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


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


(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


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


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)


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


[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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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



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


119862

ℎ= (

119868

ℎ0 minus120597

ℎ

119909

0 119868

ℎminus120597

ℎ

119910

0 0 minusΔ

ℎ

) (14)


ℎ From


119871

ℎ119862

ℎ= (

minusΔ

ℎ0 0

0 minusΔ

ℎ0

120597

ℎ

119909120597

ℎ

119910119888ℎ

2Δ

2

ℎminus Δ

2ℎ

) (15)


2ℎare

Δ

2

ℎ=

1

ℎ

4

[

[

[

[

[

[

[

[

[

1

2 minus8 2


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)


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


1) minus exp (119894120579

1)


2))

=

1

ℎ

2(4 minus 2 cos 120579

1minus 2 cos 120579

2)

(18)

120597

ℎ

119909(

997888

120579) =

1

2ℎ


1)) =

1

ℎ

119894 sin 1205791

(19)

120597

ℎ

119910(

997888

120579) =

1

2ℎ


2)) =

1

ℎ

119894 sin 1205792

(20)


119876

2ℎ

ℎ

1003816

1003816

1003816

1003816

10038161198652ℎ(997888120579 )=

_119876

2ℎ


4times4

(21)

where_119876

2ℎ


119876

2ℎ



120588 (119899119863


(120588(

_119876

2ℎ

ℎ(

_119878 ℎ(

997888

120579 120596))

119899

))

1119899

(22)


_119878 ℎ(

997888






120596opt =2


120588opt =119878max minus 119878min


(23)



2ℎ

ℎ

_119878 ℎ(

997888


120588 (119899 119871

ℎ) = max 120588 (119899 minusΔ

ℎ) 120588 (119899 119888ℎ

2Δ

2

ℎminus Δ

2ℎ) (24)






119878

119877119861

ℎ

1003816

1003816

1003816

1003816

10038161198652ℎ(997888120579 )=

_119878

119877119861

ℎ(

997888


(25)

where_119878

119877119861

ℎ(

997888



_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)



997888

120579

997888120572


(00)]

ℎ


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


997888

119904 = (119904

1 119904

2) isin 119878low = [0 12]

2



120596

119900119901119905=

16

15

120588

119900119901119905=

1

5

(29)


ℎ= minusΔ



_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)


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)


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)


Next 120588(119899 119888ℎ2Δ2ℎminus Δ




2ℎwith 119888 gt




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)


119863

ℎ= 119888ℎ

2Δ

2

ℎminus Δ

2ℎ

(34)


119863

ℎ(

997888

120579) =

1

ℎ

2[4119888 (2 minus cos 120579

1minus cos 120579

2)

2

+ sin21205791+ sin2120579

2]

(35)


_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


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)


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)



(119904

1 119904

2) isin (0 12)


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]


119896le 1 with 119896 = 1 4


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)


From (33) 12 le 120588opt(119888ℎ2Δ

2

ℎminusΔ



1

2

le 120588opt (119871ℎ)


2

ℎminus Δ

2ℎ)

= 120588opt (119888ℎ2Δ

2

ℎminus Δ

2ℎ) lt 1

(42)

4 Conclusions




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











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


119862

ℎ= (

119868

ℎ0 minus120597

ℎ

119909

0 119868

ℎminus120597

ℎ

119910

0 0 minusΔ

ℎ

) (14)


ℎ From


119871

ℎ119862

ℎ= (

minusΔ

ℎ0 0

0 minusΔ

ℎ0

120597

ℎ

119909120597

ℎ

119910119888ℎ

2Δ

2

ℎminus Δ

2ℎ

) (15)


2ℎare

Δ

2

ℎ=

1

ℎ

4

[

[

[

[

[

[

[

[

[

1

2 minus8 2


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)


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


1) minus exp (119894120579

1)


2))

=

1

ℎ

2(4 minus 2 cos 120579

1minus 2 cos 120579

2)

(18)

120597

ℎ

119909(

997888

120579) =

1

2ℎ


1)) =

1

ℎ

119894 sin 1205791

(19)

120597

ℎ

119910(

997888

120579) =

1

2ℎ


2)) =

1

ℎ

119894 sin 1205792

(20)


119876

2ℎ

ℎ

1003816

1003816

1003816

1003816

10038161198652ℎ(997888120579 )=

_119876

2ℎ


4times4

(21)

where_119876

2ℎ


119876

2ℎ



120588 (119899119863


(120588(

_119876

2ℎ

ℎ(

_119878 ℎ(

997888

120579 120596))

119899

))

1119899

(22)


_119878 ℎ(

997888






120596opt =2


120588opt =119878max minus 119878min


(23)



2ℎ

ℎ

_119878 ℎ(

997888


120588 (119899 119871

ℎ) = max 120588 (119899 minusΔ

ℎ) 120588 (119899 119888ℎ

2Δ

2

ℎminus Δ

2ℎ) (24)






119878

119877119861

ℎ

1003816

1003816

1003816

1003816

10038161198652ℎ(997888120579 )=

_119878

119877119861

ℎ(

997888


(25)

where_119878

119877119861

ℎ(

997888



_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)



997888

120579

997888120572


(00)]

ℎ


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


997888

119904 = (119904

1 119904

2) isin 119878low = [0 12]

2



120596

119900119901119905=

16

15

120588

119900119901119905=

1

5

(29)


ℎ= minusΔ



_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)


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)


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)


Next 120588(119899 119888ℎ2Δ2ℎminus Δ




2ℎwith 119888 gt




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)


119863

ℎ= 119888ℎ

2Δ

2

ℎminus Δ

2ℎ

(34)


119863

ℎ(

997888

120579) =

1

ℎ

2[4119888 (2 minus cos 120579

1minus cos 120579

2)

2

+ sin21205791+ sin2120579

2]

(35)


_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


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)


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)



(119904

1 119904

2) isin (0 12)


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]


119896le 1 with 119896 = 1 4


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)


From (33) 12 le 120588opt(119888ℎ2Δ

2

ℎminusΔ



1

2

le 120588opt (119871ℎ)


2

ℎminus Δ

2ℎ)

= 120588opt (119888ℎ2Δ

2

ℎminus Δ

2ℎ) lt 1

(42)

4 Conclusions




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119878

119877119861

ℎ

1003816

1003816

1003816

1003816

10038161198652ℎ(997888120579 )=

_119878

119877119861

ℎ(

997888


(25)

where_119878

119877119861

ℎ(

997888



_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)



997888

120579

997888120572


(00)]

ℎ


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


997888

119904 = (119904

1 119904

2) isin 119878low = [0 12]

2



120596

119900119901119905=

16

15

120588

119900119901119905=

1

5

(29)


ℎ= minusΔ



_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)


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)


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)


Next 120588(119899 119888ℎ2Δ2ℎminus Δ




2ℎwith 119888 gt




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)


119863

ℎ= 119888ℎ

2Δ

2

ℎminus Δ

2ℎ

(34)


119863

ℎ(

997888

120579) =

1

ℎ

2[4119888 (2 minus cos 120579

1minus cos 120579

2)

2

+ sin21205791+ sin2120579

2]

(35)


_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


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)


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)



(119904

1 119904

2) isin (0 12)


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]


119896le 1 with 119896 = 1 4


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)


From (33) 12 le 120588opt(119888ℎ2Δ

2

ℎminusΔ



1

2

le 120588opt (119871ℎ)


2

ℎminus Δ

2ℎ)

= 120588opt (119888ℎ2Δ

2

ℎminus Δ

2ℎ) lt 1

(42)

4 Conclusions




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











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)


119863

ℎ= 119888ℎ

2Δ

2

ℎminus Δ

2ℎ

(34)


119863

ℎ(

997888

120579) =

1

ℎ

2[4119888 (2 minus cos 120579

1minus cos 120579

2)

2

+ sin21205791+ sin2120579

2]

(35)


_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


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)


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)



(119904

1 119904

2) isin (0 12)


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]


119896le 1 with 119896 = 1 4


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)


From (33) 12 le 120588opt(119888ℎ2Δ

2

ℎminusΔ



1

2

le 120588opt (119871ℎ)


2

ℎminus Δ

2ℎ)

= 120588opt (119888ℎ2Δ

2

ℎminus Δ

2ℎ) lt 1

(42)

4 Conclusions




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119904

1 119904

2) isin (0 12)


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]


119896le 1 with 119896 = 1 4


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)


From (33) 12 le 120588opt(119888ℎ2Δ

2

ℎminusΔ



1

2

le 120588opt (119871ℎ)


2

ℎminus Δ

2ℎ)

= 120588opt (119888ℎ2Δ

2

ℎminus Δ

2ℎ) lt 1

(42)

4 Conclusions




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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