)radic
1 + 80119888
2+ 24119888 + (minus64119888
2+ 64119888 + 4) (119904
2
1+ 119904
2
2)
minus (80119888 + 4) (119904
1+ 119904
2) + (128119888
2+ 32119888) 119904
1119904
2
]
]
]
]
]
]
]
(39)
6 Mathematical Problems in Engineering
By using the MATLAB and Mathematica software withcylindrical algebraic decomposition function [19] for 997888119904 =
(119904
1 119904
2) isin (0 12)
2 there is no extreme value for (39) when0 lt 119888 le 132 one of extreme values of (38) is obtained as
119904
lowast
1=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
119904
lowast
2=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
(40)
Thus for 997888119904 isin 119878low = [0 12]
2 besides (40) the possibleextreme values of the eigenvalues of matrix (36) are placedon the boundary of 119878low From minus1 le 120582
119896le 1 with 119896 = 1 4
then 0 lt 119888 le 112 Noting that (40) exists with 0 lt 119888 le 132From (38)ndash(40) when 0 lt 119888 le 112 for 997888119904 isin 119878low the maxand min eigenvalues of (36) are yielded as
119878max = 12058234 (0 0) =1 + 4119888
1 + 20119888
119878min =
120582
34(
1
2
1
2
) =
4119888 minus 1
1 + 20119888
0 lt 119888 le
1
32
120582
2(0 0) = minus (
32119888
20119888 + 1
)
21
32
lt 119888 le
1
12
(41)
Substituting (41) into (23) (33) is obtainedTheorem 2 holds
From (33) 12 le 120588opt(119888ℎ2Δ
2
ℎminusΔ
2ℎ) lt 1holdswith 0 lt 119888 le
112 Therefore fromTheorems 1 and 2 when 0 lt 119888 le 112the smoothing factor of (6) with the distributive relaxation(14) is as
1
2
le 120588opt (119871ℎ)
= max 120588opt (minusΔ ℎ) 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ)
= 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ) lt 1
(42)
4 Conclusions
The smoothing analysis process of the distributive red-black Jacobi point relaxation for solving 2D Stokes flow isanalytically presented Applying (28) the Fouriermodes withthe trigonometric functions for the discrete operator andrelaxation are mapped to rational functions So it is possibleto apply the cylindrical algebraic decomposition function intheMathematica software to realize complex smoothing anal-ysis and the computation process is simplifiedThe analyticalexpressions of the smoothing factor for the distributive red-black Jacobi point relaxation are obtained which is an upperbound for the smoothing rates and is independent of themesh size with the parameter 119888 Obviously it is valuable tounderstand numerical experiments in multigrid method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors were supported by the National Natural ScienceFoundation of China (NSFC) (Grant no 51279071) and theDoctoral Foundation of Ministry of Education of China(Grant no 20135314130002)
References
[1] U Trottenberg C W Oosterlee and A Schuller MultigridAcademic Press San Diego Calif USA 2001
[2] R Wienands and W Joppich Practical Fourier Analysis forMultigrid Methods Chapman amp Hall CRC Press 2005
[3] W L Briggs V E Henson and S McCormick A MultigridTutorial Society for Industrial and Applied Mathematics 2ndedition 2000
[4] W Hackbusch Multigrid Methods and Applications SpringerBerlin Germany 1985
[5] P Wesseling An Introduction to Multigrid Methods JohnWileyamp Sons Chichester UK 1992
[6] K Stuben and U Trottenberg ldquoMultigrid methods fundamen-tal algorithms model problem analysis and applicationsrdquo inMultigridMethods W Hackbusch andU Trottenberg Eds vol960 of Lectwe Notes in Mathematics pp 1ndash176 Springer BerlinGermany 1982
[7] A Brandt and O E Livne 1984 Guide to Multigrid Develop-ment in Multigrid Methods Society for Industrial and AppliedMathematics 2011 httpwwwwisdomweizmannacilsimachiclassicspdf
[8] C W Oosterlee and F J G Lorenz ldquoMultigrid methods for thestokes systemrdquoComputing in Science and Engineering vol 8 no6 Article ID 1717313 pp 34ndash43 2006
[9] A Brandt and N Dinar Multigrid Solutions to Elliptic LlowProblems Institute for Computer Applications in Science andEngineering NASA Langley Research Center 1979
[10] G Wittum ldquoMulti-grid methods for stokes and navier-stokesequationsrdquoNumerische Mathematik vol 54 no 5 pp 543ndash5631989
[11] M Wang and L Chen ldquoMultigrid methods for the Stokesequations using distributive Gauss-Seidel relaxations based onthe least squares commutatorrdquo Journal of Scientific Computingvol 56 no 2 pp 409ndash431 2013
[12] M ur Rehman T Geenen C Vuik G Segal and S PMacLachlan ldquoOn iterative methods for the incompressibleStokes problemrdquo International Journal for Numerical Methodsin Fluids vol 65 no 10 pp 1180ndash1200 2011
[13] C Bacuta P S Vassilevski and S Zhang ldquoA new approachfor solving Stokes systems arising from a distributive relaxationmethodrdquo Numerical Methods for Partial Differential Equationsvol 27 no 4 pp 898ndash914 2011
[14] R Wienands F J Gaspar F J Lisbona and C W OosterleeldquoAn efficient multigrid solver based on distributive smoothingfor poroelasticity equationsrdquo Computing vol 73 no 2 pp 99ndash119 2004
[15] W Liao B Diskin Y Peng and L-S Luo ldquoTextbook-efficiencymultigrid solver for three-dimensional unsteady compressibleNavier-Stokes equationsrdquo Journal of Computational Physics vol227 no 15 pp 7160ndash7177 2008
[16] V Pillwein and S Takacs ldquoA local Fourier convergence analysisof a multigrid method using symbolic computationrdquo Journal ofSymbolic Computation vol 63 pp 1ndash20 2014
Mathematical Problems in Engineering 7
[17] S Takacs All-at-once multigrid methods for optimality systemsarising from optimal control problems [PhD thesis] JohannesKepler University Linz Doctoral Program ComputationalMathematics 2012
[18] V Pillwein and S Takacs ldquoSmoothing analysis of an all-at-oncemultigrid approach for optimal control problems using symboliccomputationrdquo inNumerical and Symbolic Scientific ComputingProgress and Prospects U Langer and P Paule Eds SpringerWien Austria 2011
[19] M Kauers ldquoHow to use cylindrical algebraic decompositionrdquoSeminaire Lotharingien de Combinatoire vol 65 article B65app 1ndash16 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
smoothing problems for Stokes flow From [1 2 14] the oper-ator 119878119877119861ℎ
makes the 2ℎ-harmonics subspace (13) invariant thatis
119878
119877119861
ℎ
1003816
1003816
1003816
1003816
10038161198652ℎ(997888120579 )=
_119878
119877119861
ℎ(
997888
120579) isin C4times4
(25)
where_119878
119877119861
ℎ(
997888
120579 ) is the Fourier representation of 119878119877119861ℎ(120596) with
relaxation parameter 120596 = 1 and is given as
_119878
119877119861
ℎ(
997888
120579 1)
=
_119878
119877119861
ℎ(
997888
120579)
=
_119878
119861
ℎ(
997888
120579)
_119878
119877
ℎ(
997888
120579)
=
1
2
(
119860
00+ 1 minus119860
11+ 1 0 0
minus119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 minus119860
01+ 1
0 0 minus119860
10+ 1 119860
01+ 1
)
sdot
1
2
(
119860
00+ 1 119860
11minus 1 0 0
119860
00minus 1 119860
11+ 1 0 0
0 0 119860
10+ 1 119860
01minus 1
0 0 119860
10minus 1 119860
01+ 1
)
(26)
in which
119860997888120572= 1 minus
119863
ℎ(
997888
120579
997888120572
)
119863
0
ℎ(
997888
120579
997888120572
)
(27)
denotes the Fourier mode of the point Jacobi relaxation for
the discrete operator (9) on subspace (13) and 1198630ℎ(
997888
120579
997888120572
) is theFourier mode of the discrete operator with the stencil [119897
(00)]
ℎ
in (9) For the sake of convenient discussion in the followingtwo variables are introduced as
119904
1= sin2
120579
0
1
2
= sin2 12057912
119904
2= sin2
120579
0
2
2
= sin2 12057922
(28)
Thus997888
120579 = (120579
1 120579
2) isin Θ
2ℎ
low = (minus1205872 1205872]2 is transformed to
997888
119904 = (119904
1 119904
2) isin 119878low = [0 12]
2
Theorem 1 For the Poisson operator minusΔℎ the optimal one-
stage relaxation parameter and related smoothing factor of thered-black Jacobi point relaxation are stated as
120596
119900119901119905=
16
15
120588
119900119901119905=
1
5
(29)
Proof For the red-black Jacobi point relaxation for thePoisson operator 119863
ℎ= minusΔ
ℎ substituting (12) (18) and (28)
into (26) and (27) and from (5) the product of (21) and (25)is written as
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878 ℎ(
997888
120579)
=
1
2
(
0 0 0 0
(119904
1+ 119904
2) (1 minus 119904
1minus 119904
2) (119904
1+ 119904
2) (119904
1+ 119904
2minus 1) 0 0
0 0 (119904
1minus 119904
2) (119904
1minus 119904
2+ 1) (119904
2minus 119904
1) (119904
1minus 119904
2+ 1)
0 0 (119904
1minus 119904
2) (119904
2minus 119904
1+ 1) (119904
2minus 119904
1) (119904
2minus 119904
1+ 1)
)
(30)
Thus eigenvalues of (30) are obtained as
120582
1= 0 120582
2= (119904
1minus 119904
2)
2
120582
3= 0 120582
4=
(119904
1+ 119904
2) (119904
1+ 119904
2minus 1)
2
(31)
From (31) the max andmin eigenvalues of (30) are yielded as
119878max = max(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = max(11990411199042)isin[012]
2
120582
2=
1
4
119878min = min(11990411199042)isin[012]
2
120582
1 120582
2 120582
3 120582
4 = min(11990411199042)isin[012]
2
120582
4= minus
1
8
(32)
From (23) and (32) (29) is obtained Theorem 1 holds
Next 120588(119899 119888ℎ2Δ2ℎminus Δ
2ℎ) for the red-black Jacobi point
relaxation need to be computed Meanwhile the smoothingfactor of distributive relaxation (15) is given as follows
Theorem 2 For the discrete operator 119888ℎ2Δ2ℎminus Δ
2ℎwith 119888 gt
0 the optimal one-stage relaxation parameter and related
Mathematical Problems in Engineering 5
smoothing factor of the red-black Jacobi point relaxation aregiven by
120596
119900119901119905=
1 + 20119888
1 + 16119888
0 lt 119888 le
1
32
2 (1 + 20119888)
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
120588
119900119901119905=
1
1 + 16119888
0 lt 119888 le
1
32
1 + 24119888 + 1104119888
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
(33)
Proof For the discrete operator
119863
ℎ= 119888ℎ
2Δ
2
ℎminus Δ
2ℎ
(34)
from (17)ndash(20) the Fourier mode of (34) is given by
119863
ℎ(
997888
120579) =
1
ℎ
2[4119888 (2 minus cos 120579
1minus cos 120579
2)
2
+ sin21205791+ sin2120579
2]
(35)
Thus when the red-black point relaxation is applied to (34)from (16) substituting (12) (28) and (35) into (26) and (27)the product of (21) and (25) is
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579) =
1
4
diag (11987711 119877
22)
(36)
where both 11987711
and 11987722
are 2 times 2 square matrices whoseexpressions are below
119877
11= (
0 0
0 1
) sdot (
119860
119861
00+ 1 minus119860
119861
11+ 1
minus119860
119861
00minus 1 119860
119861
11+ 1
) sdot (
119860
119877
00+ 1 119860
119877
11minus 1
119860
119877
00minus 1 119860
119877
11+ 1
)
=
4
(1 + 20119888)
2
sdot (
0 0
1 + 4 [minus1 minus 36119888 + 16119888 (119904
1+ 119904
2)]
sdot [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (119904
1+ 119904
2)
2
]
64119888 (minus1 + 119904
1+ 119904
2) [
119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2]
)
119877
22= (
1 0
0 1
) sdot (
119860
119861
10+ 1 minus119860
119861
01+ 1
minus119860
119861
10+ 1 119860
119861
01+ 1
) sdot (
119860
119877
10+ 1 119860
119877
01minus 1
119860
119877
10minus 1 119860
119877
01+ 1
)
=
4
(1 + 20119888)
2
sdot
(
(
(
(
(
(
(
(
(
(
(
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
+384119888
2(119904
1minus 119904
2) + (192119888
2+ 4) (119904
2
1+ 119904
2
2)
minus (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
2
+128119888119904
2
2+ 32119888119904
1119904
2(2119904
2minus 2119904
1minus 1) (1 minus 12119888)
]
]
]
]
]
]
minus64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
+ 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
minus 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
minus384119888
2(119904
1minus 119904
2) + (4 + 192119888
2) (119904
2
1+ 119904
2
2)
+ (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
1
+128119888119904
2
1+ 32119888119904
1119904
2(2119904
1minus 2119904
2+ 1) (1 minus 12119888)
]
]
]
]
]
]
)
)
)
)
)
)
)
)
)
)
)
(37)
Thus the eigenvalues of matrix (36) are obtained as
120582
1= 0
120582
2=
64119888
(1 + 20119888)
2(minus1 + 119904
1+ 119904
2) [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2
]
(38)
120582
34=
1
(1 + 20119888)
2
[
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (4 + 80119888) (119904
1+ 119904
2)
+ (4 + 64119888 + 192119888
2) (119904
2
1+ 119904
2
2) + (32119888 minus 384119888
2) 119904
1119904
2
plusmn32119888 (119904
1minus 119904
2)radic
1 + 80119888
2+ 24119888 + (minus64119888
2+ 64119888 + 4) (119904
2
1+ 119904
2
2)
minus (80119888 + 4) (119904
1+ 119904
2) + (128119888
2+ 32119888) 119904
1119904
2
]
]
]
]
]
]
]
(39)
6 Mathematical Problems in Engineering
By using the MATLAB and Mathematica software withcylindrical algebraic decomposition function [19] for 997888119904 =
(119904
1 119904
2) isin (0 12)
2 there is no extreme value for (39) when0 lt 119888 le 132 one of extreme values of (38) is obtained as
119904
lowast
1=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
119904
lowast
2=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
(40)
Thus for 997888119904 isin 119878low = [0 12]
2 besides (40) the possibleextreme values of the eigenvalues of matrix (36) are placedon the boundary of 119878low From minus1 le 120582
119896le 1 with 119896 = 1 4
then 0 lt 119888 le 112 Noting that (40) exists with 0 lt 119888 le 132From (38)ndash(40) when 0 lt 119888 le 112 for 997888119904 isin 119878low the maxand min eigenvalues of (36) are yielded as
119878max = 12058234 (0 0) =1 + 4119888
1 + 20119888
119878min =
120582
34(
1
2
1
2
) =
4119888 minus 1
1 + 20119888
0 lt 119888 le
1
32
120582
2(0 0) = minus (
32119888
20119888 + 1
)
21
32
lt 119888 le
1
12
(41)
Substituting (41) into (23) (33) is obtainedTheorem 2 holds
From (33) 12 le 120588opt(119888ℎ2Δ
2
ℎminusΔ
2ℎ) lt 1holdswith 0 lt 119888 le
112 Therefore fromTheorems 1 and 2 when 0 lt 119888 le 112the smoothing factor of (6) with the distributive relaxation(14) is as
1
2
le 120588opt (119871ℎ)
= max 120588opt (minusΔ ℎ) 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ)
= 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ) lt 1
(42)
4 Conclusions
The smoothing analysis process of the distributive red-black Jacobi point relaxation for solving 2D Stokes flow isanalytically presented Applying (28) the Fouriermodes withthe trigonometric functions for the discrete operator andrelaxation are mapped to rational functions So it is possibleto apply the cylindrical algebraic decomposition function intheMathematica software to realize complex smoothing anal-ysis and the computation process is simplifiedThe analyticalexpressions of the smoothing factor for the distributive red-black Jacobi point relaxation are obtained which is an upperbound for the smoothing rates and is independent of themesh size with the parameter 119888 Obviously it is valuable tounderstand numerical experiments in multigrid method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors were supported by the National Natural ScienceFoundation of China (NSFC) (Grant no 51279071) and theDoctoral Foundation of Ministry of Education of China(Grant no 20135314130002)
References
[1] U Trottenberg C W Oosterlee and A Schuller MultigridAcademic Press San Diego Calif USA 2001
[2] R Wienands and W Joppich Practical Fourier Analysis forMultigrid Methods Chapman amp Hall CRC Press 2005
[3] W L Briggs V E Henson and S McCormick A MultigridTutorial Society for Industrial and Applied Mathematics 2ndedition 2000
[4] W Hackbusch Multigrid Methods and Applications SpringerBerlin Germany 1985
[5] P Wesseling An Introduction to Multigrid Methods JohnWileyamp Sons Chichester UK 1992
[6] K Stuben and U Trottenberg ldquoMultigrid methods fundamen-tal algorithms model problem analysis and applicationsrdquo inMultigridMethods W Hackbusch andU Trottenberg Eds vol960 of Lectwe Notes in Mathematics pp 1ndash176 Springer BerlinGermany 1982
[7] A Brandt and O E Livne 1984 Guide to Multigrid Develop-ment in Multigrid Methods Society for Industrial and AppliedMathematics 2011 httpwwwwisdomweizmannacilsimachiclassicspdf
[8] C W Oosterlee and F J G Lorenz ldquoMultigrid methods for thestokes systemrdquoComputing in Science and Engineering vol 8 no6 Article ID 1717313 pp 34ndash43 2006
[9] A Brandt and N Dinar Multigrid Solutions to Elliptic LlowProblems Institute for Computer Applications in Science andEngineering NASA Langley Research Center 1979
[10] G Wittum ldquoMulti-grid methods for stokes and navier-stokesequationsrdquoNumerische Mathematik vol 54 no 5 pp 543ndash5631989
[11] M Wang and L Chen ldquoMultigrid methods for the Stokesequations using distributive Gauss-Seidel relaxations based onthe least squares commutatorrdquo Journal of Scientific Computingvol 56 no 2 pp 409ndash431 2013
[12] M ur Rehman T Geenen C Vuik G Segal and S PMacLachlan ldquoOn iterative methods for the incompressibleStokes problemrdquo International Journal for Numerical Methodsin Fluids vol 65 no 10 pp 1180ndash1200 2011
[13] C Bacuta P S Vassilevski and S Zhang ldquoA new approachfor solving Stokes systems arising from a distributive relaxationmethodrdquo Numerical Methods for Partial Differential Equationsvol 27 no 4 pp 898ndash914 2011
[14] R Wienands F J Gaspar F J Lisbona and C W OosterleeldquoAn efficient multigrid solver based on distributive smoothingfor poroelasticity equationsrdquo Computing vol 73 no 2 pp 99ndash119 2004
[15] W Liao B Diskin Y Peng and L-S Luo ldquoTextbook-efficiencymultigrid solver for three-dimensional unsteady compressibleNavier-Stokes equationsrdquo Journal of Computational Physics vol227 no 15 pp 7160ndash7177 2008
[16] V Pillwein and S Takacs ldquoA local Fourier convergence analysisof a multigrid method using symbolic computationrdquo Journal ofSymbolic Computation vol 63 pp 1ndash20 2014
Mathematical Problems in Engineering 7
[17] S Takacs All-at-once multigrid methods for optimality systemsarising from optimal control problems [PhD thesis] JohannesKepler University Linz Doctoral Program ComputationalMathematics 2012
[18] V Pillwein and S Takacs ldquoSmoothing analysis of an all-at-oncemultigrid approach for optimal control problems using symboliccomputationrdquo inNumerical and Symbolic Scientific ComputingProgress and Prospects U Langer and P Paule Eds SpringerWien Austria 2011
[19] M Kauers ldquoHow to use cylindrical algebraic decompositionrdquoSeminaire Lotharingien de Combinatoire vol 65 article B65app 1ndash16 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
smoothing factor of the red-black Jacobi point relaxation aregiven by
120596
119900119901119905=
1 + 20119888
1 + 16119888
0 lt 119888 le
1
32
2 (1 + 20119888)
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
120588
119900119901119905=
1
1 + 16119888
0 lt 119888 le
1
32
1 + 24119888 + 1104119888
2
1 + 56119888 + 1744119888
2
1
32
lt 119888 le
1
12
(33)
Proof For the discrete operator
119863
ℎ= 119888ℎ
2Δ
2
ℎminus Δ
2ℎ
(34)
from (17)ndash(20) the Fourier mode of (34) is given by
119863
ℎ(
997888
120579) =
1
ℎ
2[4119888 (2 minus cos 120579
1minus cos 120579
2)
2
+ sin21205791+ sin2120579
2]
(35)
Thus when the red-black point relaxation is applied to (34)from (16) substituting (12) (28) and (35) into (26) and (27)the product of (21) and (25) is
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579 1) =
_119876
2ℎ
ℎ
_119878
119877119861
ℎ(
997888
120579) =
1
4
diag (11987711 119877
22)
(36)
where both 11987711
and 11987722
are 2 times 2 square matrices whoseexpressions are below
119877
11= (
0 0
0 1
) sdot (
119860
119861
00+ 1 minus119860
119861
11+ 1
minus119860
119861
00minus 1 119860
119861
11+ 1
) sdot (
119860
119877
00+ 1 119860
119877
11minus 1
119860
119877
00minus 1 119860
119877
11+ 1
)
=
4
(1 + 20119888)
2
sdot (
0 0
1 + 4 [minus1 minus 36119888 + 16119888 (119904
1+ 119904
2)]
sdot [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (119904
1+ 119904
2)
2
]
64119888 (minus1 + 119904
1+ 119904
2) [
119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2]
)
119877
22= (
1 0
0 1
) sdot (
119860
119861
10+ 1 minus119860
119861
01+ 1
minus119860
119861
10+ 1 119860
119861
01+ 1
) sdot (
119860
119877
10+ 1 119860
119877
01minus 1
119860
119877
10minus 1 119860
119877
01+ 1
)
=
4
(1 + 20119888)
2
sdot
(
(
(
(
(
(
(
(
(
(
(
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
+384119888
2(119904
1minus 119904
2) + (192119888
2+ 4) (119904
2
1+ 119904
2
2)
minus (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
2
+128119888119904
2
2+ 32119888119904
1119904
2(2119904
2minus 2119904
1minus 1) (1 minus 12119888)
]
]
]
]
]
]
minus64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
+ 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
64119888 (119904
1minus 119904
2) [
4119888 (119904
1minus 119904
2)
2
minus 8119888 (119904
1minus 119904
2)
+4119888 + 119904
1minus 119904
2
1+ 119904
2minus 119904
2
2
]
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (48119888 + 4) (119904
1+ 119904
2)
minus384119888
2(119904
1minus 119904
2) + (4 + 192119888
2) (119904
2
1+ 119904
2
2)
+ (256119888
2minus 64119888) (119904
3
1minus 119904
3
2) minus 64119888119904
1
+128119888119904
2
1+ 32119888119904
1119904
2(2119904
1minus 2119904
2+ 1) (1 minus 12119888)
]
]
]
]
]
]
)
)
)
)
)
)
)
)
)
)
)
(37)
Thus the eigenvalues of matrix (36) are obtained as
120582
1= 0
120582
2=
64119888
(1 + 20119888)
2(minus1 + 119904
1+ 119904
2) [119904
1minus 119904
2
1+ 119904
2minus 119904
2
2+ 4119888 (minus2 + 119904
1+ 119904
2)
2
]
(38)
120582
34=
1
(1 + 20119888)
2
[
[
[
[
[
[
[
1 + 24119888 + 80119888
2minus (4 + 80119888) (119904
1+ 119904
2)
+ (4 + 64119888 + 192119888
2) (119904
2
1+ 119904
2
2) + (32119888 minus 384119888
2) 119904
1119904
2
plusmn32119888 (119904
1minus 119904
2)radic
1 + 80119888
2+ 24119888 + (minus64119888
2+ 64119888 + 4) (119904
2
1+ 119904
2
2)
minus (80119888 + 4) (119904
1+ 119904
2) + (128119888
2+ 32119888) 119904
1119904
2
]
]
]
]
]
]
]
(39)
6 Mathematical Problems in Engineering
By using the MATLAB and Mathematica software withcylindrical algebraic decomposition function [19] for 997888119904 =
(119904
1 119904
2) isin (0 12)
2 there is no extreme value for (39) when0 lt 119888 le 132 one of extreme values of (38) is obtained as
119904
lowast
1=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
119904
lowast
2=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
(40)
Thus for 997888119904 isin 119878low = [0 12]
2 besides (40) the possibleextreme values of the eigenvalues of matrix (36) are placedon the boundary of 119878low From minus1 le 120582
119896le 1 with 119896 = 1 4
then 0 lt 119888 le 112 Noting that (40) exists with 0 lt 119888 le 132From (38)ndash(40) when 0 lt 119888 le 112 for 997888119904 isin 119878low the maxand min eigenvalues of (36) are yielded as
119878max = 12058234 (0 0) =1 + 4119888
1 + 20119888
119878min =
120582
34(
1
2
1
2
) =
4119888 minus 1
1 + 20119888
0 lt 119888 le
1
32
120582
2(0 0) = minus (
32119888
20119888 + 1
)
21
32
lt 119888 le
1
12
(41)
Substituting (41) into (23) (33) is obtainedTheorem 2 holds
From (33) 12 le 120588opt(119888ℎ2Δ
2
ℎminusΔ
2ℎ) lt 1holdswith 0 lt 119888 le
112 Therefore fromTheorems 1 and 2 when 0 lt 119888 le 112the smoothing factor of (6) with the distributive relaxation(14) is as
1
2
le 120588opt (119871ℎ)
= max 120588opt (minusΔ ℎ) 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ)
= 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ) lt 1
(42)
4 Conclusions
The smoothing analysis process of the distributive red-black Jacobi point relaxation for solving 2D Stokes flow isanalytically presented Applying (28) the Fouriermodes withthe trigonometric functions for the discrete operator andrelaxation are mapped to rational functions So it is possibleto apply the cylindrical algebraic decomposition function intheMathematica software to realize complex smoothing anal-ysis and the computation process is simplifiedThe analyticalexpressions of the smoothing factor for the distributive red-black Jacobi point relaxation are obtained which is an upperbound for the smoothing rates and is independent of themesh size with the parameter 119888 Obviously it is valuable tounderstand numerical experiments in multigrid method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors were supported by the National Natural ScienceFoundation of China (NSFC) (Grant no 51279071) and theDoctoral Foundation of Ministry of Education of China(Grant no 20135314130002)
References
[1] U Trottenberg C W Oosterlee and A Schuller MultigridAcademic Press San Diego Calif USA 2001
[2] R Wienands and W Joppich Practical Fourier Analysis forMultigrid Methods Chapman amp Hall CRC Press 2005
[3] W L Briggs V E Henson and S McCormick A MultigridTutorial Society for Industrial and Applied Mathematics 2ndedition 2000
[4] W Hackbusch Multigrid Methods and Applications SpringerBerlin Germany 1985
[5] P Wesseling An Introduction to Multigrid Methods JohnWileyamp Sons Chichester UK 1992
[6] K Stuben and U Trottenberg ldquoMultigrid methods fundamen-tal algorithms model problem analysis and applicationsrdquo inMultigridMethods W Hackbusch andU Trottenberg Eds vol960 of Lectwe Notes in Mathematics pp 1ndash176 Springer BerlinGermany 1982
[7] A Brandt and O E Livne 1984 Guide to Multigrid Develop-ment in Multigrid Methods Society for Industrial and AppliedMathematics 2011 httpwwwwisdomweizmannacilsimachiclassicspdf
[8] C W Oosterlee and F J G Lorenz ldquoMultigrid methods for thestokes systemrdquoComputing in Science and Engineering vol 8 no6 Article ID 1717313 pp 34ndash43 2006
[9] A Brandt and N Dinar Multigrid Solutions to Elliptic LlowProblems Institute for Computer Applications in Science andEngineering NASA Langley Research Center 1979
[10] G Wittum ldquoMulti-grid methods for stokes and navier-stokesequationsrdquoNumerische Mathematik vol 54 no 5 pp 543ndash5631989
[11] M Wang and L Chen ldquoMultigrid methods for the Stokesequations using distributive Gauss-Seidel relaxations based onthe least squares commutatorrdquo Journal of Scientific Computingvol 56 no 2 pp 409ndash431 2013
[12] M ur Rehman T Geenen C Vuik G Segal and S PMacLachlan ldquoOn iterative methods for the incompressibleStokes problemrdquo International Journal for Numerical Methodsin Fluids vol 65 no 10 pp 1180ndash1200 2011
[13] C Bacuta P S Vassilevski and S Zhang ldquoA new approachfor solving Stokes systems arising from a distributive relaxationmethodrdquo Numerical Methods for Partial Differential Equationsvol 27 no 4 pp 898ndash914 2011
[14] R Wienands F J Gaspar F J Lisbona and C W OosterleeldquoAn efficient multigrid solver based on distributive smoothingfor poroelasticity equationsrdquo Computing vol 73 no 2 pp 99ndash119 2004
[15] W Liao B Diskin Y Peng and L-S Luo ldquoTextbook-efficiencymultigrid solver for three-dimensional unsteady compressibleNavier-Stokes equationsrdquo Journal of Computational Physics vol227 no 15 pp 7160ndash7177 2008
[16] V Pillwein and S Takacs ldquoA local Fourier convergence analysisof a multigrid method using symbolic computationrdquo Journal ofSymbolic Computation vol 63 pp 1ndash20 2014
Mathematical Problems in Engineering 7
[17] S Takacs All-at-once multigrid methods for optimality systemsarising from optimal control problems [PhD thesis] JohannesKepler University Linz Doctoral Program ComputationalMathematics 2012
[18] V Pillwein and S Takacs ldquoSmoothing analysis of an all-at-oncemultigrid approach for optimal control problems using symboliccomputationrdquo inNumerical and Symbolic Scientific ComputingProgress and Prospects U Langer and P Paule Eds SpringerWien Austria 2011
[19] M Kauers ldquoHow to use cylindrical algebraic decompositionrdquoSeminaire Lotharingien de Combinatoire vol 65 article B65app 1ndash16 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
By using the MATLAB and Mathematica software withcylindrical algebraic decomposition function [19] for 997888119904 =
(119904
1 119904
2) isin (0 12)
2 there is no extreme value for (39) when0 lt 119888 le 132 one of extreme values of (38) is obtained as
119904
lowast
1=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
119904
lowast
2=
radic
64119888
2+ 3 + 40119888 minus 3
48119888 minus 6
(40)
Thus for 997888119904 isin 119878low = [0 12]
2 besides (40) the possibleextreme values of the eigenvalues of matrix (36) are placedon the boundary of 119878low From minus1 le 120582
119896le 1 with 119896 = 1 4
then 0 lt 119888 le 112 Noting that (40) exists with 0 lt 119888 le 132From (38)ndash(40) when 0 lt 119888 le 112 for 997888119904 isin 119878low the maxand min eigenvalues of (36) are yielded as
119878max = 12058234 (0 0) =1 + 4119888
1 + 20119888
119878min =
120582
34(
1
2
1
2
) =
4119888 minus 1
1 + 20119888
0 lt 119888 le
1
32
120582
2(0 0) = minus (
32119888
20119888 + 1
)
21
32
lt 119888 le
1
12
(41)
Substituting (41) into (23) (33) is obtainedTheorem 2 holds
From (33) 12 le 120588opt(119888ℎ2Δ
2
ℎminusΔ
2ℎ) lt 1holdswith 0 lt 119888 le
112 Therefore fromTheorems 1 and 2 when 0 lt 119888 le 112the smoothing factor of (6) with the distributive relaxation(14) is as
1
2
le 120588opt (119871ℎ)
= max 120588opt (minusΔ ℎ) 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ)
= 120588opt (119888ℎ2Δ
2
ℎminus Δ
2ℎ) lt 1
(42)
4 Conclusions
The smoothing analysis process of the distributive red-black Jacobi point relaxation for solving 2D Stokes flow isanalytically presented Applying (28) the Fouriermodes withthe trigonometric functions for the discrete operator andrelaxation are mapped to rational functions So it is possibleto apply the cylindrical algebraic decomposition function intheMathematica software to realize complex smoothing anal-ysis and the computation process is simplifiedThe analyticalexpressions of the smoothing factor for the distributive red-black Jacobi point relaxation are obtained which is an upperbound for the smoothing rates and is independent of themesh size with the parameter 119888 Obviously it is valuable tounderstand numerical experiments in multigrid method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors were supported by the National Natural ScienceFoundation of China (NSFC) (Grant no 51279071) and theDoctoral Foundation of Ministry of Education of China(Grant no 20135314130002)
References
[1] U Trottenberg C W Oosterlee and A Schuller MultigridAcademic Press San Diego Calif USA 2001
[2] R Wienands and W Joppich Practical Fourier Analysis forMultigrid Methods Chapman amp Hall CRC Press 2005
[3] W L Briggs V E Henson and S McCormick A MultigridTutorial Society for Industrial and Applied Mathematics 2ndedition 2000
[4] W Hackbusch Multigrid Methods and Applications SpringerBerlin Germany 1985
[5] P Wesseling An Introduction to Multigrid Methods JohnWileyamp Sons Chichester UK 1992
[6] K Stuben and U Trottenberg ldquoMultigrid methods fundamen-tal algorithms model problem analysis and applicationsrdquo inMultigridMethods W Hackbusch andU Trottenberg Eds vol960 of Lectwe Notes in Mathematics pp 1ndash176 Springer BerlinGermany 1982
[7] A Brandt and O E Livne 1984 Guide to Multigrid Develop-ment in Multigrid Methods Society for Industrial and AppliedMathematics 2011 httpwwwwisdomweizmannacilsimachiclassicspdf
[8] C W Oosterlee and F J G Lorenz ldquoMultigrid methods for thestokes systemrdquoComputing in Science and Engineering vol 8 no6 Article ID 1717313 pp 34ndash43 2006
[9] A Brandt and N Dinar Multigrid Solutions to Elliptic LlowProblems Institute for Computer Applications in Science andEngineering NASA Langley Research Center 1979
[10] G Wittum ldquoMulti-grid methods for stokes and navier-stokesequationsrdquoNumerische Mathematik vol 54 no 5 pp 543ndash5631989
[11] M Wang and L Chen ldquoMultigrid methods for the Stokesequations using distributive Gauss-Seidel relaxations based onthe least squares commutatorrdquo Journal of Scientific Computingvol 56 no 2 pp 409ndash431 2013
[12] M ur Rehman T Geenen C Vuik G Segal and S PMacLachlan ldquoOn iterative methods for the incompressibleStokes problemrdquo International Journal for Numerical Methodsin Fluids vol 65 no 10 pp 1180ndash1200 2011
[13] C Bacuta P S Vassilevski and S Zhang ldquoA new approachfor solving Stokes systems arising from a distributive relaxationmethodrdquo Numerical Methods for Partial Differential Equationsvol 27 no 4 pp 898ndash914 2011
[14] R Wienands F J Gaspar F J Lisbona and C W OosterleeldquoAn efficient multigrid solver based on distributive smoothingfor poroelasticity equationsrdquo Computing vol 73 no 2 pp 99ndash119 2004
[15] W Liao B Diskin Y Peng and L-S Luo ldquoTextbook-efficiencymultigrid solver for three-dimensional unsteady compressibleNavier-Stokes equationsrdquo Journal of Computational Physics vol227 no 15 pp 7160ndash7177 2008
[16] V Pillwein and S Takacs ldquoA local Fourier convergence analysisof a multigrid method using symbolic computationrdquo Journal ofSymbolic Computation vol 63 pp 1ndash20 2014
Mathematical Problems in Engineering 7
[17] S Takacs All-at-once multigrid methods for optimality systemsarising from optimal control problems [PhD thesis] JohannesKepler University Linz Doctoral Program ComputationalMathematics 2012
[18] V Pillwein and S Takacs ldquoSmoothing analysis of an all-at-oncemultigrid approach for optimal control problems using symboliccomputationrdquo inNumerical and Symbolic Scientific ComputingProgress and Prospects U Langer and P Paule Eds SpringerWien Austria 2011
[19] M Kauers ldquoHow to use cylindrical algebraic decompositionrdquoSeminaire Lotharingien de Combinatoire vol 65 article B65app 1ndash16 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
[17] S Takacs All-at-once multigrid methods for optimality systemsarising from optimal control problems [PhD thesis] JohannesKepler University Linz Doctoral Program ComputationalMathematics 2012
[18] V Pillwein and S Takacs ldquoSmoothing analysis of an all-at-oncemultigrid approach for optimal control problems using symboliccomputationrdquo inNumerical and Symbolic Scientific ComputingProgress and Prospects U Langer and P Paule Eds SpringerWien Austria 2011
[19] M Kauers ldquoHow to use cylindrical algebraic decompositionrdquoSeminaire Lotharingien de Combinatoire vol 65 article B65app 1ndash16 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of