Research ArticleA Multilevel Finite Element Variational MultiscaleMethod for Incompressible Navier-Stokes EquationsBased on Two Local Gauss Integrations

Yamiao Zhang Biwu Huang Jiazhong Zhang and Zexia Zhang

School of Energy and Power Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Jiazhong Zhang jzzhangmailxjtueducn

Received 22 May 2017 Revised 23 July 2017 Accepted 5 September 2017 Published 26 November 2017

Academic Editor Xesus Nogueira

Copyright copy 2017 Yamiao Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A multilevel finite element variational multiscale method is proposed and applied to the numerical simulation of incompressibleNavier-Stokes equations This method combines the finite element variational multiscale method based on two local Gaussintegrations with the multilevel discretization using Newton correction on each stepThemain idea of the multilevel finite elementvariational multiscale method is that the equations are first solved on a single coarse grid by finite element variational multiscalemethod then finite element variational multiscale approximations are generated on a succession of refined grids by solving alinearized problemMoreover the stability analysis and error estimate of themultilevel finite element variationalmultiscalemethodare given Finally some numerical examples are presented to support the theoretical analysis and to check the efficiency of theproposed method The results show that the multilevel finite element variational multiscale method is more efficient than the one-level finite element variational multiscale method and for an appropriate choice of meshes the multilevel finite element variationalmultiscale method is not only time-saving but also highly accurate

1 Introduction

The finite element method is one of the most general tech-niques for incompressible Navier-Stokes equations How-ever in numerical simulations of the incompressible Navier-Stokes equations at higher Reynolds number the standardGalerkin finite element method is often failed due to thedomination of convection term [1ndash3] This defect can beovercome to some extent by a variety of stabilizing tech-niques including Galerkin least square (GLS) method [45] streamline-upwind Petrov-Galerkin (SUPG) method [6]defect-correctionmethod [7 8] local projection stabilization(LPS) method [9] discontinuous Galerkin method [10ndash12]and variational multiscale (VMS) method Among them theVMS method is one of some successful stabilized methodsthat was originally introduced and developed by Hughes etal [13ndash16] as a technique in the construction of subgrid-scale model in large eddy simulation for incompressible andcompressible flows Subsequently it was also used as a subgrid

stabilized technique to improve the accuracy of the solutionwhich was obtained by the Galerkin finite element approx-imation for the convection dominated problems [17ndash19] Inthe VMS method the governing equations are rewritten as acoupled systemwith two types of scales large scales and smallscales Furthermore according to Collis [20] the governingequations can also be rewritten as a coupled system withthree types of scales resolved large scales resolved smallscales and unresolved scales As solving the above coupledsystem by finite element method the traditional Galerkinapproximation can be improved by adding an asymptoticallyconsistent artificial diffusion term on the subgrid scalesUnder the circumstance it is necessary to define an appropri-ate operator function spaces and some additional dependentvariables Recently a finite element VMS method based ontwo local Gauss integrations was proposed and developedby the authors of [21ndash25] This finite element VMS methodis simpler than the common finite element VMS method

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 4917054 13 pageshttpsdoiorg10115520174917054

2 Mathematical Problems in Engineering

which does not need additional dependent variables and theprojection operator but keeps the same efficiency

However when solving nonlinear partial differentialequation directly using this finite element VMS methodbased on two local Gauss integrations it needs too muchcomputing time especially for conditions requiring a rela-tively fine-grid discretization A treatment for this problem isthe multilevel discretizationThemultilevel method is a well-established and efficient method for solving nonlinear partialdifferential equations Some classical multilevel methods canbe found in [26ndash32] The main idea of the multilevel methodis to solve the nonlinear equations first on a coarse gridand then solve a linearized problem corresponding to thisnonlinear governing equation on a succession of refinedgrids Hence the multilevel method can save much comput-ing time compared with the one-level method Neverthelessthere is a shortcoming for these classical multilevel methodsin discretization of the convection dominated Navier-Stokesequations that is the nonlinear Navier-Stokes equationsneed to be first solved on the coarsest grid by standardGalerkin finite element method This may lead to numericaloscillations in the coarse grid approximation and transmitremarkable error to the subsequent refined grid numericalapproximation and hence cannot yield a correct approxima-tion

In this paper by combining the multilevel discretizationstrategy with the finite element VMS method based ontwo local Gauss integrations we propose a multilevel finiteelement VMS method for the convection dominated Navier-Stokes equations In this method the nonlinear governingequations are first solved on a single coarse grid by finiteelement VMSmethod based on two local Gauss integrationsand then finite element VMS approximations are generatedon a succession of refined grids using Newton correctionon each step by solving a linearized problem correspondingto the nonlinear governing equations Compared with theclassical multilevel methods the proposed method can solvethe convection dominated Navier-Stokes equations moreaccurately While compared with the one-level finite elementVMS method the proposed method can save plenty ofcomputing time in fine-grid discretization

It is noted that a multilevel VMS method has beenreported in [33] However there are two essential differencesbetween the proposed method and the one studied in [33]First the proposedVMSmethod in this paper is based on twolocal Gauss integrations while the one in [33] is the residual-based VMSmethod Second the purpose and the refinementstrategy are different The meshes in the proposed methodare global refinement in domains with a Newton correctionat each level to increase the efficiency of numerical schemebased on the finite element discretization while the meshesin [33] are local adaptive refinement in domains via a clearhierarchy multilevel ℎ119901-formulation for further developingthe finite cell methodology Moreover the two-level case ofthe proposed method is also different from those in [34 35]Compared with the two-level projection-based VMSmethodin [34] the proposed method in this paper does not needadditional dependent variables and the projection operatorso it is simpler Compared with the two-level VMSmethod in

[35] the proposed method uses Newton correction to obtaina better approximation than the Stokes correction used in[35] And what is more the theoretical analysis and resultsof [35] are limited to the small data condition while the con-dition in this paper is no longer needed so that the proposedmethod has a wider effective range than the method in [35]

This paper is organized as follows In Section 2 theNavier-Stokes equations notations and well-known resultsused throughout this paper are given In Section 3 the finiteelement VMS method is introduced Then the multilevelfinite element VMSmethod is presented and the correspond-ing stability and convergence error are analyzed in Section 4In Section 5 three numerical examples are given to verifythe theoretical analysis Finally conclusions are drawn inSection 6

2 Preliminaries

In this paper the following steady Navier-Stokes equationswill be considered

minus]Δ119906 + (119906 sdot nabla) 119906 + nabla119901 = 119891 in Ωdiv 119906 = 0 in Ω119906 = 0 on 120597Ω

(1)

where Ω is a bounded domain in 1198772 with a Lipschitzcontinuous boundary 120597Ω 119906 = (1199061 1199062) the fluid velocity 119901the pressure 119891 the prescribed body force and ] gt 0 thekinematic viscosity which represents the inverse of Reynoldsnumber Re For the mathematical setting of problem (1) thefollowing Hilbert spaces are needed

119883 = 11986710 (Ω)2

119872 = 11987120 (Ω) = 119902 isin 1198712 (Ω) intΩ119902 119889119909 = 0

(2)

The norm and the inner product in 1198712(Ω) are denoted by sdot0and (sdot sdot) respectively sdot 119896 is the norm of the Sobolev space119867119896(Ω) or 119867119896(Ω)2 119896 = 1 2 Using the Poincare inequalitythe norm in space 119883 equipped with nabla sdot 0 and sdot 1 canbe considered to be equivalent We define the continuousbilinear forms 119886(sdot sdot) and 119889(sdot sdot) and the trilinear term 119887(sdot sdot sdot)by

119886 (119906 V) = (nabla119906 nablaV) forall119906 V isin 119883119889 (V 119902) = (119902 divV) forall (V 119902) isin (119883 times119872)

119887 (119906 V 119908) = ((119906 sdot nabla) V 119908) + 12 ((nabla sdot 119906) V 119908)

= 12 ((119906 sdot nabla) V 119908) minus12 ((119906 sdot nabla)119908 V)

forall119906 V 119908 isin 119883

(3)

Mathematical Problems in Engineering 3

respectively The trilinear term 119887(sdot sdot sdot) satisfies the followingestimates [36]

119887 (119906 V V) = 0 forall119906 V 119908 isin 119883 (4)

|119887 (119906 V 119908)| le 119873 nabla1199060 nablaV0 nabla1199080 forall119906 V 119908 isin 119883 (5)

where 119873 is a positive constant depending only on ΩSubsequently 119888 and 119896 (with a subscript) will denote a positiveconstant which may stand for different values at its differentoccurrences respectively

With the above notations the variational formulation ofproblem (1) reads as follows find (119906 119901) isin 119883times119872 for all (V 119902) isin119883 times119872 satisfying

]119886 (119906 V) minus 119889 (V 119901) + 119889 (119906 119902) + 119887 (119906 119906 V) = (119891 V) (6)

Thenonsingular solution of (1) is defined as follows [37]119906is a nonsingular solution of (1) if there is a constant120590(119906 ]) gt 0such that

infVisin119881

sup119908isin119881

]119886 (V 119908) + 119887 (119906 V 119908) + 119887 (V 119906 119908)V1 1199081 ge 120590 (119906 ]) (7)

Here 119881 = V isin 119883 (nabla sdot V 119902) = 0 forall119902 isin 119872 If 119906 is a nonsin-gular solution of the Navier-Stokes equations 119906 satisfies theestimate [37 38]

1199061 le ]minus1 10038171003817100381710038171198911003817100381710038171003817minus1 10038171003817100381710038171198911003817100381710038171003817minus1 = supVisin119883V =0

1003816100381610038161003816(119891 V)1003816100381610038161003816V1 (8)

Lemma 1 is also needed more details of it can be found in[39 40]

Lemma 1 Suppose that

inf119902isin119872

supVisin119883

119889 (V 119902)100381710038171003817100381711990210038171003817100381710038170 V1 ge 120573 gt 0 (9)

and that the bilinear form 119888(119906 V) satisfies

inf119906isin119881

supVisin119881

119888 (119906 V)1199061 V1 ge 1205720 gt 0 (10)

Then the continuous bilinear form 119865 (119883119872) times (119883119872) rarr 119877defined by

119865 [(119906 119901) (V 119902)] fl 119888 (119906 V) minus 119889 (V 119901) + 119889 (119906 119902) (11)

satisfies the inf-sup condition on the product space (119883119872) times(119883119872) which means that there exists a constant 120574lowast =120574lowast(1205720 120573 119888(sdot sdot)) gt 0 such that

inf(119906119901)isin(119883119872)

sup(V119902)isin(119883119872)

119865 [(119906 119901) (V 119902)]lsaquo (119906 119901) lsaquolsaquo (V 119902) lsaquo ge 120574

lowast (12)

For the finite element discretization let 119870ℎ be the reg-ular triangulations of the domain Ω The mesh defined by

ℎ = maxΩ119890isin119870ℎdiam(Ω119890) is a real-positive parameter tendingto 0 We choose the conforming velocity-pressure finiteelement space (119883ℎ119872ℎ) sub (119883119872) and consider the Taylor-Hood elements [41]

119883ℎ= Vℎ isin 1198620 (Ω)2 cap 119883 Vℎ1003816100381610038161003816Ω119890 isin 1198752 (Ω119890)2 forallΩ119890 isin 119870ℎ 119872ℎ= 119902ℎ isin 1198620 (Ω) cap119872 119902ℎ1003816100381610038161003816Ω119890 isin 1198751 (Ω119890) forallΩ119890 isin 119870ℎ

(13)

where 119875119895(Ω119890) (119895 = 1 2) is the space of 119895th order polynomialson Ω119890 The Taylor-Hood elements on both triangles andrectangles are proved to satisfy the following discrete inf-supcondition for compatibility of the velocity-pressure spaces

inf119902ℎisin119872ℎ

supVℎisin119883ℎ

119889 (Vℎ 119902ℎ)1003817100381710038171003817119902ℎ10038171003817100381710038170 1003817100381710038171003817Vℎ10038171003817100381710038171 ge 120573 gt 0 (14)

3 Finite Element VariationalMultiscale Method

The finite element VMS method proposed by John and Kaya[19 42] for the steady Navier-Stokes equations reads find(119906ℎ 119901ℎ) isin (119883ℎ119872ℎ) and 119892ℎ isin 119871ℎ for all (Vℎ 119902ℎ) isin (119883ℎ119872ℎ)satisfying

(] + 120572) 119886 (119906ℎ Vℎ) minus 120572 (119892ℎ nablaVℎ) + 119887 (119906ℎ 119906ℎ Vℎ)minus 119889 (Vℎ 119901ℎ) + 119889 (119906ℎ 119902ℎ) = (119891 Vℎ)

(119892ℎ minus nabla119906ℎ 119897ℎ) = 0 forall119897ℎ isin 119871ℎ(15)

This system is determined by the choices of 119871ℎ and 120572 Thestabilization parameter 0 lt 120572 lt 1 acts only on the small scaleswhich can be chosen as the scale of 119874(ℎ119904) (119904 ge 1) in order tostabilize the convective term appropriately

An orthogonal projection operator is defined as Π 119871 rarr 119871ℎ which is 1198712 orthogonal projection with the followingproperties

(Πnabla119906 V) = (nabla119906 V) forall119906 isin 119883 V isin 119871ℎ (H1)ΠnablaV0 le 119862 nablaV0 forallV isin 119883 (H2)

(119868 minus Π)nablaV0 le 119862 nablaV0 forallV isin 119883 (H3)(119868 minus Π)nablaV0 le 119862ℎ V2 forallV isin 1198672 (Ω)2times2 cap 119883 (H4)

where 119862 is a positive constant 119871 = 1198712(Ω) and 119871ℎ sub 119871 (fordetails refer to [18])

Based on the properties of the orthogonal projectionoperator Π the finite element variational multiscale method

4 Mathematical Problems in Engineering

proposed by Zheng et al [21] is as follows find (119906ℎ 119901ℎ) isin(119883ℎ119872ℎ) for all (Vℎ 119902ℎ) isin (119883ℎ119872ℎ) satisfying]119886 (119906ℎ Vℎ) minus 119889 (Vℎ 119901ℎ) + 119889 (119906ℎ 119902ℎ) + 119887 (119906ℎ 119906ℎ Vℎ)

+ 119866 (119906ℎ Vℎ) = (119891 Vℎ) (16)

Here

119866 (119906ℎ Vℎ) = 120572 ((119868 minus Π)nabla119906ℎ (119868 minus Π)nablaVℎ)= 120572 sumΩ119890isin120591ℎ

intΩ119890119896nabla119906ℎnablaVℎ 119889119909 minus int

Ω1198901nabla119906ℎnablaVℎ 119889119909 (17)

where intΩ119890119894(sdot) 119889119909 denotes an appropriate Gauss integral over

Ω119890 which is exact for polynomials of degree 119894 119894 = 1 119896 (119896 ge2)

From [24 35] problem (16) has the following stability anderror estimate

Theorem2 Suppose the finite element space (119883ℎ119872ℎ) satisfiesthe LBB condition (14) Let (119906 119901) isin (1198672(Ω)2 cap1198831198671(Ω)cap119872)be a nonsingular solution of (6)Then the approximate solution(119906ℎ 119901ℎ) given by (16) satisfies

1003817100381710038171003817119906ℎ10038171003817100381710038171 le ]minus1 10038171003817100381710038171198911003817100381710038171003817minus1 (18)

and the following error estimate

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171 + 1003817100381710038171003817119901 minus 119901ℎ10038171003817100381710038170 le 119888 (120572 + ℎ) (19)

The natural energy norm given by

lsaquo (V 119902) lsaquo fl (V21 + 1003817100381710038171003817119902100381710038171003817100381720)12 (20)

will be used in the next section for the theoretical analysis ofthe proposed multilevel finite element VMS method

4 Multilevel Finite Element VMS Method

In this section the multilevel finite element VMSmethod for(16) will be presented The stability and error estimate of itwill be given With mesh widths ℎ1 gt ℎ2 gt sdot sdot sdot gt ℎ119869 the finiteelement space pairs (119883ℎ119895 119872ℎ119895) 119895 = 1 119869 satisfy that

(119883ℎ1 119872ℎ1) sub (119883ℎ2 119872ℎ2) sub sdot sdot sdot sub (119883ℎ119869 119872ℎ119869) (21)

And (119906ℎ119895 119901ℎ119895) isin (119883ℎ119895 119872ℎ119895) is the solution obtained by theproposedmultilevel finite element VMSmethodThe generalsteps of the proposed multilevel finite element VMS methodfor approximating the solution of (16) can be summarized asfollows

Step 1 Solve the following nonlinear system on the coarsestmesh ℎ1 Find (119906ℎ1 119901ℎ1) isin (119883ℎ1 119872ℎ1) for all (V 119902) isin (119883ℎ1 119872ℎ1) satisfying

]119886 (119906ℎ1 V) minus 119889 (V 119901ℎ1) + 119889 (119906ℎ1 119902) + 119866 (119906ℎ1 V)+ 119887 (119906ℎ1 119906ℎ1 V) = (119891 V)

(22)

Step 2 Update on fine mesh with Newton correction at eachlevel

Give (119906ℎ119895minus1 119901ℎ119895minus1) and find (119906ℎ119895 119901ℎ119895) isin (119883ℎ119895 119872ℎ119895) suchthat

]119886 (119906ℎ119895 V) minus 119889 (V 119901ℎ119895) + 119889 (119906ℎ119895 119902) + 119887 (119906ℎ119895 119906ℎ119895minus1 V)+ 119887 (119906ℎ119895minus1 119906ℎ119895 V) + 119866 (119906ℎ119895 V)

= (119891 V) + 119887 (119906ℎ119895minus1 119906ℎ119895minus1 V)(23)

for all (V 119902) isin (119883ℎ119895 119872ℎ119895) with 119895 = 2 119869Obviously the solution (119906ℎ1 119901ℎ1) after first step of the

multilevel finite element VMS method satisfies the resultsin Theorem 2 Before giving the stability analysis and errorestimates the continuous bilinear form B (119883119872) times(119883119872) rarr 119877 is given by

B [(119906 119901) (V 119902)] fl ]119886 (119906 V) minus 119889 (V 119901) + 119889 (119906 119902)+ 119887 (119906ℎ119895 119906 V) + 119887 (119906 119906ℎ119895 V)

(24)

Here we assume that 119906 is a nonsingular solution of (1) and119906ℎ119895 is close enough to 119906 Thus by Lemma 1 and (7) thecontinuous bilinear form B satisfies the inf-sup conditionsnamely there is a constant 120574 gt 0 such that [40]

inf(119906119901)isin(119883ℎ119895 119872ℎ119895 )

sup(V119902)isin(119883ℎ119895 119872ℎ119895 )

B [(119906 119901) (V 119902)]lsaquo (119906 119901) lsaquolsaquo (V 119902) lsaquo ge 120574

119895 = 1 2 119869(25)

The stability analysis and error estimate of the proposedmultilevel finite element method will be discussed in Theo-rems 3 and 4

Theorem 3 Suppose 119883ℎ1 sub 119883ℎ2 sub 119883 119872ℎ1 sub 119872ℎ2 sub 119872 andeach pair (119883ℎ1 119872ℎ1) (119883ℎ2 119872ℎ2) satisfies the LBB condition(14) Let (119906 119901) isin (1198672(Ω)2 cap 1198831198671(Ω) cap 119872) be a nonsingularsolution of (6) Suppose 0 lt ℎ2 lt ℎ1 and ℎ1 is sufficiently small

Mathematical Problems in Engineering 5

Then the approximate solution (119906ℎ2 119901ℎ2) isin (119883ℎ2 119872ℎ2) given bythe multilevel finite element method (23) with 119895 = 2 satisfies

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 11989612 (26)

Moreover if there holds the scaled relation ℎ2 = 119874(ℎ21) it hasthe following error estimate

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le 11989622 (120572 + ℎ2) (27)

Proof Using (5) (18) (23) (25) and (H3) gives

120574lsaquo (119906ℎ2 119901ℎ2) lsaquo le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ2 119901ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ1 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2100381710038171003817100381711989110038171003817100381710038172minus1 + 119862120572lsaquo (119906ℎ2 119901ℎ2) lsaquo

(28)

Choosing 120572 = 119874(ℎ1199042) for sufficiently small ℎ2 120575 = 120574minus119862120572 gt 0Then we have

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2

100381710038171003817100381711989110038171003817100381710038172minus1) (29)

Letting 11989612 = (119891minus1 + (119873]2)1198912minus1)120575minus1 then (26) holdsSubtracting (23) with 119895 = 2 from (6) yields

B [(119906 minus 119906ℎ2 119906 minus 119901ℎ2) (V 119902)]= 119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)

(30)

The right terms in (30) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ1 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ1)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ2 V)10038161003816100381610038161003816 le 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ2100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(31)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ2 119872ℎ2) as interpolation of (119906 119901) into(119883ℎ2 119872ℎ2) we have

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) lsaquo le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

B [(119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 119888 (120572 + ℎ1)2 + 119888120572

(32)

Then together with the relation ℎ2 = 119874(ℎ21) denoting theconstant as 11989622 yields (27)Theorem 4 Suppose 119883ℎ1 sub 119883ℎ2 sub sdot sdot sdot sub 119883ℎ119869 sub 119883 119872ℎ1 sub119872ℎ2 sub sdot sdot sdot sub 119872ℎ119869 sub 119872 and each pair (119883ℎ1 119872ℎ1) (119883ℎ119869 119872ℎ119869) satisfies the LBB condition Let (119906 119901) isin (1198672(Ω)2 cap1198831198671(Ω) cap 119872) be a nonsingular solution of (6) If ℎ119895 gt 0is sufficient small then the approximate solution (119906ℎ119895 119901ℎ119895) isin(119883ℎ119895 119872ℎ119895) given by the multilevel finite element method (23)satisfies

lsaquo (119906ℎ119895 119901ℎ119895) lsaquo le 1198961119895 (33)

Moreover if there holds the scaled relation ℎ119895 = 119874(ℎ2119895minus1) 119895 =2 3 119869 it has the following error estimate

lsaquo (119906 minus 119906ℎ119895 119901 minus 119901ℎ119895) lsaquo1 le 1198962119895 (120572 + ℎ119895) (34)

with 119895 = 2 119869Proof This theorem will be proved by the induction methodFromTheorem 3 it is obvious that it holds for 119895 = 2 AssumeTheorem 4 is true for 119895 = 119898 we need to prove that it is alsotrue for 119895 = 119898 + 1 with 1 lt 119898 le 119869 minus 1

Using (5) (23) (25) and (H3) we obtain

120574lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ119898+1 119901ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ119898 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

6 Mathematical Problems in Engineering

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1119898100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898 + 119862120572lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo(35)

Choosing 120572 = 119874(ℎ119904119898+1) for sufficiently small ℎ119898+1 120575 = 120574 minus119862120572 gt 0 Then we have

lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898) (36)

Letting 1198961119895 = (119891minus1 + 11987311989621119898)120575minus1 then (33) holds for 119895 =119898 + 1

Next the convergence error of the multilevel finite ele-ment method is analyzed Subtracting (23) with 119895 = 119898 + 1from (6) yields

B [(119906 minus 119906ℎ119898+1 119906 minus 119901ℎ119898+1) (V 119902)]= 119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)

(37)

The right terms in (37) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ119898 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ119898)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ119898+1 V)10038161003816100381610038161003816 = 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ119898+1100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(38)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ119898+1 119872ℎ119898+1) as interpolation of (119906 119901)into (119883ℎ119898+1 119872ℎ119898+1) we have

lsaquo (119906 minus 119906ℎ119898+1 119901 minus 119901ℎ119898+1) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) lsaquo le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

B [(119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 119888 (120572 + ℎ119898)2 + 119888120572

(39)

Then together with the relation ℎ119898+1 = 119874(ℎ2119898) denoting theconstant as 1198962119898+1 yields (34) for 119895 = 119898 + 1 Thus Theorem 4follows from the induction principle

5 Numerical Examples

In this section three numerical examples are given thefirst one is an exact solution problem the second one isthe lid-driven cavity problem and the third one is thebackward facing step problem The mentioned methods inthis paper are all implemented using Freefem++ [43] and theconvergence tolerance is set equal to 10minus6 In the numericalexamples the triangular mesh with the Taylor-Hood elementpair is used for the finite element discretization For the sakeof simplicity the one-level and the proposed multilevel finiteelement VMS methods are written as 119895-level finite elementVMS method in the numerical examples The nonlinearsystems in these methods are all linearized by the Oseeniterative method (see [44 45])

51 Problem with an Exact Solution This problem is a steadyflow of incompressible viscous Newtonian fluid in boundeddomain Ω = (0 1) times (0 1) with 119906 = (0 0) on the boundariesIt has the following exact solution

1199061 = 101199092 (119909 minus 1)2 119910 (119910 minus 1) (2119910 minus 1) 1199062 = minus10119909 (119909 minus 1) (2119909 minus 1) 1199102 (119910 minus 1)2 119901 = 10 (2119909 minus 1) (2119910 minus 1)

(40)

where 119891 = (1198911 1198912) is determined by (1) The Taylor-Hoodelements predict a convergence rate of 119874(ℎ2) in the energynorm (ie 1198671-norm for the velocity and 1198712-norm for thepressure) so the eddy viscosity parameter is chosen as 120572 =01ℎ2119895 for multilevel finite element VMS method which is thesame as that used in [21]

First ] = 01 is considered The meshes for the one-levelfinite element VMS method are chosen as ℎ119869 = [182 11021122 1142 1162] and these meshes are also the finestmeshes for the multilevel finite element VMS method Thechoices for the coarse meshes of the multilevel finite elementVMS method are very important If the coarse meshes arenot fine enough the optimal convergence orders of velocityand pressure can not be reached According to Theorems 3and 4 the coarse meshes ℎ119895minus1 and the fine mesh ℎ119895 should bechosen as ℎ119895 sim ℎ2119895minus1 119895 = 2 3 119869 which ensure optimalaccuracy of the approximate solution Here we consider themultilevel finite element VMS method for 119895 = 2 3 4 andmeshes are chosen as ℎ119895 = ℎ2119895minus1 respectively The numericalresults using the one-level and the proposed multilevel finiteelement VMSmethods are shown in Figure 1 which indicatesthat the proposed multilevel finite element VMS method canreach the convergence order of119874(ℎ2) for velocity in1198671-normand pressure in 1198712-norm and the optimal convergence orderof119874(ℎ3) for velocity in the 1198712-normThese numerical resultsare in agreement with the theoretical predictions Moreover

Mathematical Problems in Engineering 7

Table 1 Comparison of these 119895-level finite element VMS methods

Method 1ℎ CPU(s)

100381710038171003817100381710038171003817119906 minus 119906ℎ1198951003817100381710038171003817100381710038170

1199060100381710038171003817100381710038171003817119906 minus 119906ℎ119895

10038171003817100381710038171003817100381711199061

100381710038171003817100381710038171003817119901 minus 119901ℎ1198951003817100381710038171003817100381710038170100381710038171003817100381711990110038171003817100381710038170

1-level 128 198614 137319119890 minus 005 0000776894 472788119890 minus 0052-level 64-128 71389 102514119890 minus 005 0000351524 472788119890 minus 0053-level 32-64-128 37617 211498119890 minus 005 0000342189 472788119890 minus 0054-level 16-32-64-128 29495 0000110742 0000526664 47279119890 minus 0051-level 144 25086 85448119890 minus 006 0000554574 373553119890 minus 0052-level 72-144 92251 550229119890 minus 006 0000226809 373553119890 minus 0053-level 36-72-144 45212 930361119890 minus 006 0000206441 373553119890 minus 0054-level 18-36-72-144 35437 43288119890 minus 005 0000275735 373553119890 minus 0051-level 256 925036 902168119890 minus 007 0000107772 118393119890 minus 0052-level 128-256 307479 285225119890 minus 007 466203119890 minus 005 118403119890 minus 0053-level 64-128-256 141673 202856119890 minus 007 45437119890 minus 005 118383119890 minus 0054-level 32-64-128-256 99477 146991119890 minus 007 450338119890 minus 005 118391119890 minus 005

both the proposedmultilevel finite elementVMSmethod andthe one-level finite element VMS method obtain nearly thesame approximation results but the former significantly savescomputing time compared with the later

Then ] = 00001 is considered The mesh CPU-time the1198712-error of the velocity1198671-error of the velocity and 1198712-errorof the pressure for comparison of these 119895-level (119895 = 1 2 3 4)finite element VMS methods are tabulated in Table 1 Here(119906ℎ119895 119901ℎ119895) refers to the approximation solution obtained by 119895-level finite elementVMSmethod Table 1 indicates that 119895-level(119895 ge 2) finite element VMS method saves much computingtime compared with 119895 minus 1-level finite element VMS methodat the same fine mesh and the finer the mesh becomes thesmaller the error achieves Furthermore when scrutinizingthe error estimates of velocity and the pressure in Table 1 itis easily observed that multilevel (119895 = 2 3 4) finite elementVMS method for the velocity and the pressure is better thanone-level (119895 = 1) finite element VMS method

52 Lid-Driven Cavity Problem Lid-driven cavity problemoften serves as a standard benchmark problem for incom-pressible Navier-Stokes equations because the benchmarkdata is available for comparison In this problem the compu-tation is carried out in the region Ω = (119909 119910) | 0 lt 119909 119910 lt 1The velocity is 119906 = (1 0) on the top side (119909 1) 0 lt 119909 lt 1and 119906 = (0 0) on the rest of the boundary

We compute the approximate solutions for Re =(1000 5000 10000) and then compare the results of 1-levelfinite element VMS method (1ℎ = 256) 2-level finiteelement VMSmethod (1ℎ1 = 128 1ℎ2 = 256) 3-level finiteelement VMS method (1ℎ1 = 64 1ℎ2 = 128 1ℎ3 = 256)and 4-level finite element VMS method (1ℎ1 = 64 1ℎ2 =64 1ℎ3 = 128 1ℎ4 = 256) with that of Ghia et al [1]Figures 2ndash4 plot the horizontal component of the velocityalong the vertical line passing through the geometrical centerof the cavity and the vertical component of the velocityalong the horizontal line passing through the geometricalcenter of the cavity respectively As shown in Figure 2the results of multilevel finite element VMS method are

Table 2 Computing time of these 119895-level finite element VMSmethods

Re 1-level 2-level 3-level 4-level1000 162509 501335 193591 1159115000 298647 888397 452884 34708910000 439963 156004 670384 322642

indistinguishable from 1-level finite element VMS methodand the benchmark data in [1] However in Figures 3 and 4the 4-level finite element VMS method at Re = 5000 and the3-level and 4-level finite element VMSmethod at Re = 10000provide unsatisfactory results This is because the nonlinearsolution on the initial coarse mesh is not accurate enoughActually with an appropriate choice of initial coarsemesh themultilevel finite element VMS method can get better resultsthan one-level finite element VMS method Table 2 presentsthe computing time of these 119895-level (119895 = 1 2 3 4) finiteelement VMS methods and demonstrates that 119895-level finiteelement VMSmethod savesmuch computing time comparedwith 119895 minus 1-level (119895 = 2 3 4) finite element VMS method

53 Backward Facing Step Problem Backward facing stepproblem is also a standard numerical example which involvesa steady viscous incompressible flow over an isothermaltwo-dimensional backward facing step It is well-knownthat the backward facing step problem possesses a cornersingularity thus it is often adopted by many researchersto verify their numerical methods The geometry and theboundary conditions are shown in Figure 5 as in [21 46]

In this example the Reynolds number is 150 which isbased on the maximum inlet velocity V119909(max) = 1 and theheight of the inlet We compute an approximate solution onthe following number of triangle elements 9344 (for 1-levelfinite element VMS method) 5298-9344 (for 2-level finiteelement VMS method) 2392-5298-9344 (for 3-level finiteelement VMSmethod) and 646-2392-5298-9344 (for 4-levelfinite element VMS method) Figures 6(a)ndash6(d) show the

8 Mathematical Problems in Engineering

1-level2-level

3-level4-level

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

log(

erro

r)

ℎ3

minus5 minus45 minus4minus55log(h)

(a)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(b)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

minus75

minus7

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(c)

1

15

2

25

3

35

4

45

5

55

log(

CPU

(s))

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

(d)

Figure 1 Convergence analysis for the velocity and the pressure using 119895-level (119895 = 1 2 3 4) finite element VMS method (a) 1198712 error for thevelocity (b)1198671 error for the velocity (c) 1198712 error for the pressure (d) computing time

pressure field and the streamlines obtained by these 119895-level(119895 = 1 2 3 4) finite element VMS methods respectivelyand these results are in agreement with those of [21 46] Inorder to observe these results more clearly a comparison ofthe computed velocity pressure and vorticity is given acrossthe channel at 119909 = 15 in Figures 7(a)ndash7(d) it is obviousthat the results of these 119895-level (119895 = 1 2 3 4) finite element

VMS methods nearly have no differences with each otherin velocity and vorticity but the pressure computed by 4-level finite element VMS method has a clear difference fromothers The reason is that the initial grid chosen as 646 in4-level finite element VMS method is too coarse Table 3presents the computing time of these 119895-level (119895 = 1 2 3 4)finite element VMS methods it shows that the 119895-level finite

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

Mathematical Problems in Engineering 3

respectively The trilinear term 119887(sdot sdot sdot) satisfies the followingestimates [36]

119887 (119906 V V) = 0 forall119906 V 119908 isin 119883 (4)

|119887 (119906 V 119908)| le 119873 nabla1199060 nablaV0 nabla1199080 forall119906 V 119908 isin 119883 (5)

where 119873 is a positive constant depending only on ΩSubsequently 119888 and 119896 (with a subscript) will denote a positiveconstant which may stand for different values at its differentoccurrences respectively

With the above notations the variational formulation ofproblem (1) reads as follows find (119906 119901) isin 119883times119872 for all (V 119902) isin119883 times119872 satisfying

]119886 (119906 V) minus 119889 (V 119901) + 119889 (119906 119902) + 119887 (119906 119906 V) = (119891 V) (6)

Thenonsingular solution of (1) is defined as follows [37]119906is a nonsingular solution of (1) if there is a constant120590(119906 ]) gt 0such that

infVisin119881

sup119908isin119881

]119886 (V 119908) + 119887 (119906 V 119908) + 119887 (V 119906 119908)V1 1199081 ge 120590 (119906 ]) (7)

Here 119881 = V isin 119883 (nabla sdot V 119902) = 0 forall119902 isin 119872 If 119906 is a nonsin-gular solution of the Navier-Stokes equations 119906 satisfies theestimate [37 38]

1199061 le ]minus1 10038171003817100381710038171198911003817100381710038171003817minus1 10038171003817100381710038171198911003817100381710038171003817minus1 = supVisin119883V =0

1003816100381610038161003816(119891 V)1003816100381610038161003816V1 (8)

Lemma 1 is also needed more details of it can be found in[39 40]

Lemma 1 Suppose that

inf119902isin119872

supVisin119883

119889 (V 119902)100381710038171003817100381711990210038171003817100381710038170 V1 ge 120573 gt 0 (9)

and that the bilinear form 119888(119906 V) satisfies

inf119906isin119881

supVisin119881

119888 (119906 V)1199061 V1 ge 1205720 gt 0 (10)

Then the continuous bilinear form 119865 (119883119872) times (119883119872) rarr 119877defined by

119865 [(119906 119901) (V 119902)] fl 119888 (119906 V) minus 119889 (V 119901) + 119889 (119906 119902) (11)

satisfies the inf-sup condition on the product space (119883119872) times(119883119872) which means that there exists a constant 120574lowast =120574lowast(1205720 120573 119888(sdot sdot)) gt 0 such that

inf(119906119901)isin(119883119872)

sup(V119902)isin(119883119872)

119865 [(119906 119901) (V 119902)]lsaquo (119906 119901) lsaquolsaquo (V 119902) lsaquo ge 120574

lowast (12)

For the finite element discretization let 119870ℎ be the reg-ular triangulations of the domain Ω The mesh defined by

ℎ = maxΩ119890isin119870ℎdiam(Ω119890) is a real-positive parameter tendingto 0 We choose the conforming velocity-pressure finiteelement space (119883ℎ119872ℎ) sub (119883119872) and consider the Taylor-Hood elements [41]

119883ℎ= Vℎ isin 1198620 (Ω)2 cap 119883 Vℎ1003816100381610038161003816Ω119890 isin 1198752 (Ω119890)2 forallΩ119890 isin 119870ℎ 119872ℎ= 119902ℎ isin 1198620 (Ω) cap119872 119902ℎ1003816100381610038161003816Ω119890 isin 1198751 (Ω119890) forallΩ119890 isin 119870ℎ

(13)

where 119875119895(Ω119890) (119895 = 1 2) is the space of 119895th order polynomialson Ω119890 The Taylor-Hood elements on both triangles andrectangles are proved to satisfy the following discrete inf-supcondition for compatibility of the velocity-pressure spaces

inf119902ℎisin119872ℎ

supVℎisin119883ℎ

119889 (Vℎ 119902ℎ)1003817100381710038171003817119902ℎ10038171003817100381710038170 1003817100381710038171003817Vℎ10038171003817100381710038171 ge 120573 gt 0 (14)

3 Finite Element VariationalMultiscale Method

The finite element VMS method proposed by John and Kaya[19 42] for the steady Navier-Stokes equations reads find(119906ℎ 119901ℎ) isin (119883ℎ119872ℎ) and 119892ℎ isin 119871ℎ for all (Vℎ 119902ℎ) isin (119883ℎ119872ℎ)satisfying

(] + 120572) 119886 (119906ℎ Vℎ) minus 120572 (119892ℎ nablaVℎ) + 119887 (119906ℎ 119906ℎ Vℎ)minus 119889 (Vℎ 119901ℎ) + 119889 (119906ℎ 119902ℎ) = (119891 Vℎ)

(119892ℎ minus nabla119906ℎ 119897ℎ) = 0 forall119897ℎ isin 119871ℎ(15)

This system is determined by the choices of 119871ℎ and 120572 Thestabilization parameter 0 lt 120572 lt 1 acts only on the small scaleswhich can be chosen as the scale of 119874(ℎ119904) (119904 ge 1) in order tostabilize the convective term appropriately

An orthogonal projection operator is defined as Π 119871 rarr 119871ℎ which is 1198712 orthogonal projection with the followingproperties

(Πnabla119906 V) = (nabla119906 V) forall119906 isin 119883 V isin 119871ℎ (H1)ΠnablaV0 le 119862 nablaV0 forallV isin 119883 (H2)

(119868 minus Π)nablaV0 le 119862 nablaV0 forallV isin 119883 (H3)(119868 minus Π)nablaV0 le 119862ℎ V2 forallV isin 1198672 (Ω)2times2 cap 119883 (H4)

where 119862 is a positive constant 119871 = 1198712(Ω) and 119871ℎ sub 119871 (fordetails refer to [18])

Based on the properties of the orthogonal projectionoperator Π the finite element variational multiscale method

4 Mathematical Problems in Engineering

proposed by Zheng et al [21] is as follows find (119906ℎ 119901ℎ) isin(119883ℎ119872ℎ) for all (Vℎ 119902ℎ) isin (119883ℎ119872ℎ) satisfying]119886 (119906ℎ Vℎ) minus 119889 (Vℎ 119901ℎ) + 119889 (119906ℎ 119902ℎ) + 119887 (119906ℎ 119906ℎ Vℎ)

+ 119866 (119906ℎ Vℎ) = (119891 Vℎ) (16)

Here

119866 (119906ℎ Vℎ) = 120572 ((119868 minus Π)nabla119906ℎ (119868 minus Π)nablaVℎ)= 120572 sumΩ119890isin120591ℎ

intΩ119890119896nabla119906ℎnablaVℎ 119889119909 minus int

Ω1198901nabla119906ℎnablaVℎ 119889119909 (17)

where intΩ119890119894(sdot) 119889119909 denotes an appropriate Gauss integral over

Ω119890 which is exact for polynomials of degree 119894 119894 = 1 119896 (119896 ge2)

From [24 35] problem (16) has the following stability anderror estimate

Theorem2 Suppose the finite element space (119883ℎ119872ℎ) satisfiesthe LBB condition (14) Let (119906 119901) isin (1198672(Ω)2 cap1198831198671(Ω)cap119872)be a nonsingular solution of (6)Then the approximate solution(119906ℎ 119901ℎ) given by (16) satisfies

1003817100381710038171003817119906ℎ10038171003817100381710038171 le ]minus1 10038171003817100381710038171198911003817100381710038171003817minus1 (18)

and the following error estimate

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171 + 1003817100381710038171003817119901 minus 119901ℎ10038171003817100381710038170 le 119888 (120572 + ℎ) (19)

The natural energy norm given by

lsaquo (V 119902) lsaquo fl (V21 + 1003817100381710038171003817119902100381710038171003817100381720)12 (20)

will be used in the next section for the theoretical analysis ofthe proposed multilevel finite element VMS method

4 Multilevel Finite Element VMS Method

In this section the multilevel finite element VMSmethod for(16) will be presented The stability and error estimate of itwill be given With mesh widths ℎ1 gt ℎ2 gt sdot sdot sdot gt ℎ119869 the finiteelement space pairs (119883ℎ119895 119872ℎ119895) 119895 = 1 119869 satisfy that

(119883ℎ1 119872ℎ1) sub (119883ℎ2 119872ℎ2) sub sdot sdot sdot sub (119883ℎ119869 119872ℎ119869) (21)

And (119906ℎ119895 119901ℎ119895) isin (119883ℎ119895 119872ℎ119895) is the solution obtained by theproposedmultilevel finite element VMSmethodThe generalsteps of the proposed multilevel finite element VMS methodfor approximating the solution of (16) can be summarized asfollows

Step 1 Solve the following nonlinear system on the coarsestmesh ℎ1 Find (119906ℎ1 119901ℎ1) isin (119883ℎ1 119872ℎ1) for all (V 119902) isin (119883ℎ1 119872ℎ1) satisfying

]119886 (119906ℎ1 V) minus 119889 (V 119901ℎ1) + 119889 (119906ℎ1 119902) + 119866 (119906ℎ1 V)+ 119887 (119906ℎ1 119906ℎ1 V) = (119891 V)

(22)

Step 2 Update on fine mesh with Newton correction at eachlevel

Give (119906ℎ119895minus1 119901ℎ119895minus1) and find (119906ℎ119895 119901ℎ119895) isin (119883ℎ119895 119872ℎ119895) suchthat

]119886 (119906ℎ119895 V) minus 119889 (V 119901ℎ119895) + 119889 (119906ℎ119895 119902) + 119887 (119906ℎ119895 119906ℎ119895minus1 V)+ 119887 (119906ℎ119895minus1 119906ℎ119895 V) + 119866 (119906ℎ119895 V)

= (119891 V) + 119887 (119906ℎ119895minus1 119906ℎ119895minus1 V)(23)

for all (V 119902) isin (119883ℎ119895 119872ℎ119895) with 119895 = 2 119869Obviously the solution (119906ℎ1 119901ℎ1) after first step of the

multilevel finite element VMS method satisfies the resultsin Theorem 2 Before giving the stability analysis and errorestimates the continuous bilinear form B (119883119872) times(119883119872) rarr 119877 is given by

B [(119906 119901) (V 119902)] fl ]119886 (119906 V) minus 119889 (V 119901) + 119889 (119906 119902)+ 119887 (119906ℎ119895 119906 V) + 119887 (119906 119906ℎ119895 V)

(24)

Here we assume that 119906 is a nonsingular solution of (1) and119906ℎ119895 is close enough to 119906 Thus by Lemma 1 and (7) thecontinuous bilinear form B satisfies the inf-sup conditionsnamely there is a constant 120574 gt 0 such that [40]

inf(119906119901)isin(119883ℎ119895 119872ℎ119895 )

sup(V119902)isin(119883ℎ119895 119872ℎ119895 )

B [(119906 119901) (V 119902)]lsaquo (119906 119901) lsaquolsaquo (V 119902) lsaquo ge 120574

119895 = 1 2 119869(25)

The stability analysis and error estimate of the proposedmultilevel finite element method will be discussed in Theo-rems 3 and 4

Theorem 3 Suppose 119883ℎ1 sub 119883ℎ2 sub 119883 119872ℎ1 sub 119872ℎ2 sub 119872 andeach pair (119883ℎ1 119872ℎ1) (119883ℎ2 119872ℎ2) satisfies the LBB condition(14) Let (119906 119901) isin (1198672(Ω)2 cap 1198831198671(Ω) cap 119872) be a nonsingularsolution of (6) Suppose 0 lt ℎ2 lt ℎ1 and ℎ1 is sufficiently small

Mathematical Problems in Engineering 5

Then the approximate solution (119906ℎ2 119901ℎ2) isin (119883ℎ2 119872ℎ2) given bythe multilevel finite element method (23) with 119895 = 2 satisfies

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 11989612 (26)

Moreover if there holds the scaled relation ℎ2 = 119874(ℎ21) it hasthe following error estimate

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le 11989622 (120572 + ℎ2) (27)

Proof Using (5) (18) (23) (25) and (H3) gives

120574lsaquo (119906ℎ2 119901ℎ2) lsaquo le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ2 119901ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ1 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2100381710038171003817100381711989110038171003817100381710038172minus1 + 119862120572lsaquo (119906ℎ2 119901ℎ2) lsaquo

(28)

Choosing 120572 = 119874(ℎ1199042) for sufficiently small ℎ2 120575 = 120574minus119862120572 gt 0Then we have

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2

100381710038171003817100381711989110038171003817100381710038172minus1) (29)

Letting 11989612 = (119891minus1 + (119873]2)1198912minus1)120575minus1 then (26) holdsSubtracting (23) with 119895 = 2 from (6) yields

B [(119906 minus 119906ℎ2 119906 minus 119901ℎ2) (V 119902)]= 119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)

(30)

The right terms in (30) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ1 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ1)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ2 V)10038161003816100381610038161003816 le 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ2100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(31)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ2 119872ℎ2) as interpolation of (119906 119901) into(119883ℎ2 119872ℎ2) we have

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) lsaquo le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

B [(119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 119888 (120572 + ℎ1)2 + 119888120572

(32)

Then together with the relation ℎ2 = 119874(ℎ21) denoting theconstant as 11989622 yields (27)Theorem 4 Suppose 119883ℎ1 sub 119883ℎ2 sub sdot sdot sdot sub 119883ℎ119869 sub 119883 119872ℎ1 sub119872ℎ2 sub sdot sdot sdot sub 119872ℎ119869 sub 119872 and each pair (119883ℎ1 119872ℎ1) (119883ℎ119869 119872ℎ119869) satisfies the LBB condition Let (119906 119901) isin (1198672(Ω)2 cap1198831198671(Ω) cap 119872) be a nonsingular solution of (6) If ℎ119895 gt 0is sufficient small then the approximate solution (119906ℎ119895 119901ℎ119895) isin(119883ℎ119895 119872ℎ119895) given by the multilevel finite element method (23)satisfies

lsaquo (119906ℎ119895 119901ℎ119895) lsaquo le 1198961119895 (33)

Moreover if there holds the scaled relation ℎ119895 = 119874(ℎ2119895minus1) 119895 =2 3 119869 it has the following error estimate

lsaquo (119906 minus 119906ℎ119895 119901 minus 119901ℎ119895) lsaquo1 le 1198962119895 (120572 + ℎ119895) (34)

with 119895 = 2 119869Proof This theorem will be proved by the induction methodFromTheorem 3 it is obvious that it holds for 119895 = 2 AssumeTheorem 4 is true for 119895 = 119898 we need to prove that it is alsotrue for 119895 = 119898 + 1 with 1 lt 119898 le 119869 minus 1

Using (5) (23) (25) and (H3) we obtain

120574lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ119898+1 119901ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ119898 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

6 Mathematical Problems in Engineering

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1119898100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898 + 119862120572lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo(35)

Choosing 120572 = 119874(ℎ119904119898+1) for sufficiently small ℎ119898+1 120575 = 120574 minus119862120572 gt 0 Then we have

lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898) (36)

Letting 1198961119895 = (119891minus1 + 11987311989621119898)120575minus1 then (33) holds for 119895 =119898 + 1

Next the convergence error of the multilevel finite ele-ment method is analyzed Subtracting (23) with 119895 = 119898 + 1from (6) yields

B [(119906 minus 119906ℎ119898+1 119906 minus 119901ℎ119898+1) (V 119902)]= 119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)

(37)

The right terms in (37) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ119898 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ119898)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ119898+1 V)10038161003816100381610038161003816 = 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ119898+1100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(38)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ119898+1 119872ℎ119898+1) as interpolation of (119906 119901)into (119883ℎ119898+1 119872ℎ119898+1) we have

lsaquo (119906 minus 119906ℎ119898+1 119901 minus 119901ℎ119898+1) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) lsaquo le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

B [(119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 119888 (120572 + ℎ119898)2 + 119888120572

(39)

Then together with the relation ℎ119898+1 = 119874(ℎ2119898) denoting theconstant as 1198962119898+1 yields (34) for 119895 = 119898 + 1 Thus Theorem 4follows from the induction principle

5 Numerical Examples

In this section three numerical examples are given thefirst one is an exact solution problem the second one isthe lid-driven cavity problem and the third one is thebackward facing step problem The mentioned methods inthis paper are all implemented using Freefem++ [43] and theconvergence tolerance is set equal to 10minus6 In the numericalexamples the triangular mesh with the Taylor-Hood elementpair is used for the finite element discretization For the sakeof simplicity the one-level and the proposed multilevel finiteelement VMS methods are written as 119895-level finite elementVMS method in the numerical examples The nonlinearsystems in these methods are all linearized by the Oseeniterative method (see [44 45])

51 Problem with an Exact Solution This problem is a steadyflow of incompressible viscous Newtonian fluid in boundeddomain Ω = (0 1) times (0 1) with 119906 = (0 0) on the boundariesIt has the following exact solution

1199061 = 101199092 (119909 minus 1)2 119910 (119910 minus 1) (2119910 minus 1) 1199062 = minus10119909 (119909 minus 1) (2119909 minus 1) 1199102 (119910 minus 1)2 119901 = 10 (2119909 minus 1) (2119910 minus 1)

(40)

where 119891 = (1198911 1198912) is determined by (1) The Taylor-Hoodelements predict a convergence rate of 119874(ℎ2) in the energynorm (ie 1198671-norm for the velocity and 1198712-norm for thepressure) so the eddy viscosity parameter is chosen as 120572 =01ℎ2119895 for multilevel finite element VMS method which is thesame as that used in [21]

First ] = 01 is considered The meshes for the one-levelfinite element VMS method are chosen as ℎ119869 = [182 11021122 1142 1162] and these meshes are also the finestmeshes for the multilevel finite element VMS method Thechoices for the coarse meshes of the multilevel finite elementVMS method are very important If the coarse meshes arenot fine enough the optimal convergence orders of velocityand pressure can not be reached According to Theorems 3and 4 the coarse meshes ℎ119895minus1 and the fine mesh ℎ119895 should bechosen as ℎ119895 sim ℎ2119895minus1 119895 = 2 3 119869 which ensure optimalaccuracy of the approximate solution Here we consider themultilevel finite element VMS method for 119895 = 2 3 4 andmeshes are chosen as ℎ119895 = ℎ2119895minus1 respectively The numericalresults using the one-level and the proposed multilevel finiteelement VMSmethods are shown in Figure 1 which indicatesthat the proposed multilevel finite element VMS method canreach the convergence order of119874(ℎ2) for velocity in1198671-normand pressure in 1198712-norm and the optimal convergence orderof119874(ℎ3) for velocity in the 1198712-normThese numerical resultsare in agreement with the theoretical predictions Moreover

Mathematical Problems in Engineering 7

Table 1 Comparison of these 119895-level finite element VMS methods

Method 1ℎ CPU(s)

100381710038171003817100381710038171003817119906 minus 119906ℎ1198951003817100381710038171003817100381710038170

1199060100381710038171003817100381710038171003817119906 minus 119906ℎ119895

10038171003817100381710038171003817100381711199061

100381710038171003817100381710038171003817119901 minus 119901ℎ1198951003817100381710038171003817100381710038170100381710038171003817100381711990110038171003817100381710038170

1-level 128 198614 137319119890 minus 005 0000776894 472788119890 minus 0052-level 64-128 71389 102514119890 minus 005 0000351524 472788119890 minus 0053-level 32-64-128 37617 211498119890 minus 005 0000342189 472788119890 minus 0054-level 16-32-64-128 29495 0000110742 0000526664 47279119890 minus 0051-level 144 25086 85448119890 minus 006 0000554574 373553119890 minus 0052-level 72-144 92251 550229119890 minus 006 0000226809 373553119890 minus 0053-level 36-72-144 45212 930361119890 minus 006 0000206441 373553119890 minus 0054-level 18-36-72-144 35437 43288119890 minus 005 0000275735 373553119890 minus 0051-level 256 925036 902168119890 minus 007 0000107772 118393119890 minus 0052-level 128-256 307479 285225119890 minus 007 466203119890 minus 005 118403119890 minus 0053-level 64-128-256 141673 202856119890 minus 007 45437119890 minus 005 118383119890 minus 0054-level 32-64-128-256 99477 146991119890 minus 007 450338119890 minus 005 118391119890 minus 005

both the proposedmultilevel finite elementVMSmethod andthe one-level finite element VMS method obtain nearly thesame approximation results but the former significantly savescomputing time compared with the later

Then ] = 00001 is considered The mesh CPU-time the1198712-error of the velocity1198671-error of the velocity and 1198712-errorof the pressure for comparison of these 119895-level (119895 = 1 2 3 4)finite element VMS methods are tabulated in Table 1 Here(119906ℎ119895 119901ℎ119895) refers to the approximation solution obtained by 119895-level finite elementVMSmethod Table 1 indicates that 119895-level(119895 ge 2) finite element VMS method saves much computingtime compared with 119895 minus 1-level finite element VMS methodat the same fine mesh and the finer the mesh becomes thesmaller the error achieves Furthermore when scrutinizingthe error estimates of velocity and the pressure in Table 1 itis easily observed that multilevel (119895 = 2 3 4) finite elementVMS method for the velocity and the pressure is better thanone-level (119895 = 1) finite element VMS method

52 Lid-Driven Cavity Problem Lid-driven cavity problemoften serves as a standard benchmark problem for incom-pressible Navier-Stokes equations because the benchmarkdata is available for comparison In this problem the compu-tation is carried out in the region Ω = (119909 119910) | 0 lt 119909 119910 lt 1The velocity is 119906 = (1 0) on the top side (119909 1) 0 lt 119909 lt 1and 119906 = (0 0) on the rest of the boundary

We compute the approximate solutions for Re =(1000 5000 10000) and then compare the results of 1-levelfinite element VMS method (1ℎ = 256) 2-level finiteelement VMSmethod (1ℎ1 = 128 1ℎ2 = 256) 3-level finiteelement VMS method (1ℎ1 = 64 1ℎ2 = 128 1ℎ3 = 256)and 4-level finite element VMS method (1ℎ1 = 64 1ℎ2 =64 1ℎ3 = 128 1ℎ4 = 256) with that of Ghia et al [1]Figures 2ndash4 plot the horizontal component of the velocityalong the vertical line passing through the geometrical centerof the cavity and the vertical component of the velocityalong the horizontal line passing through the geometricalcenter of the cavity respectively As shown in Figure 2the results of multilevel finite element VMS method are

Table 2 Computing time of these 119895-level finite element VMSmethods

Re 1-level 2-level 3-level 4-level1000 162509 501335 193591 1159115000 298647 888397 452884 34708910000 439963 156004 670384 322642

indistinguishable from 1-level finite element VMS methodand the benchmark data in [1] However in Figures 3 and 4the 4-level finite element VMS method at Re = 5000 and the3-level and 4-level finite element VMSmethod at Re = 10000provide unsatisfactory results This is because the nonlinearsolution on the initial coarse mesh is not accurate enoughActually with an appropriate choice of initial coarsemesh themultilevel finite element VMS method can get better resultsthan one-level finite element VMS method Table 2 presentsthe computing time of these 119895-level (119895 = 1 2 3 4) finiteelement VMS methods and demonstrates that 119895-level finiteelement VMSmethod savesmuch computing time comparedwith 119895 minus 1-level (119895 = 2 3 4) finite element VMS method

53 Backward Facing Step Problem Backward facing stepproblem is also a standard numerical example which involvesa steady viscous incompressible flow over an isothermaltwo-dimensional backward facing step It is well-knownthat the backward facing step problem possesses a cornersingularity thus it is often adopted by many researchersto verify their numerical methods The geometry and theboundary conditions are shown in Figure 5 as in [21 46]

In this example the Reynolds number is 150 which isbased on the maximum inlet velocity V119909(max) = 1 and theheight of the inlet We compute an approximate solution onthe following number of triangle elements 9344 (for 1-levelfinite element VMS method) 5298-9344 (for 2-level finiteelement VMS method) 2392-5298-9344 (for 3-level finiteelement VMSmethod) and 646-2392-5298-9344 (for 4-levelfinite element VMS method) Figures 6(a)ndash6(d) show the

8 Mathematical Problems in Engineering

1-level2-level

3-level4-level

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

log(

erro

r)

ℎ3

minus5 minus45 minus4minus55log(h)

(a)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(b)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

minus75

minus7

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(c)

1

15

2

25

3

35

4

45

5

55

log(

CPU

(s))

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

(d)

Figure 1 Convergence analysis for the velocity and the pressure using 119895-level (119895 = 1 2 3 4) finite element VMS method (a) 1198712 error for thevelocity (b)1198671 error for the velocity (c) 1198712 error for the pressure (d) computing time

pressure field and the streamlines obtained by these 119895-level(119895 = 1 2 3 4) finite element VMS methods respectivelyand these results are in agreement with those of [21 46] Inorder to observe these results more clearly a comparison ofthe computed velocity pressure and vorticity is given acrossthe channel at 119909 = 15 in Figures 7(a)ndash7(d) it is obviousthat the results of these 119895-level (119895 = 1 2 3 4) finite element

VMS methods nearly have no differences with each otherin velocity and vorticity but the pressure computed by 4-level finite element VMS method has a clear difference fromothers The reason is that the initial grid chosen as 646 in4-level finite element VMS method is too coarse Table 3presents the computing time of these 119895-level (119895 = 1 2 3 4)finite element VMS methods it shows that the 119895-level finite

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

4 Mathematical Problems in Engineering

proposed by Zheng et al [21] is as follows find (119906ℎ 119901ℎ) isin(119883ℎ119872ℎ) for all (Vℎ 119902ℎ) isin (119883ℎ119872ℎ) satisfying]119886 (119906ℎ Vℎ) minus 119889 (Vℎ 119901ℎ) + 119889 (119906ℎ 119902ℎ) + 119887 (119906ℎ 119906ℎ Vℎ)

+ 119866 (119906ℎ Vℎ) = (119891 Vℎ) (16)

Here

119866 (119906ℎ Vℎ) = 120572 ((119868 minus Π)nabla119906ℎ (119868 minus Π)nablaVℎ)= 120572 sumΩ119890isin120591ℎ

intΩ119890119896nabla119906ℎnablaVℎ 119889119909 minus int

Ω1198901nabla119906ℎnablaVℎ 119889119909 (17)

where intΩ119890119894(sdot) 119889119909 denotes an appropriate Gauss integral over

Ω119890 which is exact for polynomials of degree 119894 119894 = 1 119896 (119896 ge2)

From [24 35] problem (16) has the following stability anderror estimate

Theorem2 Suppose the finite element space (119883ℎ119872ℎ) satisfiesthe LBB condition (14) Let (119906 119901) isin (1198672(Ω)2 cap1198831198671(Ω)cap119872)be a nonsingular solution of (6)Then the approximate solution(119906ℎ 119901ℎ) given by (16) satisfies

1003817100381710038171003817119906ℎ10038171003817100381710038171 le ]minus1 10038171003817100381710038171198911003817100381710038171003817minus1 (18)

and the following error estimate

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171 + 1003817100381710038171003817119901 minus 119901ℎ10038171003817100381710038170 le 119888 (120572 + ℎ) (19)

The natural energy norm given by

lsaquo (V 119902) lsaquo fl (V21 + 1003817100381710038171003817119902100381710038171003817100381720)12 (20)

will be used in the next section for the theoretical analysis ofthe proposed multilevel finite element VMS method

4 Multilevel Finite Element VMS Method

In this section the multilevel finite element VMSmethod for(16) will be presented The stability and error estimate of itwill be given With mesh widths ℎ1 gt ℎ2 gt sdot sdot sdot gt ℎ119869 the finiteelement space pairs (119883ℎ119895 119872ℎ119895) 119895 = 1 119869 satisfy that

(119883ℎ1 119872ℎ1) sub (119883ℎ2 119872ℎ2) sub sdot sdot sdot sub (119883ℎ119869 119872ℎ119869) (21)

And (119906ℎ119895 119901ℎ119895) isin (119883ℎ119895 119872ℎ119895) is the solution obtained by theproposedmultilevel finite element VMSmethodThe generalsteps of the proposed multilevel finite element VMS methodfor approximating the solution of (16) can be summarized asfollows

Step 1 Solve the following nonlinear system on the coarsestmesh ℎ1 Find (119906ℎ1 119901ℎ1) isin (119883ℎ1 119872ℎ1) for all (V 119902) isin (119883ℎ1 119872ℎ1) satisfying

]119886 (119906ℎ1 V) minus 119889 (V 119901ℎ1) + 119889 (119906ℎ1 119902) + 119866 (119906ℎ1 V)+ 119887 (119906ℎ1 119906ℎ1 V) = (119891 V)

(22)

Step 2 Update on fine mesh with Newton correction at eachlevel

Give (119906ℎ119895minus1 119901ℎ119895minus1) and find (119906ℎ119895 119901ℎ119895) isin (119883ℎ119895 119872ℎ119895) suchthat

]119886 (119906ℎ119895 V) minus 119889 (V 119901ℎ119895) + 119889 (119906ℎ119895 119902) + 119887 (119906ℎ119895 119906ℎ119895minus1 V)+ 119887 (119906ℎ119895minus1 119906ℎ119895 V) + 119866 (119906ℎ119895 V)

= (119891 V) + 119887 (119906ℎ119895minus1 119906ℎ119895minus1 V)(23)

for all (V 119902) isin (119883ℎ119895 119872ℎ119895) with 119895 = 2 119869Obviously the solution (119906ℎ1 119901ℎ1) after first step of the

multilevel finite element VMS method satisfies the resultsin Theorem 2 Before giving the stability analysis and errorestimates the continuous bilinear form B (119883119872) times(119883119872) rarr 119877 is given by

B [(119906 119901) (V 119902)] fl ]119886 (119906 V) minus 119889 (V 119901) + 119889 (119906 119902)+ 119887 (119906ℎ119895 119906 V) + 119887 (119906 119906ℎ119895 V)

(24)

Here we assume that 119906 is a nonsingular solution of (1) and119906ℎ119895 is close enough to 119906 Thus by Lemma 1 and (7) thecontinuous bilinear form B satisfies the inf-sup conditionsnamely there is a constant 120574 gt 0 such that [40]

inf(119906119901)isin(119883ℎ119895 119872ℎ119895 )

sup(V119902)isin(119883ℎ119895 119872ℎ119895 )

B [(119906 119901) (V 119902)]lsaquo (119906 119901) lsaquolsaquo (V 119902) lsaquo ge 120574

119895 = 1 2 119869(25)

The stability analysis and error estimate of the proposedmultilevel finite element method will be discussed in Theo-rems 3 and 4

Theorem 3 Suppose 119883ℎ1 sub 119883ℎ2 sub 119883 119872ℎ1 sub 119872ℎ2 sub 119872 andeach pair (119883ℎ1 119872ℎ1) (119883ℎ2 119872ℎ2) satisfies the LBB condition(14) Let (119906 119901) isin (1198672(Ω)2 cap 1198831198671(Ω) cap 119872) be a nonsingularsolution of (6) Suppose 0 lt ℎ2 lt ℎ1 and ℎ1 is sufficiently small

Mathematical Problems in Engineering 5

Then the approximate solution (119906ℎ2 119901ℎ2) isin (119883ℎ2 119872ℎ2) given bythe multilevel finite element method (23) with 119895 = 2 satisfies

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 11989612 (26)

Moreover if there holds the scaled relation ℎ2 = 119874(ℎ21) it hasthe following error estimate

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le 11989622 (120572 + ℎ2) (27)

Proof Using (5) (18) (23) (25) and (H3) gives

120574lsaquo (119906ℎ2 119901ℎ2) lsaquo le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ2 119901ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ1 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2100381710038171003817100381711989110038171003817100381710038172minus1 + 119862120572lsaquo (119906ℎ2 119901ℎ2) lsaquo

(28)

Choosing 120572 = 119874(ℎ1199042) for sufficiently small ℎ2 120575 = 120574minus119862120572 gt 0Then we have

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2

100381710038171003817100381711989110038171003817100381710038172minus1) (29)

Letting 11989612 = (119891minus1 + (119873]2)1198912minus1)120575minus1 then (26) holdsSubtracting (23) with 119895 = 2 from (6) yields

B [(119906 minus 119906ℎ2 119906 minus 119901ℎ2) (V 119902)]= 119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)

(30)

The right terms in (30) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ1 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ1)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ2 V)10038161003816100381610038161003816 le 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ2100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(31)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ2 119872ℎ2) as interpolation of (119906 119901) into(119883ℎ2 119872ℎ2) we have

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) lsaquo le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

B [(119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 119888 (120572 + ℎ1)2 + 119888120572

(32)

Then together with the relation ℎ2 = 119874(ℎ21) denoting theconstant as 11989622 yields (27)Theorem 4 Suppose 119883ℎ1 sub 119883ℎ2 sub sdot sdot sdot sub 119883ℎ119869 sub 119883 119872ℎ1 sub119872ℎ2 sub sdot sdot sdot sub 119872ℎ119869 sub 119872 and each pair (119883ℎ1 119872ℎ1) (119883ℎ119869 119872ℎ119869) satisfies the LBB condition Let (119906 119901) isin (1198672(Ω)2 cap1198831198671(Ω) cap 119872) be a nonsingular solution of (6) If ℎ119895 gt 0is sufficient small then the approximate solution (119906ℎ119895 119901ℎ119895) isin(119883ℎ119895 119872ℎ119895) given by the multilevel finite element method (23)satisfies

lsaquo (119906ℎ119895 119901ℎ119895) lsaquo le 1198961119895 (33)

Moreover if there holds the scaled relation ℎ119895 = 119874(ℎ2119895minus1) 119895 =2 3 119869 it has the following error estimate

lsaquo (119906 minus 119906ℎ119895 119901 minus 119901ℎ119895) lsaquo1 le 1198962119895 (120572 + ℎ119895) (34)

with 119895 = 2 119869Proof This theorem will be proved by the induction methodFromTheorem 3 it is obvious that it holds for 119895 = 2 AssumeTheorem 4 is true for 119895 = 119898 we need to prove that it is alsotrue for 119895 = 119898 + 1 with 1 lt 119898 le 119869 minus 1

Using (5) (23) (25) and (H3) we obtain

120574lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ119898+1 119901ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ119898 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

6 Mathematical Problems in Engineering

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1119898100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898 + 119862120572lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo(35)

Choosing 120572 = 119874(ℎ119904119898+1) for sufficiently small ℎ119898+1 120575 = 120574 minus119862120572 gt 0 Then we have

lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898) (36)

Letting 1198961119895 = (119891minus1 + 11987311989621119898)120575minus1 then (33) holds for 119895 =119898 + 1

Next the convergence error of the multilevel finite ele-ment method is analyzed Subtracting (23) with 119895 = 119898 + 1from (6) yields

B [(119906 minus 119906ℎ119898+1 119906 minus 119901ℎ119898+1) (V 119902)]= 119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)

(37)

The right terms in (37) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ119898 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ119898)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ119898+1 V)10038161003816100381610038161003816 = 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ119898+1100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(38)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ119898+1 119872ℎ119898+1) as interpolation of (119906 119901)into (119883ℎ119898+1 119872ℎ119898+1) we have

lsaquo (119906 minus 119906ℎ119898+1 119901 minus 119901ℎ119898+1) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) lsaquo le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

B [(119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 119888 (120572 + ℎ119898)2 + 119888120572

(39)

Then together with the relation ℎ119898+1 = 119874(ℎ2119898) denoting theconstant as 1198962119898+1 yields (34) for 119895 = 119898 + 1 Thus Theorem 4follows from the induction principle

5 Numerical Examples

In this section three numerical examples are given thefirst one is an exact solution problem the second one isthe lid-driven cavity problem and the third one is thebackward facing step problem The mentioned methods inthis paper are all implemented using Freefem++ [43] and theconvergence tolerance is set equal to 10minus6 In the numericalexamples the triangular mesh with the Taylor-Hood elementpair is used for the finite element discretization For the sakeof simplicity the one-level and the proposed multilevel finiteelement VMS methods are written as 119895-level finite elementVMS method in the numerical examples The nonlinearsystems in these methods are all linearized by the Oseeniterative method (see [44 45])

51 Problem with an Exact Solution This problem is a steadyflow of incompressible viscous Newtonian fluid in boundeddomain Ω = (0 1) times (0 1) with 119906 = (0 0) on the boundariesIt has the following exact solution

1199061 = 101199092 (119909 minus 1)2 119910 (119910 minus 1) (2119910 minus 1) 1199062 = minus10119909 (119909 minus 1) (2119909 minus 1) 1199102 (119910 minus 1)2 119901 = 10 (2119909 minus 1) (2119910 minus 1)

(40)

where 119891 = (1198911 1198912) is determined by (1) The Taylor-Hoodelements predict a convergence rate of 119874(ℎ2) in the energynorm (ie 1198671-norm for the velocity and 1198712-norm for thepressure) so the eddy viscosity parameter is chosen as 120572 =01ℎ2119895 for multilevel finite element VMS method which is thesame as that used in [21]

First ] = 01 is considered The meshes for the one-levelfinite element VMS method are chosen as ℎ119869 = [182 11021122 1142 1162] and these meshes are also the finestmeshes for the multilevel finite element VMS method Thechoices for the coarse meshes of the multilevel finite elementVMS method are very important If the coarse meshes arenot fine enough the optimal convergence orders of velocityand pressure can not be reached According to Theorems 3and 4 the coarse meshes ℎ119895minus1 and the fine mesh ℎ119895 should bechosen as ℎ119895 sim ℎ2119895minus1 119895 = 2 3 119869 which ensure optimalaccuracy of the approximate solution Here we consider themultilevel finite element VMS method for 119895 = 2 3 4 andmeshes are chosen as ℎ119895 = ℎ2119895minus1 respectively The numericalresults using the one-level and the proposed multilevel finiteelement VMSmethods are shown in Figure 1 which indicatesthat the proposed multilevel finite element VMS method canreach the convergence order of119874(ℎ2) for velocity in1198671-normand pressure in 1198712-norm and the optimal convergence orderof119874(ℎ3) for velocity in the 1198712-normThese numerical resultsare in agreement with the theoretical predictions Moreover

Mathematical Problems in Engineering 7

Table 1 Comparison of these 119895-level finite element VMS methods

Method 1ℎ CPU(s)

100381710038171003817100381710038171003817119906 minus 119906ℎ1198951003817100381710038171003817100381710038170

1199060100381710038171003817100381710038171003817119906 minus 119906ℎ119895

10038171003817100381710038171003817100381711199061

100381710038171003817100381710038171003817119901 minus 119901ℎ1198951003817100381710038171003817100381710038170100381710038171003817100381711990110038171003817100381710038170

1-level 128 198614 137319119890 minus 005 0000776894 472788119890 minus 0052-level 64-128 71389 102514119890 minus 005 0000351524 472788119890 minus 0053-level 32-64-128 37617 211498119890 minus 005 0000342189 472788119890 minus 0054-level 16-32-64-128 29495 0000110742 0000526664 47279119890 minus 0051-level 144 25086 85448119890 minus 006 0000554574 373553119890 minus 0052-level 72-144 92251 550229119890 minus 006 0000226809 373553119890 minus 0053-level 36-72-144 45212 930361119890 minus 006 0000206441 373553119890 minus 0054-level 18-36-72-144 35437 43288119890 minus 005 0000275735 373553119890 minus 0051-level 256 925036 902168119890 minus 007 0000107772 118393119890 minus 0052-level 128-256 307479 285225119890 minus 007 466203119890 minus 005 118403119890 minus 0053-level 64-128-256 141673 202856119890 minus 007 45437119890 minus 005 118383119890 minus 0054-level 32-64-128-256 99477 146991119890 minus 007 450338119890 minus 005 118391119890 minus 005

both the proposedmultilevel finite elementVMSmethod andthe one-level finite element VMS method obtain nearly thesame approximation results but the former significantly savescomputing time compared with the later

Then ] = 00001 is considered The mesh CPU-time the1198712-error of the velocity1198671-error of the velocity and 1198712-errorof the pressure for comparison of these 119895-level (119895 = 1 2 3 4)finite element VMS methods are tabulated in Table 1 Here(119906ℎ119895 119901ℎ119895) refers to the approximation solution obtained by 119895-level finite elementVMSmethod Table 1 indicates that 119895-level(119895 ge 2) finite element VMS method saves much computingtime compared with 119895 minus 1-level finite element VMS methodat the same fine mesh and the finer the mesh becomes thesmaller the error achieves Furthermore when scrutinizingthe error estimates of velocity and the pressure in Table 1 itis easily observed that multilevel (119895 = 2 3 4) finite elementVMS method for the velocity and the pressure is better thanone-level (119895 = 1) finite element VMS method

52 Lid-Driven Cavity Problem Lid-driven cavity problemoften serves as a standard benchmark problem for incom-pressible Navier-Stokes equations because the benchmarkdata is available for comparison In this problem the compu-tation is carried out in the region Ω = (119909 119910) | 0 lt 119909 119910 lt 1The velocity is 119906 = (1 0) on the top side (119909 1) 0 lt 119909 lt 1and 119906 = (0 0) on the rest of the boundary

We compute the approximate solutions for Re =(1000 5000 10000) and then compare the results of 1-levelfinite element VMS method (1ℎ = 256) 2-level finiteelement VMSmethod (1ℎ1 = 128 1ℎ2 = 256) 3-level finiteelement VMS method (1ℎ1 = 64 1ℎ2 = 128 1ℎ3 = 256)and 4-level finite element VMS method (1ℎ1 = 64 1ℎ2 =64 1ℎ3 = 128 1ℎ4 = 256) with that of Ghia et al [1]Figures 2ndash4 plot the horizontal component of the velocityalong the vertical line passing through the geometrical centerof the cavity and the vertical component of the velocityalong the horizontal line passing through the geometricalcenter of the cavity respectively As shown in Figure 2the results of multilevel finite element VMS method are

Table 2 Computing time of these 119895-level finite element VMSmethods

Re 1-level 2-level 3-level 4-level1000 162509 501335 193591 1159115000 298647 888397 452884 34708910000 439963 156004 670384 322642

indistinguishable from 1-level finite element VMS methodand the benchmark data in [1] However in Figures 3 and 4the 4-level finite element VMS method at Re = 5000 and the3-level and 4-level finite element VMSmethod at Re = 10000provide unsatisfactory results This is because the nonlinearsolution on the initial coarse mesh is not accurate enoughActually with an appropriate choice of initial coarsemesh themultilevel finite element VMS method can get better resultsthan one-level finite element VMS method Table 2 presentsthe computing time of these 119895-level (119895 = 1 2 3 4) finiteelement VMS methods and demonstrates that 119895-level finiteelement VMSmethod savesmuch computing time comparedwith 119895 minus 1-level (119895 = 2 3 4) finite element VMS method

53 Backward Facing Step Problem Backward facing stepproblem is also a standard numerical example which involvesa steady viscous incompressible flow over an isothermaltwo-dimensional backward facing step It is well-knownthat the backward facing step problem possesses a cornersingularity thus it is often adopted by many researchersto verify their numerical methods The geometry and theboundary conditions are shown in Figure 5 as in [21 46]

In this example the Reynolds number is 150 which isbased on the maximum inlet velocity V119909(max) = 1 and theheight of the inlet We compute an approximate solution onthe following number of triangle elements 9344 (for 1-levelfinite element VMS method) 5298-9344 (for 2-level finiteelement VMS method) 2392-5298-9344 (for 3-level finiteelement VMSmethod) and 646-2392-5298-9344 (for 4-levelfinite element VMS method) Figures 6(a)ndash6(d) show the

8 Mathematical Problems in Engineering

1-level2-level

3-level4-level

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

log(

erro

r)

ℎ3

minus5 minus45 minus4minus55log(h)

(a)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(b)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

minus75

minus7

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(c)

1

15

2

25

3

35

4

45

5

55

log(

CPU

(s))

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

(d)

Figure 1 Convergence analysis for the velocity and the pressure using 119895-level (119895 = 1 2 3 4) finite element VMS method (a) 1198712 error for thevelocity (b)1198671 error for the velocity (c) 1198712 error for the pressure (d) computing time

pressure field and the streamlines obtained by these 119895-level(119895 = 1 2 3 4) finite element VMS methods respectivelyand these results are in agreement with those of [21 46] Inorder to observe these results more clearly a comparison ofthe computed velocity pressure and vorticity is given acrossthe channel at 119909 = 15 in Figures 7(a)ndash7(d) it is obviousthat the results of these 119895-level (119895 = 1 2 3 4) finite element

VMS methods nearly have no differences with each otherin velocity and vorticity but the pressure computed by 4-level finite element VMS method has a clear difference fromothers The reason is that the initial grid chosen as 646 in4-level finite element VMS method is too coarse Table 3presents the computing time of these 119895-level (119895 = 1 2 3 4)finite element VMS methods it shows that the 119895-level finite

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

Mathematical Problems in Engineering 5

Then the approximate solution (119906ℎ2 119901ℎ2) isin (119883ℎ2 119872ℎ2) given bythe multilevel finite element method (23) with 119895 = 2 satisfies

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 11989612 (26)

Moreover if there holds the scaled relation ℎ2 = 119874(ℎ21) it hasthe following error estimate

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le 11989622 (120572 + ℎ2) (27)

Proof Using (5) (18) (23) (25) and (H3) gives

120574lsaquo (119906ℎ2 119901ℎ2) lsaquo le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ2 119901ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ1 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2100381710038171003817100381711989110038171003817100381710038172minus1 + 119862120572lsaquo (119906ℎ2 119901ℎ2) lsaquo

(28)

Choosing 120572 = 119874(ℎ1199042) for sufficiently small ℎ2 120575 = 120574minus119862120572 gt 0Then we have

lsaquo (119906ℎ2 119901ℎ2) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 119873]2

100381710038171003817100381711989110038171003817100381710038172minus1) (29)

Letting 11989612 = (119891minus1 + (119873]2)1198912minus1)120575minus1 then (26) holdsSubtracting (23) with 119895 = 2 from (6) yields

B [(119906 minus 119906ℎ2 119906 minus 119901ℎ2) (V 119902)]= 119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)

(30)

The right terms in (30) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ1 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ1)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ2 V)10038161003816100381610038161003816 le 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ2100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ2100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(31)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ2 119872ℎ2) as interpolation of (119906 119901) into(119883ℎ2 119872ℎ2) we have

lsaquo (119906 minus 119906ℎ2 119901 minus 119901ℎ2) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) lsaquo le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

B [(119868ℎ119906 minus 119906ℎ2 119869ℎ119901 minus 119906ℎ2) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 1120574

sdot sup(V119902)isin(119883ℎ2 119872ℎ2 )

10038161003816100381610038161003816119887 (119906ℎ1 minus 119906 119906 minus 119906ℎ1 V) + 119866 (119906ℎ2 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ2) + 119888 (120572 + ℎ1)2 + 119888120572

(32)

Then together with the relation ℎ2 = 119874(ℎ21) denoting theconstant as 11989622 yields (27)Theorem 4 Suppose 119883ℎ1 sub 119883ℎ2 sub sdot sdot sdot sub 119883ℎ119869 sub 119883 119872ℎ1 sub119872ℎ2 sub sdot sdot sdot sub 119872ℎ119869 sub 119872 and each pair (119883ℎ1 119872ℎ1) (119883ℎ119869 119872ℎ119869) satisfies the LBB condition Let (119906 119901) isin (1198672(Ω)2 cap1198831198671(Ω) cap 119872) be a nonsingular solution of (6) If ℎ119895 gt 0is sufficient small then the approximate solution (119906ℎ119895 119901ℎ119895) isin(119883ℎ119895 119872ℎ119895) given by the multilevel finite element method (23)satisfies

lsaquo (119906ℎ119895 119901ℎ119895) lsaquo le 1198961119895 (33)

Moreover if there holds the scaled relation ℎ119895 = 119874(ℎ2119895minus1) 119895 =2 3 119869 it has the following error estimate

lsaquo (119906 minus 119906ℎ119895 119901 minus 119901ℎ119895) lsaquo1 le 1198962119895 (120572 + ℎ119895) (34)

with 119895 = 2 119869Proof This theorem will be proved by the induction methodFromTheorem 3 it is obvious that it holds for 119895 = 2 AssumeTheorem 4 is true for 119895 = 119898 we need to prove that it is alsotrue for 119895 = 119898 + 1 with 1 lt 119898 le 119869 minus 1

Using (5) (23) (25) and (H3) we obtain

120574lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

B [(119906ℎ119898+1 119901ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le sup(V119902)isin(119883ℎ 119872ℎ)

10038161003816100381610038161003816(119891 V) + 119887 (119906ℎ119898 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

6 Mathematical Problems in Engineering

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1119898100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898 + 119862120572lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo(35)

Choosing 120572 = 119874(ℎ119904119898+1) for sufficiently small ℎ119898+1 120575 = 120574 minus119862120572 gt 0 Then we have

lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898) (36)

Letting 1198961119895 = (119891minus1 + 11987311989621119898)120575minus1 then (33) holds for 119895 =119898 + 1

Next the convergence error of the multilevel finite ele-ment method is analyzed Subtracting (23) with 119895 = 119898 + 1from (6) yields

B [(119906 minus 119906ℎ119898+1 119906 minus 119901ℎ119898+1) (V 119902)]= 119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)

(37)

The right terms in (37) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ119898 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ119898)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ119898+1 V)10038161003816100381610038161003816 = 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ119898+1100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(38)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ119898+1 119872ℎ119898+1) as interpolation of (119906 119901)into (119883ℎ119898+1 119872ℎ119898+1) we have

lsaquo (119906 minus 119906ℎ119898+1 119901 minus 119901ℎ119898+1) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) lsaquo le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

B [(119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 119888 (120572 + ℎ119898)2 + 119888120572

(39)

Then together with the relation ℎ119898+1 = 119874(ℎ2119898) denoting theconstant as 1198962119898+1 yields (34) for 119895 = 119898 + 1 Thus Theorem 4follows from the induction principle

5 Numerical Examples

In this section three numerical examples are given thefirst one is an exact solution problem the second one isthe lid-driven cavity problem and the third one is thebackward facing step problem The mentioned methods inthis paper are all implemented using Freefem++ [43] and theconvergence tolerance is set equal to 10minus6 In the numericalexamples the triangular mesh with the Taylor-Hood elementpair is used for the finite element discretization For the sakeof simplicity the one-level and the proposed multilevel finiteelement VMS methods are written as 119895-level finite elementVMS method in the numerical examples The nonlinearsystems in these methods are all linearized by the Oseeniterative method (see [44 45])

51 Problem with an Exact Solution This problem is a steadyflow of incompressible viscous Newtonian fluid in boundeddomain Ω = (0 1) times (0 1) with 119906 = (0 0) on the boundariesIt has the following exact solution

1199061 = 101199092 (119909 minus 1)2 119910 (119910 minus 1) (2119910 minus 1) 1199062 = minus10119909 (119909 minus 1) (2119909 minus 1) 1199102 (119910 minus 1)2 119901 = 10 (2119909 minus 1) (2119910 minus 1)

(40)

where 119891 = (1198911 1198912) is determined by (1) The Taylor-Hoodelements predict a convergence rate of 119874(ℎ2) in the energynorm (ie 1198671-norm for the velocity and 1198712-norm for thepressure) so the eddy viscosity parameter is chosen as 120572 =01ℎ2119895 for multilevel finite element VMS method which is thesame as that used in [21]

First ] = 01 is considered The meshes for the one-levelfinite element VMS method are chosen as ℎ119869 = [182 11021122 1142 1162] and these meshes are also the finestmeshes for the multilevel finite element VMS method Thechoices for the coarse meshes of the multilevel finite elementVMS method are very important If the coarse meshes arenot fine enough the optimal convergence orders of velocityand pressure can not be reached According to Theorems 3and 4 the coarse meshes ℎ119895minus1 and the fine mesh ℎ119895 should bechosen as ℎ119895 sim ℎ2119895minus1 119895 = 2 3 119869 which ensure optimalaccuracy of the approximate solution Here we consider themultilevel finite element VMS method for 119895 = 2 3 4 andmeshes are chosen as ℎ119895 = ℎ2119895minus1 respectively The numericalresults using the one-level and the proposed multilevel finiteelement VMSmethods are shown in Figure 1 which indicatesthat the proposed multilevel finite element VMS method canreach the convergence order of119874(ℎ2) for velocity in1198671-normand pressure in 1198712-norm and the optimal convergence orderof119874(ℎ3) for velocity in the 1198712-normThese numerical resultsare in agreement with the theoretical predictions Moreover

Mathematical Problems in Engineering 7

Table 1 Comparison of these 119895-level finite element VMS methods

Method 1ℎ CPU(s)

100381710038171003817100381710038171003817119906 minus 119906ℎ1198951003817100381710038171003817100381710038170

1199060100381710038171003817100381710038171003817119906 minus 119906ℎ119895

10038171003817100381710038171003817100381711199061

100381710038171003817100381710038171003817119901 minus 119901ℎ1198951003817100381710038171003817100381710038170100381710038171003817100381711990110038171003817100381710038170

1-level 128 198614 137319119890 minus 005 0000776894 472788119890 minus 0052-level 64-128 71389 102514119890 minus 005 0000351524 472788119890 minus 0053-level 32-64-128 37617 211498119890 minus 005 0000342189 472788119890 minus 0054-level 16-32-64-128 29495 0000110742 0000526664 47279119890 minus 0051-level 144 25086 85448119890 minus 006 0000554574 373553119890 minus 0052-level 72-144 92251 550229119890 minus 006 0000226809 373553119890 minus 0053-level 36-72-144 45212 930361119890 minus 006 0000206441 373553119890 minus 0054-level 18-36-72-144 35437 43288119890 minus 005 0000275735 373553119890 minus 0051-level 256 925036 902168119890 minus 007 0000107772 118393119890 minus 0052-level 128-256 307479 285225119890 minus 007 466203119890 minus 005 118403119890 minus 0053-level 64-128-256 141673 202856119890 minus 007 45437119890 minus 005 118383119890 minus 0054-level 32-64-128-256 99477 146991119890 minus 007 450338119890 minus 005 118391119890 minus 005

both the proposedmultilevel finite elementVMSmethod andthe one-level finite element VMS method obtain nearly thesame approximation results but the former significantly savescomputing time compared with the later

Then ] = 00001 is considered The mesh CPU-time the1198712-error of the velocity1198671-error of the velocity and 1198712-errorof the pressure for comparison of these 119895-level (119895 = 1 2 3 4)finite element VMS methods are tabulated in Table 1 Here(119906ℎ119895 119901ℎ119895) refers to the approximation solution obtained by 119895-level finite elementVMSmethod Table 1 indicates that 119895-level(119895 ge 2) finite element VMS method saves much computingtime compared with 119895 minus 1-level finite element VMS methodat the same fine mesh and the finer the mesh becomes thesmaller the error achieves Furthermore when scrutinizingthe error estimates of velocity and the pressure in Table 1 itis easily observed that multilevel (119895 = 2 3 4) finite elementVMS method for the velocity and the pressure is better thanone-level (119895 = 1) finite element VMS method

52 Lid-Driven Cavity Problem Lid-driven cavity problemoften serves as a standard benchmark problem for incom-pressible Navier-Stokes equations because the benchmarkdata is available for comparison In this problem the compu-tation is carried out in the region Ω = (119909 119910) | 0 lt 119909 119910 lt 1The velocity is 119906 = (1 0) on the top side (119909 1) 0 lt 119909 lt 1and 119906 = (0 0) on the rest of the boundary

We compute the approximate solutions for Re =(1000 5000 10000) and then compare the results of 1-levelfinite element VMS method (1ℎ = 256) 2-level finiteelement VMSmethod (1ℎ1 = 128 1ℎ2 = 256) 3-level finiteelement VMS method (1ℎ1 = 64 1ℎ2 = 128 1ℎ3 = 256)and 4-level finite element VMS method (1ℎ1 = 64 1ℎ2 =64 1ℎ3 = 128 1ℎ4 = 256) with that of Ghia et al [1]Figures 2ndash4 plot the horizontal component of the velocityalong the vertical line passing through the geometrical centerof the cavity and the vertical component of the velocityalong the horizontal line passing through the geometricalcenter of the cavity respectively As shown in Figure 2the results of multilevel finite element VMS method are

Table 2 Computing time of these 119895-level finite element VMSmethods

Re 1-level 2-level 3-level 4-level1000 162509 501335 193591 1159115000 298647 888397 452884 34708910000 439963 156004 670384 322642

indistinguishable from 1-level finite element VMS methodand the benchmark data in [1] However in Figures 3 and 4the 4-level finite element VMS method at Re = 5000 and the3-level and 4-level finite element VMSmethod at Re = 10000provide unsatisfactory results This is because the nonlinearsolution on the initial coarse mesh is not accurate enoughActually with an appropriate choice of initial coarsemesh themultilevel finite element VMS method can get better resultsthan one-level finite element VMS method Table 2 presentsthe computing time of these 119895-level (119895 = 1 2 3 4) finiteelement VMS methods and demonstrates that 119895-level finiteelement VMSmethod savesmuch computing time comparedwith 119895 minus 1-level (119895 = 2 3 4) finite element VMS method

53 Backward Facing Step Problem Backward facing stepproblem is also a standard numerical example which involvesa steady viscous incompressible flow over an isothermaltwo-dimensional backward facing step It is well-knownthat the backward facing step problem possesses a cornersingularity thus it is often adopted by many researchersto verify their numerical methods The geometry and theboundary conditions are shown in Figure 5 as in [21 46]

In this example the Reynolds number is 150 which isbased on the maximum inlet velocity V119909(max) = 1 and theheight of the inlet We compute an approximate solution onthe following number of triangle elements 9344 (for 1-levelfinite element VMS method) 5298-9344 (for 2-level finiteelement VMS method) 2392-5298-9344 (for 3-level finiteelement VMSmethod) and 646-2392-5298-9344 (for 4-levelfinite element VMS method) Figures 6(a)ndash6(d) show the

8 Mathematical Problems in Engineering

1-level2-level

3-level4-level

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

log(

erro

r)

ℎ3

minus5 minus45 minus4minus55log(h)

(a)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(b)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

minus75

minus7

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(c)

1

15

2

25

3

35

4

45

5

55

log(

CPU

(s))

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

(d)

Figure 1 Convergence analysis for the velocity and the pressure using 119895-level (119895 = 1 2 3 4) finite element VMS method (a) 1198712 error for thevelocity (b)1198671 error for the velocity (c) 1198712 error for the pressure (d) computing time

pressure field and the streamlines obtained by these 119895-level(119895 = 1 2 3 4) finite element VMS methods respectivelyand these results are in agreement with those of [21 46] Inorder to observe these results more clearly a comparison ofthe computed velocity pressure and vorticity is given acrossthe channel at 119909 = 15 in Figures 7(a)ndash7(d) it is obviousthat the results of these 119895-level (119895 = 1 2 3 4) finite element

VMS methods nearly have no differences with each otherin velocity and vorticity but the pressure computed by 4-level finite element VMS method has a clear difference fromothers The reason is that the initial grid chosen as 646 in4-level finite element VMS method is too coarse Table 3presents the computing time of these 119895-level (119895 = 1 2 3 4)finite element VMS methods it shows that the 119895-level finite

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

6 Mathematical Problems in Engineering

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987310038171003817100381710038171003817119906ℎ1119898100381710038171003817100381710038172

1+ 119862120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171

le 10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898 + 119862120572lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo(35)

Choosing 120572 = 119874(ℎ119904119898+1) for sufficiently small ℎ119898+1 120575 = 120574 minus119862120572 gt 0 Then we have

lsaquo (119906ℎ119898+1 119901ℎ119898+1) lsaquo le 1120575 (10038171003817100381710038171198911003817100381710038171003817minus1 + 11987311989621119898) (36)

Letting 1198961119895 = (119891minus1 + 11987311989621119898)120575minus1 then (33) holds for 119895 =119898 + 1

Next the convergence error of the multilevel finite ele-ment method is analyzed Subtracting (23) with 119895 = 119898 + 1from (6) yields

B [(119906 minus 119906ℎ119898+1 119906 minus 119901ℎ119898+1) (V 119902)]= 119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)

(37)

The right terms in (37) are bounded according to

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V)10038161003816100381610038161003816 le 119873 10038171003817100381710038171003817119906ℎ119898 minus 119906100381710038171003817100381710038172

1V1

le 119888 (120572 + ℎ119898)2 lsaquo (V 119902) lsaquo10038161003816100381610038161003816119866 (119906ℎ119898+1 V)10038161003816100381610038161003816 = 120572 10038171003817100381710038171003817(119868 minus Π)nabla119906ℎ119898+1100381710038171003817100381710038170 (119868 minus Π)nablaV0le 119888120572 10038171003817100381710038171003817119906ℎ119898+1100381710038171003817100381710038171 V1 le 119888120572lsaquo (V 119902) lsaquo

(38)

Taking (119868ℎ119906 119869ℎ119901) isin (119883ℎ119898+1 119872ℎ119898+1) as interpolation of (119906 119901)into (119883ℎ119898+1 119872ℎ119898+1) we have

lsaquo (119906 minus 119906ℎ119898+1 119901 minus 119901ℎ119898+1) lsaquo le lsaquo (119906 minus 119868ℎ119906 119901 minus 119869ℎ119901) lsaquo+ lsaquo (119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) lsaquo le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

B [(119868ℎ119906 minus 119906ℎ119898+1 119869ℎ119901 minus 119906ℎ119898+1) (V 119902)]lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 1120574

sdot sup(V119902)isin(119883ℎ119898+1 119872ℎ119898+1 )

10038161003816100381610038161003816119887 (119906ℎ119898 minus 119906 119906 minus 119906ℎ119898 V) + 119866 (119906ℎ119898+1 V)10038161003816100381610038161003816lsaquo (V 119902) lsaquo

le 119888 (120572 + ℎ119898+1) + 119888 (120572 + ℎ119898)2 + 119888120572

(39)

Then together with the relation ℎ119898+1 = 119874(ℎ2119898) denoting theconstant as 1198962119898+1 yields (34) for 119895 = 119898 + 1 Thus Theorem 4follows from the induction principle

5 Numerical Examples

In this section three numerical examples are given thefirst one is an exact solution problem the second one isthe lid-driven cavity problem and the third one is thebackward facing step problem The mentioned methods inthis paper are all implemented using Freefem++ [43] and theconvergence tolerance is set equal to 10minus6 In the numericalexamples the triangular mesh with the Taylor-Hood elementpair is used for the finite element discretization For the sakeof simplicity the one-level and the proposed multilevel finiteelement VMS methods are written as 119895-level finite elementVMS method in the numerical examples The nonlinearsystems in these methods are all linearized by the Oseeniterative method (see [44 45])

51 Problem with an Exact Solution This problem is a steadyflow of incompressible viscous Newtonian fluid in boundeddomain Ω = (0 1) times (0 1) with 119906 = (0 0) on the boundariesIt has the following exact solution

1199061 = 101199092 (119909 minus 1)2 119910 (119910 minus 1) (2119910 minus 1) 1199062 = minus10119909 (119909 minus 1) (2119909 minus 1) 1199102 (119910 minus 1)2 119901 = 10 (2119909 minus 1) (2119910 minus 1)

(40)

where 119891 = (1198911 1198912) is determined by (1) The Taylor-Hoodelements predict a convergence rate of 119874(ℎ2) in the energynorm (ie 1198671-norm for the velocity and 1198712-norm for thepressure) so the eddy viscosity parameter is chosen as 120572 =01ℎ2119895 for multilevel finite element VMS method which is thesame as that used in [21]

First ] = 01 is considered The meshes for the one-levelfinite element VMS method are chosen as ℎ119869 = [182 11021122 1142 1162] and these meshes are also the finestmeshes for the multilevel finite element VMS method Thechoices for the coarse meshes of the multilevel finite elementVMS method are very important If the coarse meshes arenot fine enough the optimal convergence orders of velocityand pressure can not be reached According to Theorems 3and 4 the coarse meshes ℎ119895minus1 and the fine mesh ℎ119895 should bechosen as ℎ119895 sim ℎ2119895minus1 119895 = 2 3 119869 which ensure optimalaccuracy of the approximate solution Here we consider themultilevel finite element VMS method for 119895 = 2 3 4 andmeshes are chosen as ℎ119895 = ℎ2119895minus1 respectively The numericalresults using the one-level and the proposed multilevel finiteelement VMSmethods are shown in Figure 1 which indicatesthat the proposed multilevel finite element VMS method canreach the convergence order of119874(ℎ2) for velocity in1198671-normand pressure in 1198712-norm and the optimal convergence orderof119874(ℎ3) for velocity in the 1198712-normThese numerical resultsare in agreement with the theoretical predictions Moreover

Mathematical Problems in Engineering 7

Table 1 Comparison of these 119895-level finite element VMS methods

Method 1ℎ CPU(s)

100381710038171003817100381710038171003817119906 minus 119906ℎ1198951003817100381710038171003817100381710038170

1199060100381710038171003817100381710038171003817119906 minus 119906ℎ119895

10038171003817100381710038171003817100381711199061

100381710038171003817100381710038171003817119901 minus 119901ℎ1198951003817100381710038171003817100381710038170100381710038171003817100381711990110038171003817100381710038170

1-level 128 198614 137319119890 minus 005 0000776894 472788119890 minus 0052-level 64-128 71389 102514119890 minus 005 0000351524 472788119890 minus 0053-level 32-64-128 37617 211498119890 minus 005 0000342189 472788119890 minus 0054-level 16-32-64-128 29495 0000110742 0000526664 47279119890 minus 0051-level 144 25086 85448119890 minus 006 0000554574 373553119890 minus 0052-level 72-144 92251 550229119890 minus 006 0000226809 373553119890 minus 0053-level 36-72-144 45212 930361119890 minus 006 0000206441 373553119890 minus 0054-level 18-36-72-144 35437 43288119890 minus 005 0000275735 373553119890 minus 0051-level 256 925036 902168119890 minus 007 0000107772 118393119890 minus 0052-level 128-256 307479 285225119890 minus 007 466203119890 minus 005 118403119890 minus 0053-level 64-128-256 141673 202856119890 minus 007 45437119890 minus 005 118383119890 minus 0054-level 32-64-128-256 99477 146991119890 minus 007 450338119890 minus 005 118391119890 minus 005

both the proposedmultilevel finite elementVMSmethod andthe one-level finite element VMS method obtain nearly thesame approximation results but the former significantly savescomputing time compared with the later

Then ] = 00001 is considered The mesh CPU-time the1198712-error of the velocity1198671-error of the velocity and 1198712-errorof the pressure for comparison of these 119895-level (119895 = 1 2 3 4)finite element VMS methods are tabulated in Table 1 Here(119906ℎ119895 119901ℎ119895) refers to the approximation solution obtained by 119895-level finite elementVMSmethod Table 1 indicates that 119895-level(119895 ge 2) finite element VMS method saves much computingtime compared with 119895 minus 1-level finite element VMS methodat the same fine mesh and the finer the mesh becomes thesmaller the error achieves Furthermore when scrutinizingthe error estimates of velocity and the pressure in Table 1 itis easily observed that multilevel (119895 = 2 3 4) finite elementVMS method for the velocity and the pressure is better thanone-level (119895 = 1) finite element VMS method

52 Lid-Driven Cavity Problem Lid-driven cavity problemoften serves as a standard benchmark problem for incom-pressible Navier-Stokes equations because the benchmarkdata is available for comparison In this problem the compu-tation is carried out in the region Ω = (119909 119910) | 0 lt 119909 119910 lt 1The velocity is 119906 = (1 0) on the top side (119909 1) 0 lt 119909 lt 1and 119906 = (0 0) on the rest of the boundary

We compute the approximate solutions for Re =(1000 5000 10000) and then compare the results of 1-levelfinite element VMS method (1ℎ = 256) 2-level finiteelement VMSmethod (1ℎ1 = 128 1ℎ2 = 256) 3-level finiteelement VMS method (1ℎ1 = 64 1ℎ2 = 128 1ℎ3 = 256)and 4-level finite element VMS method (1ℎ1 = 64 1ℎ2 =64 1ℎ3 = 128 1ℎ4 = 256) with that of Ghia et al [1]Figures 2ndash4 plot the horizontal component of the velocityalong the vertical line passing through the geometrical centerof the cavity and the vertical component of the velocityalong the horizontal line passing through the geometricalcenter of the cavity respectively As shown in Figure 2the results of multilevel finite element VMS method are

Table 2 Computing time of these 119895-level finite element VMSmethods

Re 1-level 2-level 3-level 4-level1000 162509 501335 193591 1159115000 298647 888397 452884 34708910000 439963 156004 670384 322642

indistinguishable from 1-level finite element VMS methodand the benchmark data in [1] However in Figures 3 and 4the 4-level finite element VMS method at Re = 5000 and the3-level and 4-level finite element VMSmethod at Re = 10000provide unsatisfactory results This is because the nonlinearsolution on the initial coarse mesh is not accurate enoughActually with an appropriate choice of initial coarsemesh themultilevel finite element VMS method can get better resultsthan one-level finite element VMS method Table 2 presentsthe computing time of these 119895-level (119895 = 1 2 3 4) finiteelement VMS methods and demonstrates that 119895-level finiteelement VMSmethod savesmuch computing time comparedwith 119895 minus 1-level (119895 = 2 3 4) finite element VMS method

53 Backward Facing Step Problem Backward facing stepproblem is also a standard numerical example which involvesa steady viscous incompressible flow over an isothermaltwo-dimensional backward facing step It is well-knownthat the backward facing step problem possesses a cornersingularity thus it is often adopted by many researchersto verify their numerical methods The geometry and theboundary conditions are shown in Figure 5 as in [21 46]

In this example the Reynolds number is 150 which isbased on the maximum inlet velocity V119909(max) = 1 and theheight of the inlet We compute an approximate solution onthe following number of triangle elements 9344 (for 1-levelfinite element VMS method) 5298-9344 (for 2-level finiteelement VMS method) 2392-5298-9344 (for 3-level finiteelement VMSmethod) and 646-2392-5298-9344 (for 4-levelfinite element VMS method) Figures 6(a)ndash6(d) show the

8 Mathematical Problems in Engineering

1-level2-level

3-level4-level

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

log(

erro

r)

ℎ3

minus5 minus45 minus4minus55log(h)

(a)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(b)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

minus75

minus7

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(c)

1

15

2

25

3

35

4

45

5

55

log(

CPU

(s))

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

(d)

Figure 1 Convergence analysis for the velocity and the pressure using 119895-level (119895 = 1 2 3 4) finite element VMS method (a) 1198712 error for thevelocity (b)1198671 error for the velocity (c) 1198712 error for the pressure (d) computing time

pressure field and the streamlines obtained by these 119895-level(119895 = 1 2 3 4) finite element VMS methods respectivelyand these results are in agreement with those of [21 46] Inorder to observe these results more clearly a comparison ofthe computed velocity pressure and vorticity is given acrossthe channel at 119909 = 15 in Figures 7(a)ndash7(d) it is obviousthat the results of these 119895-level (119895 = 1 2 3 4) finite element

VMS methods nearly have no differences with each otherin velocity and vorticity but the pressure computed by 4-level finite element VMS method has a clear difference fromothers The reason is that the initial grid chosen as 646 in4-level finite element VMS method is too coarse Table 3presents the computing time of these 119895-level (119895 = 1 2 3 4)finite element VMS methods it shows that the 119895-level finite

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

Mathematical Problems in Engineering 7

Table 1 Comparison of these 119895-level finite element VMS methods

Method 1ℎ CPU(s)

100381710038171003817100381710038171003817119906 minus 119906ℎ1198951003817100381710038171003817100381710038170

1199060100381710038171003817100381710038171003817119906 minus 119906ℎ119895

10038171003817100381710038171003817100381711199061

100381710038171003817100381710038171003817119901 minus 119901ℎ1198951003817100381710038171003817100381710038170100381710038171003817100381711990110038171003817100381710038170

1-level 128 198614 137319119890 minus 005 0000776894 472788119890 minus 0052-level 64-128 71389 102514119890 minus 005 0000351524 472788119890 minus 0053-level 32-64-128 37617 211498119890 minus 005 0000342189 472788119890 minus 0054-level 16-32-64-128 29495 0000110742 0000526664 47279119890 minus 0051-level 144 25086 85448119890 minus 006 0000554574 373553119890 minus 0052-level 72-144 92251 550229119890 minus 006 0000226809 373553119890 minus 0053-level 36-72-144 45212 930361119890 minus 006 0000206441 373553119890 minus 0054-level 18-36-72-144 35437 43288119890 minus 005 0000275735 373553119890 minus 0051-level 256 925036 902168119890 minus 007 0000107772 118393119890 minus 0052-level 128-256 307479 285225119890 minus 007 466203119890 minus 005 118403119890 minus 0053-level 64-128-256 141673 202856119890 minus 007 45437119890 minus 005 118383119890 minus 0054-level 32-64-128-256 99477 146991119890 minus 007 450338119890 minus 005 118391119890 minus 005

both the proposedmultilevel finite elementVMSmethod andthe one-level finite element VMS method obtain nearly thesame approximation results but the former significantly savescomputing time compared with the later

Then ] = 00001 is considered The mesh CPU-time the1198712-error of the velocity1198671-error of the velocity and 1198712-errorof the pressure for comparison of these 119895-level (119895 = 1 2 3 4)finite element VMS methods are tabulated in Table 1 Here(119906ℎ119895 119901ℎ119895) refers to the approximation solution obtained by 119895-level finite elementVMSmethod Table 1 indicates that 119895-level(119895 ge 2) finite element VMS method saves much computingtime compared with 119895 minus 1-level finite element VMS methodat the same fine mesh and the finer the mesh becomes thesmaller the error achieves Furthermore when scrutinizingthe error estimates of velocity and the pressure in Table 1 itis easily observed that multilevel (119895 = 2 3 4) finite elementVMS method for the velocity and the pressure is better thanone-level (119895 = 1) finite element VMS method

52 Lid-Driven Cavity Problem Lid-driven cavity problemoften serves as a standard benchmark problem for incom-pressible Navier-Stokes equations because the benchmarkdata is available for comparison In this problem the compu-tation is carried out in the region Ω = (119909 119910) | 0 lt 119909 119910 lt 1The velocity is 119906 = (1 0) on the top side (119909 1) 0 lt 119909 lt 1and 119906 = (0 0) on the rest of the boundary

We compute the approximate solutions for Re =(1000 5000 10000) and then compare the results of 1-levelfinite element VMS method (1ℎ = 256) 2-level finiteelement VMSmethod (1ℎ1 = 128 1ℎ2 = 256) 3-level finiteelement VMS method (1ℎ1 = 64 1ℎ2 = 128 1ℎ3 = 256)and 4-level finite element VMS method (1ℎ1 = 64 1ℎ2 =64 1ℎ3 = 128 1ℎ4 = 256) with that of Ghia et al [1]Figures 2ndash4 plot the horizontal component of the velocityalong the vertical line passing through the geometrical centerof the cavity and the vertical component of the velocityalong the horizontal line passing through the geometricalcenter of the cavity respectively As shown in Figure 2the results of multilevel finite element VMS method are

Table 2 Computing time of these 119895-level finite element VMSmethods

Re 1-level 2-level 3-level 4-level1000 162509 501335 193591 1159115000 298647 888397 452884 34708910000 439963 156004 670384 322642

indistinguishable from 1-level finite element VMS methodand the benchmark data in [1] However in Figures 3 and 4the 4-level finite element VMS method at Re = 5000 and the3-level and 4-level finite element VMSmethod at Re = 10000provide unsatisfactory results This is because the nonlinearsolution on the initial coarse mesh is not accurate enoughActually with an appropriate choice of initial coarsemesh themultilevel finite element VMS method can get better resultsthan one-level finite element VMS method Table 2 presentsthe computing time of these 119895-level (119895 = 1 2 3 4) finiteelement VMS methods and demonstrates that 119895-level finiteelement VMSmethod savesmuch computing time comparedwith 119895 minus 1-level (119895 = 2 3 4) finite element VMS method

53 Backward Facing Step Problem Backward facing stepproblem is also a standard numerical example which involvesa steady viscous incompressible flow over an isothermaltwo-dimensional backward facing step It is well-knownthat the backward facing step problem possesses a cornersingularity thus it is often adopted by many researchersto verify their numerical methods The geometry and theboundary conditions are shown in Figure 5 as in [21 46]

In this example the Reynolds number is 150 which isbased on the maximum inlet velocity V119909(max) = 1 and theheight of the inlet We compute an approximate solution onthe following number of triangle elements 9344 (for 1-levelfinite element VMS method) 5298-9344 (for 2-level finiteelement VMS method) 2392-5298-9344 (for 3-level finiteelement VMSmethod) and 646-2392-5298-9344 (for 4-levelfinite element VMS method) Figures 6(a)ndash6(d) show the

8 Mathematical Problems in Engineering

1-level2-level

3-level4-level

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

log(

erro

r)

ℎ3

minus5 minus45 minus4minus55log(h)

(a)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(b)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

minus75

minus7

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(c)

1

15

2

25

3

35

4

45

5

55

log(

CPU

(s))

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

(d)

Figure 1 Convergence analysis for the velocity and the pressure using 119895-level (119895 = 1 2 3 4) finite element VMS method (a) 1198712 error for thevelocity (b)1198671 error for the velocity (c) 1198712 error for the pressure (d) computing time

pressure field and the streamlines obtained by these 119895-level(119895 = 1 2 3 4) finite element VMS methods respectivelyand these results are in agreement with those of [21 46] Inorder to observe these results more clearly a comparison ofthe computed velocity pressure and vorticity is given acrossthe channel at 119909 = 15 in Figures 7(a)ndash7(d) it is obviousthat the results of these 119895-level (119895 = 1 2 3 4) finite element

VMS methods nearly have no differences with each otherin velocity and vorticity but the pressure computed by 4-level finite element VMS method has a clear difference fromothers The reason is that the initial grid chosen as 646 in4-level finite element VMS method is too coarse Table 3presents the computing time of these 119895-level (119895 = 1 2 3 4)finite element VMS methods it shows that the 119895-level finite

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

8 Mathematical Problems in Engineering

1-level2-level

3-level4-level

minus19

minus18

minus17

minus16

minus15

minus14

minus13

minus12

log(

erro

r)

ℎ3

minus5 minus45 minus4minus55log(h)

(a)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(b)

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

minus75

minus7

log(

erro

r)

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

ℎ2

(c)

1

15

2

25

3

35

4

45

5

55

log(

CPU

(s))

minus5 minus45 minus4minus55log(h)

1-level2-level

3-level4-level

(d)

Figure 1 Convergence analysis for the velocity and the pressure using 119895-level (119895 = 1 2 3 4) finite element VMS method (a) 1198712 error for thevelocity (b)1198671 error for the velocity (c) 1198712 error for the pressure (d) computing time

pressure field and the streamlines obtained by these 119895-level(119895 = 1 2 3 4) finite element VMS methods respectivelyand these results are in agreement with those of [21 46] Inorder to observe these results more clearly a comparison ofthe computed velocity pressure and vorticity is given acrossthe channel at 119909 = 15 in Figures 7(a)ndash7(d) it is obviousthat the results of these 119895-level (119895 = 1 2 3 4) finite element

VMS methods nearly have no differences with each otherin velocity and vorticity but the pressure computed by 4-level finite element VMS method has a clear difference fromothers The reason is that the initial grid chosen as 646 in4-level finite element VMS method is too coarse Table 3presents the computing time of these 119895-level (119895 = 1 2 3 4)finite element VMS methods it shows that the 119895-level finite

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

Mathematical Problems in Engineering 9

Ghia et al1-level2-level

3-level4-level

minus04

minus02

0

02

04

06

08

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

(a)

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 2 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 1000

0 02 04 06 08 1y

minus05

0

05

1

x co

mpo

nent

of v

eloci

ty

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 3 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 5000

element VMSmethod savesmuch computing time comparedwith 119895 minus 1-level finite element VMS method From thisexample it can be concluded that the proposed multilevelfinite element VMSmethod is also efficient for problems witha corner singularity and it is a highly effective and time-saving method if the coarse grid is chosen appropriately

6 Conclusions

In this paper amultilevel finite element variationalmultiscalemethod based on two local Gauss integrations is proposedand applied to the numerical simulation of steady incom-pressible Navier-Stokes equations Theoretically the stability

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

10 Mathematical Problems in Engineering

minus05

0

05

1x

com

pone

nt o

f velo

city

02 04 06 08 10y

Ghia et al1-level2-level

3-level4-level

(a)

minus08

minus06

minus04

minus02

0

02

04

06

y co

mpo

nent

of v

eloci

ty

02 04 06 08 10x

Ghia et al1-level2-level

3-level4-level

(b)

Figure 4 Horizontal (a) and vertical (b) of velocity by 119895-level finite element VMS method at Re = 10000

105

1

Tractionfree

4

Vxmax = 1

Vy = 0

Vy = 0Vx = 0

Vy = 0Vx = 0

Figure 5 Problem description of the backward facing step

Y

0

05

1

1 2 3 40X

(a) 1-level finite element VMS method

Y

0

05

1

1 2 3 40X

(b) 2-level finite element VMS method

0

05

1

Y

1 2 3 40X

(c) 3-level finite element VMS method

0

05

1

Y

1 2 3 40X

(d) 4-level finite element VMS method

Figure 6 Computed pressure field and streamlines by these 119895-level (119895 = 1 2 3 4) finite element VMS methods

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

Mathematical Problems in Engineering 11

1-level2-level

3-level4-level

0

01

02

03

04

05

06

07

08

09

1

y

0 02 04 06 08 1 12minus02u1-velocity

(a) 1199061-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus01 minus008 minus006 minus004 minus002 0 002minus012u2-velocity

1-level2-level

3-level4-level

(b) 1199062-velocity at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

minus007 minus0065 minus006 minus0055 minus005 minus0045minus0075Pressure

1-level2-level

3-level4-level

(c) Pressure at 119909 = 15

0

01

02

03

04

05

06

07

08

09

1

y

0 005 01 015minus005Vorticity

1-level2-level

3-level4-level

(d) Vorticity at 119909 = 15

Figure 7 Comparison of 1199061-velocity 1199062-velocity pressure and vorticity profiles at 119909 = 15

Table 3 Computing time of these 119895-level finite element VMSmethods

Method 1-level 2-level 3-level 4-levelComputing time 157896 98153 56247 31261

and error estimate of the multilevel finite element variationalmultiscale method are analyzed Numerically three numer-ical examples are given The numerical results demonstratethat under an appropriate initial coarse mesh the multilevel

finite element variational multiscale method is more efficientthan standard one-level finite element variational multiscalemethod In addition this multilevel finite element variationalmultiscale method can be extended to solve other morecomplex fluid dynamical models which will be discussed infurther work

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

12 Mathematical Problems in Engineering

Acknowledgments

The project is supported by 973 Program (no 2012CB026002)the NSF of China (no 51305355) and the National KeyTechnology RampD Program of China (no 2013BAF01B02)

References

[1] U Ghia K N Ghia and C T Shin ldquoHigh-Re solutions forincompressible flow using the Navier-Stokes equations and amultigrid methodrdquo Journal of Computational Physics vol 48no 3 pp 387ndash411 1982

[2] E Erturk T C Corke and C Gokcol ldquoNumerical solutions of2-D steady incompressible driven cavity flow at high Reynoldsnumbersrdquo International Journal for Numerical Methods in Flu-ids vol 48 no 7 pp 747ndash774 2005

[3] Y Shang ldquoA two-level subgrid stabilizedOseen iterativemethodfor the steady Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 233 pp 210ndash226 2013

[4] L P Franca and S L Frey ldquoStabilized finite element methodsII The incompressible Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 99 no 2-3 pp 209ndash233 1992

[5] L P Franca andT J Hughes ldquoConvergence analyses ofGalerkinleast-squares methods for symmetric advective-diffusive formsof the STOkes and incompressible Navier-STOkes equationsrdquoComputer Methods in Applied Mechanics and Engineering vol105 no 2 pp 285ndash298 1993

[6] A N Brooks and T J Hughes ldquoStreamline upwindPetrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equa-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 32 no 1-3 pp 199ndash259 1982

[7] W Layton H K Lee and J Peterson ldquoA defect-correctionmethod for the incompressible Navier-Stokes equationsrdquoApplied Mathematics and Computation vol 129 no 1 pp 1ndash192002

[8] Z Si and Y He ldquoA defect-correction mixed finite ele-ment method for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2011 Article ID370192 28 pages 2011

[9] D Arndt H Dallmann and G Lube ldquoLocal projection FEMstabilization for the time-dependent incompressible Navier-Stokes problemrdquo Numerical Methods for Partial DifferentialEquations An International Journal vol 31 no 4 pp 1224ndash12502015

[10] A Montlaur S Fernandez-Mendez J Peraire and A HuertaldquoDiscontinuous Galerkin methods for the Navier-Stokes equa-tions using solenoidal approximationsrdquo International Journalfor Numerical Methods in Fluids vol 64 no 5 pp 549ndash5642010

[11] N C Nguyen J Peraire and B Cockburn ldquoAn implicit high-order hybridizable discontinuous Galerkin method for theincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 230 no 4 pp 1147ndash1170 2011

[12] B Riviere and S Sardar ldquoPenalty-free discontinuous Galerkinmethods for incompressible Navier-Stokes equationsrdquo Mathe-matical Models and Methods in Applied Sciences vol 24 no 6pp 1217ndash1236 2014

[13] T J R Hughes L Mazzei and K E Jansen ldquoLarge EddySimulation and the variational multiscale methodrdquo Computingand Visualization in Science vol 3 no 1-2 pp 47ndash59 2000

[14] T J R Hughes L Mazzei A A Oberai and A A WrayldquoThe multiscale formulation of large eddy simulation Decay ofhomogeneous isotropic turbulencerdquoPhysics of Fluids vol 13 no2 pp 505ndash512 2001

[15] T J R Hughes A A Oberai and L Mazzei ldquoLarge eddy sim-ulation of turbulent channel flows by the variational multiscalemethodrdquo Physics of Fluids vol 13 no 6 pp 1784ndash1799 2001

[16] T J R Hughes V M Calo and G Scovazzi Variationaland Multiscale Methods in Turbulence Mechanics of the 21stCentury 2005 153ndash163

[17] J-L Guermond ldquoStabilization of Galerkin approximations oftransport equations by subgrid modelingrdquo M2AN Mathemat-ical Modelling and Numerical Analysis vol 33 no 6 pp 1293ndash1316 1999

[18] W Layton ldquoA connection between subgrid scale eddy viscosityand mixed methodsrdquo Applied Mathematics and Computationvol 133 no 1 pp 147ndash157 2002

[19] V John and S Kaya ldquoA finite element variational multiscalemethod for the Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 26 no 5 pp 1485ndash1503 2005

[20] S S Collis ldquoMonitoring unresolved scales in multiscale turbu-lence modelingrdquo Physics of Fluids vol 13 no 6 pp 1800ndash18062001

[21] H Zheng Y Hou F Shi and L Song ldquoA finite elementvariational multiscale method for incompressible flows basedon two local Gauss integrationsrdquo Journal of ComputationalPhysics vol 228 no 16 pp 5961ndash5977 2009

[22] F Shi H Zheng J Yu and Y Li ldquoOn the convergence ofvariational multiscale methods based on Newtonrsquos iterationfor the incompressible flowsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 23 pp 5726ndash5742 2014

[23] J Yu H Zheng and F Shi ldquoA finite element variationalmultiscale method for incompressible flows based on theconstruction of the projection basis functionsrdquo InternationalJournal for Numerical Methods in Fluids vol 70 no 6 pp 793ndash804 2012

[24] Y Shang and S Huang ldquoA parallel subgrid stabilized finiteelement method based on two-grid discretization for simula-tion of 2D3D Steady incompressible flowsrdquo Journal of ScientificComputing vol 60 no 3 pp 564ndash583 2014

[25] Y Jiang L Mei H Wei W Tian and J Ge ldquoA finite ele-ment variational multiscale method based on two local gaussintegrations for stationary conduction-convection problemsrdquoMathematical Problems in Engineering vol 2012 Article ID747391 14 pages 2012

[26] W Layton and H W Lenferink ldquoA multilevel mesh indepen-dence principle for the Navier-Stokes equationsrdquo SIAM Journalon Numerical Analysis vol 33 no 1 pp 17ndash30 1996

[27] C Calgaro A Debussche and J Laminie ldquoOn a multilevelapproach for the two-dimensional Navier-Stokes equationswith finite elementsrdquo International Journal for Numerical Meth-ods in Fluids vol 27 no 1-4 Special Issue pp 241ndash258 1998

[28] W Layton H K Lee and J Peterson ldquoNumerical solution ofthe stationary Navier-Stokes equations using a multilevel finiteelementmethodrdquo SIAM Journal on Scientific Computing vol 20no 1 pp 1ndash12 1998

[29] Y He and K-M Liu ldquoA multilevel finite element method inspace-time for the Navier-Stokes problemrdquo Numerical Methodsfor Partial Differential Equations An International Journal vol21 no 6 pp 1052ndash1078 2005

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

Mathematical Problems in Engineering 13

[30] J Li Y He and H Xu ldquoA multi-level stabilized finite elementmethod for the stationary Navier-Stokes equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196 no 29-30 pp 2852ndash2862 2007

[31] J Li Z Chen and Y He ldquoA stabilized multi-level methodfor non-singular finite volume solutions of the stationary 3DNavier-Stokes equationsrdquoNumerische Mathematik vol 122 no2 pp 279ndash304 2012

[32] X Zhao J Li J Su and G Lei ldquoAnalysis of newton multilevelstabilized finite volume method for the three-dimensionalstationary Navier-Stokes equationsrdquo Numerical Methods forPartial Differential Equations vol 29 no 6 pp 2146ndash2160 2013

[33] P Kopp Multi-level hp-FEM and the Finite Cell Method for theNavier-Stokes equations using a Variational Multiscale Formula-tion [MS thesis] Technische Universitat Munchen 2017

[34] V John S Kaya and W Layton ldquoA two-level variational mul-tiscale method for convection-dominated convection-diffusionequationsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 195 no 33-36 pp 4594ndash4603 2006

[35] Y Li L Mei Y Li and K Zhao ldquoA two-level variationalmultiscale method for incompressible flows based on two localGauss integrationsrdquo Numerical Methods for Partial DifferentialEquations vol 29 no 6 pp 1986ndash2003 2013

[36] R Temam Navier-Stokes Equations North-Holland Amster-dam The Netherlands 1984

[37] S Kaya W Layton and B Rivire ldquoSubgrid stabilized defectcorrection methods for the Navier-Stokes equationsrdquo SIAMJournal on Numerical Analysis vol 44 no 4 pp 1639ndash16542006

[38] Y He ldquoStability and convergence of iterative methods relatedto viscosities for the 2D3D Steady Navier-Stokes equationsrdquoJournal of Mathematical Analysis and Applications vol 423 no2 pp 1129ndash1149 2015

[39] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 1991

[40] W Layton and L Tobiska ldquoA two-level method with back-tracking for the Navier-Stokes equationsrdquo SIAM Journal onNumerical Analysis vol 35 no 5 pp 2035ndash2054 1998

[41] C Taylor and P Hood ldquoA numerical solution of the Navier-Stokes equations using the finite element techniquerdquoComputersand Fluids vol 1 no 1 pp 73ndash100 1973

[42] V John and S Kaya ldquoFinite element error analysis of avariational multiscale method for the Navier-Stokes equationsrdquoAdvances in Computational Mathematics vol 28 no 1 pp 43ndash61 2008

[43] ldquoFreefem++rdquo version 338 httpwwwfreefemorg[44] Y He and J Li ldquoConvergence of three iterative methods based

on the finite element discretization for the stationary Navier-Stokes equationsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 15-16 pp 1351ndash1359 2009

[45] R An ldquoComparisons of STOkesOseenNewton iterationmethods for NAVier-STOkes equations with friction boundaryconditionsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol38 no 23 pp 5535ndash5544 2014

[46] A Masud and R A Khurram ldquoA multiscale finite elementmethod for the incompressible Navier-Stokes equationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 195no 13-16 pp 1750ndash1777 2006

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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