Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 951692 10 pageshttpdxdoiorg1011552013951692

Research ArticleA New Characteristic Nonconforming Mixed Finite ElementScheme for Convection-Dominated Diffusion Problem

Dongyang Shi1 Qili Tang12 and Yadong Zhang3

1 Department of Mathematics Zhengzhou University Zhengzhou 450001 China2 School of Mathematics and Statistics Henan University of Science and Technology Luoyang 471003 China3 School of Mathematics and Statistics Xuchang University Xuchang 461000 China

Correspondence should be addressed to Qili Tang tql132163com

Received 14 December 2012 Accepted 23 March 2013

Academic Editor Junjie Wei

Copyright copy 2013 Dongyang Shi 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 characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusionproblem based on a new mixed variational formulation The optimal order error estimates for both the original variable 119906 andthe auxiliary variable 120590 with respect to the space are obtained by employing some typical characters of the interpolation operatorinstead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis Atlast we give some numerical results to confirm the theoretical analysis

1 Introduction

Consider the following convection-dominated diffusion pro-blem

119906119905+ a (119909 119910) sdot nabla119906 minus nabla sdot (119887 (119909 119910) nabla119906)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(1)

whereΩ is a bounded polygonal domain inR2 with Lipschitzcontinuous boundary 120597Ω 119869 = (0 119879] 0 lt 119879 lt +infin nabla andnablasdot denote the gradient and the divergence operators respec-tively

Model (1) has been widely used to describe the conduc-tion of heat in fluid the diffusion of soluble minerals or pol-lutants in groundwater the incompressiblemiscible displace-ment in porous media and so on The parameters appearingin (1) satisfy the following assumptions [1 2]

(A1) 119906 denotes for example the concentration or satura-tion of soluble substances

(A2) a(119909 119910) = (1198861(119909 119910) 119886

2(119909 119910)) represents Darcy veloc-

ity of mixed fluid and 119891 a source term(A3) 119887(119909 119910) is sufficiently smooth and there exist constants

1198871and 1198872 such that

0 lt 1198871le 119887 (119909 119910) le 119887

2lt +infin forall (119909 119910) isin Ω (2)

It is well known that convection dominated-diffusionproblem (1) often presents serious numerical difficultiesThe standard numerical methods such as finite differencemethod (FDM) FEMandMFEM usually produce numericaldiffusion along sharp fronts In order to overcome this fataldefect Douglas et al [3] combined the method of chara-cteristics with FE procedures so as to reduce the truncationerror and it allows us to use large time steps without lose ofaccuracy Moreover there have appeared many effective dis-cretization schemes concentrating on the hyperbolic natureof the equation for example characteristic FD streamline dif-fusion method [4 5] Eulerian-Lagrangian method [6 7]characteristic-finite volume element method [2 8 9]

2 Journal of Applied Mathematics

characteristics-mixed covolume method [10 11] the mod-ified method of characteristic-Galerkin FE procedure [12]characteristic nonconforming FEM [13ndash15] characteristicMFEM [16ndash19] and expanded characteristic MFEM [1 20]and so forth

As for the characteristic MFEM or expanded character-istic MFEM the convergence rates of 119906 and 120590 in existingliterature were suboptimal [11 18 21 22] and the convergenceanalysis was valid only to the case of the lowest order MFEapproximation [10 17] So far to our best knowledgethere are few studies on the optimal order error estimatesexcept for [23] in which a family of characteristic MFEMwith arbitrary degree of Raviart-Thomas-Nedelec space in[24 25] for transient convection diffusion equations wasstudied

Recently based on the low regularity requirement of theflux variable in practical problems a new mixed variationalform for second elliptic problem was proposed in [26] It hastwo typical advantages the flux space belongs to the squareintegrable space instead of the traditional119867( div Ω) whichmakes the choices ofMFE spaces sufficiently simple and easythe LBB condition is automatically satisfiedwhen the gradientof approximation space for the original variable is includedin approximation space for the flux variable Motivated bythis idea this paper will construct a characteristic noncon-forming MFE scheme for (1) with a new mixed variationalformulation Similar to the expanded characteristic MFEMthe coefficient 119887 of (1) in this proposed scheme does notneed to be inverted therefore it is also suitable for the casewhen 119887 is small By employing some distinct characters ofthe interpolation operators on the element instead of themixed or expandedmixed elliptic projection used in [1 17 20]which is an indispensable tool in the traditional characteristicMFEM analysis the 119874(ℎ2) order error estimate in 119871

2-normfor original variable 119906 which is one order higher than [1 20]and half order higher than [18] is derived and the optimalerror estimates with order119874(ℎ) for auxiliary variable 120590 in 1198712-normand for119906 in broken1198671-normare obtained respectivelyIt seems that the result for 119906 in broken 119867

1-norm has neverbeen seen in the existing literature by making full use of thehigh-accuracy estimates of the lowest order Raviart-Thomaselement proved by the technique of integral identities in [27]and the special properties of nonconforming 119864119876rot

1element

(see Lemma 1 below)The paper is organized as follows Section 2 is devoted

to the introduction of the nonconforming FE approximationspaces and their corresponding interpolation operators InSection 3 we will give the construction of the new charac-teristic nonconformingMFE scheme and two important lem-mas and the existence and uniqueness of the discrete schemesolutionwill be proved In Section 4 the convergence analysisand optimal error estimates for both the original variable119906 and the flux variable 120590 are obtained In Section 5 somenumerical results are provided to illustrate the effectivenessof our proposed method

Throughout this paper119862 denotes a generic positive cons-tant independent of the mesh parameters ℎ and Δ119905 withrespect to domainΩ and time 119905

2 Construction of Nonconforming MFEs

As in [28] we frequently employ the space 1198712(Ω) of squareintegrable functions with scalar product and norm

(119906 V) = (119906 V)1198712(Ω)

= (int

Ω

119906V119889119909119889119910)12

V = V1198712(Ω)

= (int

Ω

V2119889119909119889119910)

12

(3)

We also employ the Sobolev space 119867119898(Ω) 119898 ge 1 of func-tions V such that 119863120573V isin 1198712(Ω) for all |120573| le 119898 equipped withthe norm and seminorm

V119898Ω

= V119867119898(Ω)

= ( sum

|120573|le119898

10038171003817100381710038171003817119863120573V10038171003817100381710038171003817

2

)

12

|V|119898Ω

= |V|119867119898(Ω)

= ( sum

|120573|=119898

10038171003817100381710038171003817119863120573V10038171003817100381710038171003817

2

)

12

(4)

The space 11986710(Ω) denotes the closure of the set of infinitely

differentiable functions with compact supports inΩ For anySobolev space 119884 119871119901(0 119879 119884) is the space of measurable 119884-valued functions Φ of 119905 isin (0 119879) such that int119879

0Φ(sdot 119905)

119901

119884119889119905 lt

infin if 1 le 119901 lt infin or such that ess sup0lt119905lt119879

Φ(sdot 119905)119884lt infin if

119901 = infinWe now introduce the nonconforming MFE space des-

cribed in [29] for and summarize it as followsLet Ω sub R2 be a polygon domain with edges parallel to

the coordinate axes on 119909119910 plane and let 119879ℎbe a rectangular

subdivision of Ω satisfying the regular condition [30] For agiven element 119890 isin 119879

ℎ denote the barycenter of element 119890 by

(119909119890 119910119890) denote the length of edges parallel to 119909-axis and 119910-

axis by 2ℎ119909119890

and 2ℎ119910119890

respectively ℎ119890= max

119890isin119879ℎ

ℎ119909119890

ℎ119910119890

ℎ =

max119890isin119879ℎ

ℎ119890

Let 119890 = [minus1 1] times [minus1 1] be the reference element on 119909119910plane and four vertices

1198891= (minus1 minus1) 119889

2= (1 minus1) 119889

3=

(1 1) and 1198894= (minus1 1) the four edges 119897

1=

11988911198892 1198972=

11988921198893

1198973=11988931198894 and 119897

4=11988941198891 Then there exists an affine mapping

119865119890 119890 rarr 119890 as

119909 = 119909119890+ ℎ119909119890

119909

119910 = 119910119890+ ℎ119910119890

119910

(5)

Define the FE spaces (119890 119875119894 sum119894

) (119894 = 1 2 3) bysum

1

= V1 V2 V3 V4 V5

1= span 1 119909 119910 120601 (119909) 120601 (119910)

sum

2

= 1199011 1199012

2= span 1 119909

sum

3

= 1199021 1199022

3= span 1 119910

(6)

where V119894= (1|

119897119894|) int119894

V119889119904 (119894 = 1 2 3 4) V5

= (1|119890|)

int119890V119889119909 119889119910 120601(119905) = (12)(3119905

2minus 1) 119901

119894= (1|

1198972119894|) int2119894

119901119889119904 119902119894=

(1|1198972119894minus1

|) int2119894minus1

119902119889119904 (119894 = 1 2)

Journal of Applied Mathematics 3

The interpolation operators on 119890 are defined as follows

Π1 V isin 119867

1(119890) 997888rarr Π

1V isin 1

int

119894

(Π1V minus V) 119889119904 = 0 (119894 = 1 2 3 4)

int

119890

(Π1V minus V) 119889119909 119889119910 = 0

Π2 119901 isin 119867

1(119890) 997888rarr Π

2119901 isin

2

int

2119894

(Π2119901 minus 119901) 119889119904 = 0 (119894 = 1 2)

Π3 119902 isin 119867

1(119890) 997888rarr Π

3119902 isin 3

int

2119894minus1

(Π3119902 minus 119902) 119889119904 = 0 (119894 = 1 2)

(7)

Then the associated nonconforming 119864119876rot1

element space119872ℎ

[29] and lowest order Raviart-Thomas element space Vℎ[25

27] are defined as

119872ℎ= Vℎ Vℎ|119890= V ∘ 119865

minus1

119890 V isin

1

int

119865

[Vℎ] 119889119904 = 0 119865 sub 120597119890

Vℎ= wℎ= (119908ℎ1 119908ℎ2)

wℎ|119890= (1199081∘ 119865minus1

119890 1199082∘ 119865minus1

119890)

w = (1199081 1199082) isin 2times 3

(8)

respectively where [120593] represents the jump value of 120593 acrossthe boundary 119865 and [120593] = 120593 if 119865 sub 120597Ω

Similarly the interpolation operators 1205871

ℎand 120587

2

ℎare

defined as

1205871

ℎ 1198671(Ω) 997888rarr 119872

ℎ 120587

1

ℎ

10038161003816100381610038161003816119890= 1205871

119890

1205871

119890V = (Π

1V) ∘ 119865

minus1

119890 forallV isin 119867

1(Ω)

1205872

ℎ (1198671(Ω))

2

997888rarr Vℎ 1205872

ℎ|119890= 1205872

119890

1205872

119890w = ((Π

21199081) ∘ 119865minus1

119890 (Π31199082) ∘ 119865minus1

119890)

forallw = (1199081 1199082) isin (119867

1(Ω))

2

(9)

3 New Characteristic Nonconforming MFEScheme and Two Lemmas

Let 120595(119909 119910) = (1 + |a(119909 119910)|2)12 and 120591 = 120591(119909 119910) be the chara-cteristic direction associated with 119906

119905+ a(119909 119910) sdot nabla119906 such that

120597

120597120591

=

1

120595 (119909 119910)

120597

120597119905

+

a (119909 119910)120595 (119909 119910)

sdot nabla (10)

Then (1) can be put in the following system

120595 (119909 119910)

120597119906

120597120591

minus nabla sdot (119887 (119909 119910) nabla119906) = 119891 (119909 119910 119905)

forall (119909 119910 119905) isin Ω times (0 119879]

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879]

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(11)

By introducing 120590 = minus119887(119909 119910)nabla119906 and using Greenrsquosformula we obtain the new characteristic mixed form of (11)Find (119906 120590) (0 119879] rarr 119867

1

0(Ω) times (119871

2(Ω))

2 such that

(120595 (119909 119910)

120597119906

120597120591

V) minus (120590 nablaV) = (119891 (119909 119910 119905) V) forallV isin 1198671

0(Ω)

(120590w) + (119887 (119909 119910) nabla119906w) = 0 forallw isin (1198712(Ω))

2

(12)

Let Δ119905 gt 0119873 = 119879Δ119905 isin Z 119905119899 = 119899Δ119905 and 120601119899 = 120601(119909 119910 119905119899)

When solving 119906119899+1

ℎ we would like to make the scheme as

implicit as possible by using of the characteristic vector 120591Denote119883 = (119909 119910) isin Ω and

119883 = 119883 minus a (119909 119910) Δ119905 (13)

similar to [1 3] and then we have the following approxima-tion

120595 (119909 119910)

120597119906

120597120591

10038161003816100381610038161003816100381610038161003816119905119899

asymp 120595 (119909 119910)

119906 (119883 119905119899) minus 119906 (119883 119905

119899minus1)

radic(119883 minus 119883)

2

+ (Δ119905)2

=

119906 (119883 119905119899) minus 119906 (119883 119905

119899minus1)

Δ119905

=

119906119899minus 119906119899minus1

Δ119905

(14)

This leads to the following characteristic nonconformingMFE scheme Find (119906

ℎ 120590ℎ) 1199050 1199051 119905119873 rarr 119872

ℎtimesVℎ such

that

(

119906119899

ℎminus 119906119899minus1

ℎ

Δ119905

Vℎ) minus (120590

119899

ℎ nablaVℎ)ℎ= (119891119899 Vℎ) forallV

ℎisin 119872ℎ

(15a)

(120590119899

ℎwℎ) + (119887nabla119906

119899

ℎwℎ)ℎ= 0 forallw

ℎisin Vℎ (15b)

1199060

ℎ= 1205871

ℎ1199060(119909 119910) 120590

0

ℎ= 1205872

ℎ(119887nabla1199060(119909 119910)) forall (119909 119910) isin Ω

(15c)

where 119906119899ℎ= 119906ℎ(119883 119905119899) (119906 V)

ℎ= sum119890isin119879ℎ

int119890119906V119889119909119889119910 Generally

speaking 119906119899minus1ℎ

(119899 = 2 119873) are not node values and shouldbe derived by interpolation formulas on 119906119899minus1

ℎ

Remark 1 In [1] the expanded characteristic MFE schemewas presented by introducing two new auxiliary variableswhich avoided the inversion of the coefficient 119887 when 119887 issmallThe newmixed schemes (15a) (15b) and (15c) not onlykeep the advantage of expanded characteristic MFE schemebut also donot need to solve three variables

4 Journal of Applied Mathematics

Now we prove the existence and uniqueness of the solu-tion of (15a) (15b) and (15c)

Theorem 1 Under assumption (A3) there exists a uniquesolution (119906

ℎ 120590ℎ) isin 119872

ℎtimes Vℎto the schemes (15a) (15b) and

(15c)

Proof The linear system generated by (15a) (15b) and (15c)is square so the existence of the solution is implied by its uni-queness From (15a) (15b) and (15c) we have

(

119906119899

ℎ

Δ119905

Vℎ) minus (120590

119899

ℎ nablaVℎ)ℎ= (

119906119899minus1

ℎ

Δ119905

Vℎ) + (119891

119899 Vℎ) forallV

ℎisin 119872ℎ

(120590119899

ℎwℎ) + (119887nabla119906

119899

ℎwℎ)ℎ= 0 forallw

ℎisin Vℎ

(16)

Let 119906119899ℎand 119891 be zero and thus 119906119899

ℎis zero too taking V

ℎ=

119906119899

ℎ wℎ= (1119887)120590

119899

ℎin (16) and adding them together we have

1

Δ119905

1003817100381710038171003817119906119899

ℎ

1003817100381710038171003817

2

+ (

1

119887

120590119899

ℎ 120590119899

ℎ) = 0 (17)

Thus assumption (A3) implies that 119906119899ℎ= 120590119899

ℎ= 0 The proof is

complete

To get error estimates we state the following two impor-tant lemmas

Lemma 1 (see [27 29 31]) Assume that 119906 isin 1198671(Ω) p isin

(1198672(Ω))

2 for all Vℎisin 119872ℎ wℎisin Vℎ and then there hold

(nabla (119906 minus 1205871

ℎ119906) nablaV

ℎ)ℎ= 0 (nabla (119906 minus 120587

1

ℎ119906) wℎ)ℎ= 0

(18)

(p minus 1205872ℎpwℎ) le 119862ℎ

2|p|2Ω

1003817100381710038171003817wℎ

1003817100381710038171003817 (19)

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119890isin119879ℎ

int

120597119890

pVℎsdot n 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862ℎ2|p|2Ω

1003817100381710038171003817Vℎ

10038171003817100381710038171ℎ

(20)

where sdot 1ℎ

= (sum119890isin119879ℎ

| sdot |1119890)12 is a norm on 119872

ℎ and n

denotes the outward unit normal vector on 120597119890

Lemma 2 (see [1 3]) Let 120593 isin 1198712(Ω) and 120593 = 120593(119883minus119892(119883)Δ119905)

where function 119892 and its gradient nabla119892 are bounded then1003817100381710038171003817120593 minus 120593

1003817100381710038171003817minus1

le 11986210038171003817100381710038171205931003817100381710038171003817Δ119905 (21)

where 120593minus1= sup

120601isin1198671(Ω)((120593 120601)120601

1Ω)

4 Convergence Analysis and Optimal OrderError Estimates

In this section we aim to analyze the convergence analysisand error estimates of characteristic nonconforming MFEMIn order to do this let

119906ℎminus 119906 = 119906

ℎminus 1205871

ℎ119906 + 1205871

ℎ119906 minus 119906 = 119890 + 120588

120590ℎminus 120590 = 120590

ℎminus 1205872

ℎ120590 + 1205872

ℎ120590 minus 120590 = 120585 + 120578

(22)

Taking 119905 = 119905119899 in (12) yields

(120595

120597119906119899

120597120591

Vℎ) minus (120590

119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n119889119904 = (119891

119899 Vℎ)

forallVℎisin 119872ℎ

(23a)

(120590119899wℎ) + (119887nabla119906

119899wℎ)ℎ= 0 forallw

ℎisin Vℎ (23b)

From (23a) (23b) (15a) (15b) and (15c) we get

(

119890119899minus 119890119899minus1

Δ119905

Vℎ) minus (120585

119899 nablaVℎ)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

Vℎ) minus (

120588119899minus 120588119899minus1

Δ119905

Vℎ)

+ (120578119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n 119889119904 forallV

ℎisin 119872ℎ

(24a)

(120585119899wℎ) + (119887nabla119890

119899wℎ)ℎ= minus (120578

119899wℎ) minus (119887nabla120588

119899wℎ)ℎ

forallwℎisin Vℎ

(24b)

We are now in a position to prove the optimal order errorestimates

Theorem 2 Let (119906 120590) and (119906119899

ℎ 120590119899

ℎ) be the solutions of (12)

(15a) (15b) and (15c) respectively (12059721199061205971205912) isin 1198712(0 119879

1198712(Ω)) 119906

119905isin 1198712(0 119879119867

2(Ω)) 119906 isin 119871

infin(0 119879119867

2(Ω)) 120590 isin

119871infin(0 119879119867

2(Ω)) and assume that Δ119905 = 119874(ℎ

2) Then under

assumption (A3) we have

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)10038171003817100381710038171ℎ

le 119862 (Δ119905 + ℎ) (25)

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ

2) (26)

max0le119899le119873

1003817100381710038171003817(120590ℎminus 120590) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ) (27)

Proof Taking Vℎ= 119890119899 in (24a) and w

ℎ= nabla119890119899 in (24b) and

adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899) minus (

120588119899minus 120588119899minus1

Δ119905

119890119899)

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899)

+ sum

119890isin119879ℎ

int

120597119890

120590119899119890119899sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899)

ℎ

=

5

sum

119894=1

(Err)119894

(28)

Journal of Applied Mathematics 5

On the one hand we consider the right hand of (28)Using the method similar to [3] we have

(Err)1le 119862

100381710038171003817100381710038171003817100381710038171003817

120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(29)

(Err)2can be estimated as

1003816100381610038161003816(Err)2

1003816100381610038161003816le

1

Δ119905

(int

Ω

(int

119905119899

119905119899minus1

120588119905119889119904)

2

119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

1

radicΔ119905

(int

Ω

int

119905119899

119905119899minus1

1205882

119905119889119904 119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

119862

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817120588119905

1003817100381710038171003817

2

119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(30)

By Lemma 2 we obtain

1003816100381610038161003816(Err)3

1003816100381610038161003816le

1

Δ119905

10038171003817100381710038171003817120588119899minus1

minus 120588119899minus110038171003817

100381710038171003817minus1

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817120588119899minus110038171003817

100381710038171003817

2

+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862ℎ410038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(31)

It follows from Lemma 1 that

1003816100381610038161003816(Err)4

1003816100381610038161003816le 119862ℎ410038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ (32)

Let 119887 = (1|119890|) int119890119887(119909 119910)119889119909 119889119910 By Lemma 1 we have

1003816100381610038161003816(Err)5

1003816100381610038161003816=

10038161003816100381610038161003816minus((119887 minus 119887) nabla120588

119899 nabla119890119899)ℎ

10038161003816100381610038161003816

le 119862ℎ|119887|1198821infin(Ω)

100381710038171003817100381712058811989910038171003817100381710038171ℎ

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862ℎ410038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(33)

On the other hand the left hand of (28) can be bounded by

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

ge

1

2Δ119905

((119890119899 119890119899) minus (119890

119899minus1 119890119899minus1

)) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

ge

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus (1 + 119862Δ119905)

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(34)

where the inequality 119890119899minus12 le (1+119862Δ119905)119890119899minus1

2 proved in [3]is used in the last step

Combining (29)ndash(34) with (28) gives

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862(Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus1 119905119899 1198712(Ω))

+

ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+ℎ4(

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω))

+ 1205761

10038171003817100381710038171198901198991003817100381710038171003817

2

+ 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

100381710038171003817100381711989011989910038171003817100381710038171ℎ

(35)

Taking 1 minus 2Δ1199051205761gt 0 multiplying (35) by 2Δ119905 summing over

from 119894 = 1 to 119894 = 119899 and noticing that 1198900 = 0 we obtain

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4int

119905119899

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+Δ119905ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)) + 119862

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

(36)

It follows from discrete Gronwallrsquos lemma that

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(37)

From (37) we get the optimal order error estimate of 119890119899rather than 119890

1198991ℎ So we start to reestimate 119890119899

1ℎin the

following manner and derive the estimation of 120585119899 simul-taneously

Firstly choosing Vℎ= ((119890119899minus 119890119899minus1

)Δ119905) in (24a) and wℎ=

nabla((119890119899minus 119890119899minus1

)Δ119905) in (24b) and adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

minus (

120588119899minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

6 Journal of Applied Mathematics

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

+ sum

119890isin119879ℎ

int

120597119890

120590119899 119890119899minus 119890119899minus1

Δ119905

sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899minus 119890119899minus1

Δ119905

)

ℎ

=

5

sum

119894=1

(Err)1015840119894

(38)The left hand can be estimated as

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

ge

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899) minus (119887nabla119890

119899minus1 nabla119890119899minus1

)]

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(39)

and (Err)1015840119894 (119894 = 1 2 3 4 5) can be bounded by

10038161003816100381610038161003816(Err)10158401

10038161003816100381610038161003816le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158402

10038161003816100381610038161003816le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158403

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158404

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158405

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

(40)From (38)ndash(40) we get

1

2

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899)ℎminus (119887nabla119890

119899minus1 nabla119890119899minus1

)ℎ]

le 119862[Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

ℎ4

Δ119905

(int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω

+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω)] + 120576Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(41)

Multiplying (41) by 2Δ119905 and summing over in time from 119894 = 1

to 119894 = 119899 yield

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+ 120576(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+

119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

(42)Secondly we takeΔ119905 rarr 0 andΔ119905must approach zero in sucha way that Δ119905 and ℎ satisfy

Δ119905 = 119874 (ℎ2) (43)

and by inverse inequality we have

(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

le 119862Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

(44)

At the same time using Lemma 2 we obtain119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

= (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899) +

119899minus1

sum

119894=1

(

119890119894minus1

minus 119890119894minus (119890119894minus1

minus 119890119894)

Δ119905

119890119894)

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

100381710038171003817100381711989011989910038171003817100381710038171ℎ

+

119899minus1

sum

119894=1

10038171003817100381710038171003817119890119894minus 119890119894minus110038171003817100381710038171003817

10038171003817100381710038171003817119890119894100381710038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(45)From (42)ndash(45) taking suitable small 120576 such that 1 minus 120576119862 gt 0we have

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(46)

Journal of Applied Mathematics 7

Finally applying discrete Gronwallrsquos lemma yields

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎle 119862[(Δ119905)

2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ ℎ2(1199062

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(47)

In order to derive (27) set wℎ= 120585119899 in (24b) and employ

Lemma 1 and assumption (A3) to give

10038171003817100381710038171205851198991003817100381710038171003817

2

= minus(119887nabla119890119899 120585119899)ℎminus (120578119899 120585119899) minus (119887nabla120588

119899 120585119899)ℎ

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4100381710038171003817100381712059011989910038171003817100381710038172Ω

)

minus ((119887 minus 119887) nabla120588119899 120585119899)ℎ+

1

4

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4(100381710038171003817100381712059011989910038171003817100381710038172Ω

+100381710038171003817100381711990611989910038171003817100381710038172Ω

)) +

1

2

10038171003817100381710038171205851198991003817100381710038171003817

2

(48)

Combining (47) with (48) yields

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ2(|119906|2

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(49)

By using of interpolation theory and the triangle inequality(37) (47) and (49) lead to (25) (26) and (27) respectivelywhich are the desired results

Remark 2 From (37) we have

Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ= Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817

(1205871

ℎ119906 minus 119906ℎ)

119894100381710038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(50)

This byproduct can be regarded as the superclose resultbetween 1205871

ℎ119906 and 119906

ℎin mean broken1198671-norm It seems that

both (25) and (50) have never been seen in the existing stud-ies At the same time by employing the new characteristicnonconforming MFE scheme we can also obtain the sameerror estimate of (27) as traditional characteristicMFEM[10]

Remark 3 From the analysis of Theorem 2 in this paperwe may see that Lemma 1 is the key result leading to the

Table 1 Numerical results of 119906 minus 119906ℎ1ℎ

119898 times 119899 119905 = 02 120572 119905 = 03 120572 119905 = 04 120572

8 times 8 075277 075017 066433 16 times 16 042984 081 041849 084 035474 09132 times 32 021758 099 021412 097 017552 102119898 times 119899 119905 = 05 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 055291 042211 040937 16 times 16 029234 092 023117 087 021120 09632 times 32 014466 102 010807 110 009343 118

Table 2 Numerical results of 119906 minus 119906ℎ

119898 times 119899 119905 = 04 120572 119905 = 05 120572 119905 = 07 120572

8 times 8 00298190 00276370 00223240 16 times 16 00073087 203 00062445 215 00048038 22232 times 32 00020769 182 00017926 180 00013309 185119898 times 119899 119905 = 08 120572 119905 = 09 120572 119905 = 10 120572

8 times 8 00198730 00175900 00154090 16 times 16 00044472 216 00041982 207 00039150 19832 times 32 00011894 190 00010738 197 00009466 205

successful optimal order error estimations If we want toget higher order accuracy similar to Lemma 1 the non-conforming finite elements for approximating 119906 should alsopossess a very special property that is the consistency errorestimates with 119874(ℎ

2) order and satisfy (18) For the famous

nonconformingWilson element [32] whose shape function isspan1 119909 119910 1199092 1199102 by a counter-example it has been provenin [32] that its consistency error estimate is of119874(ℎ) order andcannot be improved any more For the rotated bilinear 119876

1

element [33] whose shape function is span1 119909 119910 1199092 minus 1199102

although its consistency error with 119874(ℎ2) order and (nabla(119906 minus

1205871

ℎ119906) nablaV

ℎ)ℎ= 0 on squaremeshes is satisfied the second term

of (18) is not valid Thus when they are applied to (1) on newcharacteristic mixed finite element scheme up to now theoptimal order error estimates of (25) (26) and (27) cannotbe obtained directly

5 Numerical Example

In order to verify our theoretical analysis in previous sectionswe consider the convection-dominated diffusion problem (1)as follows

119906119905+ 119906119909+ 119906119910minus 10minus4(119906119909119909+ 119906119910119910)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(51)

withΩ = [0 1] times [0 1] a(119909 119910) = (1 1) and 119887(119909 119910) = 10minus4

The right hand term 119891(119909 119910 119905) is taken such that 119906 =

119890minus119905 sin(120587119909) sin(2120587119910) 120590 = minus10

minus4119890minus119905(120587 cos(120587119909) sin(2120587119910)

2120587 sin(120587119909) cos(2120587119910)) are the exact solutions

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

characteristics-mixed covolume method [10 11] the mod-ified method of characteristic-Galerkin FE procedure [12]characteristic nonconforming FEM [13ndash15] characteristicMFEM [16ndash19] and expanded characteristic MFEM [1 20]and so forth

As for the characteristic MFEM or expanded character-istic MFEM the convergence rates of 119906 and 120590 in existingliterature were suboptimal [11 18 21 22] and the convergenceanalysis was valid only to the case of the lowest order MFEapproximation [10 17] So far to our best knowledgethere are few studies on the optimal order error estimatesexcept for [23] in which a family of characteristic MFEMwith arbitrary degree of Raviart-Thomas-Nedelec space in[24 25] for transient convection diffusion equations wasstudied

Recently based on the low regularity requirement of theflux variable in practical problems a new mixed variationalform for second elliptic problem was proposed in [26] It hastwo typical advantages the flux space belongs to the squareintegrable space instead of the traditional119867( div Ω) whichmakes the choices ofMFE spaces sufficiently simple and easythe LBB condition is automatically satisfiedwhen the gradientof approximation space for the original variable is includedin approximation space for the flux variable Motivated bythis idea this paper will construct a characteristic noncon-forming MFE scheme for (1) with a new mixed variationalformulation Similar to the expanded characteristic MFEMthe coefficient 119887 of (1) in this proposed scheme does notneed to be inverted therefore it is also suitable for the casewhen 119887 is small By employing some distinct characters ofthe interpolation operators on the element instead of themixed or expandedmixed elliptic projection used in [1 17 20]which is an indispensable tool in the traditional characteristicMFEM analysis the 119874(ℎ2) order error estimate in 119871

2-normfor original variable 119906 which is one order higher than [1 20]and half order higher than [18] is derived and the optimalerror estimates with order119874(ℎ) for auxiliary variable 120590 in 1198712-normand for119906 in broken1198671-normare obtained respectivelyIt seems that the result for 119906 in broken 119867

1-norm has neverbeen seen in the existing literature by making full use of thehigh-accuracy estimates of the lowest order Raviart-Thomaselement proved by the technique of integral identities in [27]and the special properties of nonconforming 119864119876rot

1element

(see Lemma 1 below)The paper is organized as follows Section 2 is devoted

to the introduction of the nonconforming FE approximationspaces and their corresponding interpolation operators InSection 3 we will give the construction of the new charac-teristic nonconformingMFE scheme and two important lem-mas and the existence and uniqueness of the discrete schemesolutionwill be proved In Section 4 the convergence analysisand optimal error estimates for both the original variable119906 and the flux variable 120590 are obtained In Section 5 somenumerical results are provided to illustrate the effectivenessof our proposed method

Throughout this paper119862 denotes a generic positive cons-tant independent of the mesh parameters ℎ and Δ119905 withrespect to domainΩ and time 119905

2 Construction of Nonconforming MFEs

As in [28] we frequently employ the space 1198712(Ω) of squareintegrable functions with scalar product and norm

(119906 V) = (119906 V)1198712(Ω)

= (int

Ω

119906V119889119909119889119910)12

V = V1198712(Ω)

= (int

Ω

V2119889119909119889119910)

12

(3)

We also employ the Sobolev space 119867119898(Ω) 119898 ge 1 of func-tions V such that 119863120573V isin 1198712(Ω) for all |120573| le 119898 equipped withthe norm and seminorm

V119898Ω

= V119867119898(Ω)

= ( sum

|120573|le119898

10038171003817100381710038171003817119863120573V10038171003817100381710038171003817

2

)

12

|V|119898Ω

= |V|119867119898(Ω)

= ( sum

|120573|=119898

10038171003817100381710038171003817119863120573V10038171003817100381710038171003817

2

)

12

(4)

The space 11986710(Ω) denotes the closure of the set of infinitely

differentiable functions with compact supports inΩ For anySobolev space 119884 119871119901(0 119879 119884) is the space of measurable 119884-valued functions Φ of 119905 isin (0 119879) such that int119879

0Φ(sdot 119905)

119901

119884119889119905 lt

infin if 1 le 119901 lt infin or such that ess sup0lt119905lt119879

Φ(sdot 119905)119884lt infin if

119901 = infinWe now introduce the nonconforming MFE space des-

cribed in [29] for and summarize it as followsLet Ω sub R2 be a polygon domain with edges parallel to

the coordinate axes on 119909119910 plane and let 119879ℎbe a rectangular

subdivision of Ω satisfying the regular condition [30] For agiven element 119890 isin 119879

ℎ denote the barycenter of element 119890 by

(119909119890 119910119890) denote the length of edges parallel to 119909-axis and 119910-

axis by 2ℎ119909119890

and 2ℎ119910119890

respectively ℎ119890= max

119890isin119879ℎ

ℎ119909119890

ℎ119910119890

ℎ =

max119890isin119879ℎ

ℎ119890

Let 119890 = [minus1 1] times [minus1 1] be the reference element on 119909119910plane and four vertices

1198891= (minus1 minus1) 119889

2= (1 minus1) 119889

3=

(1 1) and 1198894= (minus1 1) the four edges 119897

1=

11988911198892 1198972=

11988921198893

1198973=11988931198894 and 119897

4=11988941198891 Then there exists an affine mapping

119865119890 119890 rarr 119890 as

119909 = 119909119890+ ℎ119909119890

119909

119910 = 119910119890+ ℎ119910119890

119910

(5)

Define the FE spaces (119890 119875119894 sum119894

) (119894 = 1 2 3) bysum

1

= V1 V2 V3 V4 V5

1= span 1 119909 119910 120601 (119909) 120601 (119910)

sum

2

= 1199011 1199012

2= span 1 119909

sum

3

= 1199021 1199022

3= span 1 119910

(6)

where V119894= (1|

119897119894|) int119894

V119889119904 (119894 = 1 2 3 4) V5

= (1|119890|)

int119890V119889119909 119889119910 120601(119905) = (12)(3119905

2minus 1) 119901

119894= (1|

1198972119894|) int2119894

119901119889119904 119902119894=

(1|1198972119894minus1

|) int2119894minus1

119902119889119904 (119894 = 1 2)

Journal of Applied Mathematics 3

The interpolation operators on 119890 are defined as follows

Π1 V isin 119867

1(119890) 997888rarr Π

1V isin 1

int

119894

(Π1V minus V) 119889119904 = 0 (119894 = 1 2 3 4)

int

119890

(Π1V minus V) 119889119909 119889119910 = 0

Π2 119901 isin 119867

1(119890) 997888rarr Π

2119901 isin

2

int

2119894

(Π2119901 minus 119901) 119889119904 = 0 (119894 = 1 2)

Π3 119902 isin 119867

1(119890) 997888rarr Π

3119902 isin 3

int

2119894minus1

(Π3119902 minus 119902) 119889119904 = 0 (119894 = 1 2)

(7)

Then the associated nonconforming 119864119876rot1

element space119872ℎ

[29] and lowest order Raviart-Thomas element space Vℎ[25

27] are defined as

119872ℎ= Vℎ Vℎ|119890= V ∘ 119865

minus1

119890 V isin

1

int

119865

[Vℎ] 119889119904 = 0 119865 sub 120597119890

Vℎ= wℎ= (119908ℎ1 119908ℎ2)

wℎ|119890= (1199081∘ 119865minus1

119890 1199082∘ 119865minus1

119890)

w = (1199081 1199082) isin 2times 3

(8)

respectively where [120593] represents the jump value of 120593 acrossthe boundary 119865 and [120593] = 120593 if 119865 sub 120597Ω

Similarly the interpolation operators 1205871

ℎand 120587

2

ℎare

defined as

1205871

ℎ 1198671(Ω) 997888rarr 119872

ℎ 120587

1

ℎ

10038161003816100381610038161003816119890= 1205871

119890

1205871

119890V = (Π

1V) ∘ 119865

minus1

119890 forallV isin 119867

1(Ω)

1205872

ℎ (1198671(Ω))

2

997888rarr Vℎ 1205872

ℎ|119890= 1205872

119890

1205872

119890w = ((Π

21199081) ∘ 119865minus1

119890 (Π31199082) ∘ 119865minus1

119890)

forallw = (1199081 1199082) isin (119867

1(Ω))

2

(9)

3 New Characteristic Nonconforming MFEScheme and Two Lemmas

Let 120595(119909 119910) = (1 + |a(119909 119910)|2)12 and 120591 = 120591(119909 119910) be the chara-cteristic direction associated with 119906

119905+ a(119909 119910) sdot nabla119906 such that

120597

120597120591

=

1

120595 (119909 119910)

120597

120597119905

+

a (119909 119910)120595 (119909 119910)

sdot nabla (10)

Then (1) can be put in the following system

120595 (119909 119910)

120597119906

120597120591

minus nabla sdot (119887 (119909 119910) nabla119906) = 119891 (119909 119910 119905)

forall (119909 119910 119905) isin Ω times (0 119879]

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879]

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(11)

By introducing 120590 = minus119887(119909 119910)nabla119906 and using Greenrsquosformula we obtain the new characteristic mixed form of (11)Find (119906 120590) (0 119879] rarr 119867

1

0(Ω) times (119871

2(Ω))

2 such that

(120595 (119909 119910)

120597119906

120597120591

V) minus (120590 nablaV) = (119891 (119909 119910 119905) V) forallV isin 1198671

0(Ω)

(120590w) + (119887 (119909 119910) nabla119906w) = 0 forallw isin (1198712(Ω))

2

(12)

Let Δ119905 gt 0119873 = 119879Δ119905 isin Z 119905119899 = 119899Δ119905 and 120601119899 = 120601(119909 119910 119905119899)

When solving 119906119899+1

ℎ we would like to make the scheme as

implicit as possible by using of the characteristic vector 120591Denote119883 = (119909 119910) isin Ω and

119883 = 119883 minus a (119909 119910) Δ119905 (13)

similar to [1 3] and then we have the following approxima-tion

120595 (119909 119910)

120597119906

120597120591

10038161003816100381610038161003816100381610038161003816119905119899

asymp 120595 (119909 119910)

119906 (119883 119905119899) minus 119906 (119883 119905

119899minus1)

radic(119883 minus 119883)

2

+ (Δ119905)2

=

119906 (119883 119905119899) minus 119906 (119883 119905

119899minus1)

Δ119905

=

119906119899minus 119906119899minus1

Δ119905

(14)

This leads to the following characteristic nonconformingMFE scheme Find (119906

ℎ 120590ℎ) 1199050 1199051 119905119873 rarr 119872

ℎtimesVℎ such

that

(

119906119899

ℎminus 119906119899minus1

ℎ

Δ119905

Vℎ) minus (120590

119899

ℎ nablaVℎ)ℎ= (119891119899 Vℎ) forallV

ℎisin 119872ℎ

(15a)

(120590119899

ℎwℎ) + (119887nabla119906

119899

ℎwℎ)ℎ= 0 forallw

ℎisin Vℎ (15b)

1199060

ℎ= 1205871

ℎ1199060(119909 119910) 120590

0

ℎ= 1205872

ℎ(119887nabla1199060(119909 119910)) forall (119909 119910) isin Ω

(15c)

where 119906119899ℎ= 119906ℎ(119883 119905119899) (119906 V)

ℎ= sum119890isin119879ℎ

int119890119906V119889119909119889119910 Generally

speaking 119906119899minus1ℎ

(119899 = 2 119873) are not node values and shouldbe derived by interpolation formulas on 119906119899minus1

ℎ

Remark 1 In [1] the expanded characteristic MFE schemewas presented by introducing two new auxiliary variableswhich avoided the inversion of the coefficient 119887 when 119887 issmallThe newmixed schemes (15a) (15b) and (15c) not onlykeep the advantage of expanded characteristic MFE schemebut also donot need to solve three variables

4 Journal of Applied Mathematics

Now we prove the existence and uniqueness of the solu-tion of (15a) (15b) and (15c)

Theorem 1 Under assumption (A3) there exists a uniquesolution (119906

ℎ 120590ℎ) isin 119872

ℎtimes Vℎto the schemes (15a) (15b) and

(15c)

Proof The linear system generated by (15a) (15b) and (15c)is square so the existence of the solution is implied by its uni-queness From (15a) (15b) and (15c) we have

(

119906119899

ℎ

Δ119905

Vℎ) minus (120590

119899

ℎ nablaVℎ)ℎ= (

119906119899minus1

ℎ

Δ119905

Vℎ) + (119891

119899 Vℎ) forallV

ℎisin 119872ℎ

(120590119899

ℎwℎ) + (119887nabla119906

119899

ℎwℎ)ℎ= 0 forallw

ℎisin Vℎ

(16)

Let 119906119899ℎand 119891 be zero and thus 119906119899

ℎis zero too taking V

ℎ=

119906119899

ℎ wℎ= (1119887)120590

119899

ℎin (16) and adding them together we have

1

Δ119905

1003817100381710038171003817119906119899

ℎ

1003817100381710038171003817

2

+ (

1

119887

120590119899

ℎ 120590119899

ℎ) = 0 (17)

Thus assumption (A3) implies that 119906119899ℎ= 120590119899

ℎ= 0 The proof is

complete

To get error estimates we state the following two impor-tant lemmas

Lemma 1 (see [27 29 31]) Assume that 119906 isin 1198671(Ω) p isin

(1198672(Ω))

2 for all Vℎisin 119872ℎ wℎisin Vℎ and then there hold

(nabla (119906 minus 1205871

ℎ119906) nablaV

ℎ)ℎ= 0 (nabla (119906 minus 120587

1

ℎ119906) wℎ)ℎ= 0

(18)

(p minus 1205872ℎpwℎ) le 119862ℎ

2|p|2Ω

1003817100381710038171003817wℎ

1003817100381710038171003817 (19)

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119890isin119879ℎ

int

120597119890

pVℎsdot n 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862ℎ2|p|2Ω

1003817100381710038171003817Vℎ

10038171003817100381710038171ℎ

(20)

where sdot 1ℎ

= (sum119890isin119879ℎ

| sdot |1119890)12 is a norm on 119872

ℎ and n

denotes the outward unit normal vector on 120597119890

Lemma 2 (see [1 3]) Let 120593 isin 1198712(Ω) and 120593 = 120593(119883minus119892(119883)Δ119905)

where function 119892 and its gradient nabla119892 are bounded then1003817100381710038171003817120593 minus 120593

1003817100381710038171003817minus1

le 11986210038171003817100381710038171205931003817100381710038171003817Δ119905 (21)

where 120593minus1= sup

120601isin1198671(Ω)((120593 120601)120601

1Ω)

4 Convergence Analysis and Optimal OrderError Estimates

In this section we aim to analyze the convergence analysisand error estimates of characteristic nonconforming MFEMIn order to do this let

119906ℎminus 119906 = 119906

ℎminus 1205871

ℎ119906 + 1205871

ℎ119906 minus 119906 = 119890 + 120588

120590ℎminus 120590 = 120590

ℎminus 1205872

ℎ120590 + 1205872

ℎ120590 minus 120590 = 120585 + 120578

(22)

Taking 119905 = 119905119899 in (12) yields

(120595

120597119906119899

120597120591

Vℎ) minus (120590

119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n119889119904 = (119891

119899 Vℎ)

forallVℎisin 119872ℎ

(23a)

(120590119899wℎ) + (119887nabla119906

119899wℎ)ℎ= 0 forallw

ℎisin Vℎ (23b)

From (23a) (23b) (15a) (15b) and (15c) we get

(

119890119899minus 119890119899minus1

Δ119905

Vℎ) minus (120585

119899 nablaVℎ)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

Vℎ) minus (

120588119899minus 120588119899minus1

Δ119905

Vℎ)

+ (120578119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n 119889119904 forallV

ℎisin 119872ℎ

(24a)

(120585119899wℎ) + (119887nabla119890

119899wℎ)ℎ= minus (120578

119899wℎ) minus (119887nabla120588

119899wℎ)ℎ

forallwℎisin Vℎ

(24b)

We are now in a position to prove the optimal order errorestimates

Theorem 2 Let (119906 120590) and (119906119899

ℎ 120590119899

ℎ) be the solutions of (12)

(15a) (15b) and (15c) respectively (12059721199061205971205912) isin 1198712(0 119879

1198712(Ω)) 119906

119905isin 1198712(0 119879119867

2(Ω)) 119906 isin 119871

infin(0 119879119867

2(Ω)) 120590 isin

119871infin(0 119879119867

2(Ω)) and assume that Δ119905 = 119874(ℎ

2) Then under

assumption (A3) we have

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)10038171003817100381710038171ℎ

le 119862 (Δ119905 + ℎ) (25)

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ

2) (26)

max0le119899le119873

1003817100381710038171003817(120590ℎminus 120590) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ) (27)

Proof Taking Vℎ= 119890119899 in (24a) and w

ℎ= nabla119890119899 in (24b) and

adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899) minus (

120588119899minus 120588119899minus1

Δ119905

119890119899)

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899)

+ sum

119890isin119879ℎ

int

120597119890

120590119899119890119899sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899)

ℎ

=

5

sum

119894=1

(Err)119894

(28)

Journal of Applied Mathematics 5

On the one hand we consider the right hand of (28)Using the method similar to [3] we have

(Err)1le 119862

100381710038171003817100381710038171003817100381710038171003817

120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(29)

(Err)2can be estimated as

1003816100381610038161003816(Err)2

1003816100381610038161003816le

1

Δ119905

(int

Ω

(int

119905119899

119905119899minus1

120588119905119889119904)

2

119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

1

radicΔ119905

(int

Ω

int

119905119899

119905119899minus1

1205882

119905119889119904 119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

119862

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817120588119905

1003817100381710038171003817

2

119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(30)

By Lemma 2 we obtain

1003816100381610038161003816(Err)3

1003816100381610038161003816le

1

Δ119905

10038171003817100381710038171003817120588119899minus1

minus 120588119899minus110038171003817

100381710038171003817minus1

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817120588119899minus110038171003817

100381710038171003817

2

+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862ℎ410038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(31)

It follows from Lemma 1 that

1003816100381610038161003816(Err)4

1003816100381610038161003816le 119862ℎ410038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ (32)

Let 119887 = (1|119890|) int119890119887(119909 119910)119889119909 119889119910 By Lemma 1 we have

1003816100381610038161003816(Err)5

1003816100381610038161003816=

10038161003816100381610038161003816minus((119887 minus 119887) nabla120588

119899 nabla119890119899)ℎ

10038161003816100381610038161003816

le 119862ℎ|119887|1198821infin(Ω)

100381710038171003817100381712058811989910038171003817100381710038171ℎ

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862ℎ410038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(33)

On the other hand the left hand of (28) can be bounded by

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

ge

1

2Δ119905

((119890119899 119890119899) minus (119890

119899minus1 119890119899minus1

)) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

ge

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus (1 + 119862Δ119905)

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(34)

where the inequality 119890119899minus12 le (1+119862Δ119905)119890119899minus1

2 proved in [3]is used in the last step

Combining (29)ndash(34) with (28) gives

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862(Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus1 119905119899 1198712(Ω))

+

ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+ℎ4(

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω))

+ 1205761

10038171003817100381710038171198901198991003817100381710038171003817

2

+ 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

100381710038171003817100381711989011989910038171003817100381710038171ℎ

(35)

Taking 1 minus 2Δ1199051205761gt 0 multiplying (35) by 2Δ119905 summing over

from 119894 = 1 to 119894 = 119899 and noticing that 1198900 = 0 we obtain

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4int

119905119899

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+Δ119905ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)) + 119862

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

(36)

It follows from discrete Gronwallrsquos lemma that

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(37)

From (37) we get the optimal order error estimate of 119890119899rather than 119890

1198991ℎ So we start to reestimate 119890119899

1ℎin the

following manner and derive the estimation of 120585119899 simul-taneously

Firstly choosing Vℎ= ((119890119899minus 119890119899minus1

)Δ119905) in (24a) and wℎ=

nabla((119890119899minus 119890119899minus1

)Δ119905) in (24b) and adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

minus (

120588119899minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

6 Journal of Applied Mathematics

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

+ sum

119890isin119879ℎ

int

120597119890

120590119899 119890119899minus 119890119899minus1

Δ119905

sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899minus 119890119899minus1

Δ119905

)

ℎ

=

5

sum

119894=1

(Err)1015840119894

(38)The left hand can be estimated as

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

ge

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899) minus (119887nabla119890

119899minus1 nabla119890119899minus1

)]

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(39)

and (Err)1015840119894 (119894 = 1 2 3 4 5) can be bounded by

10038161003816100381610038161003816(Err)10158401

10038161003816100381610038161003816le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158402

10038161003816100381610038161003816le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158403

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158404

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158405

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

(40)From (38)ndash(40) we get

1

2

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899)ℎminus (119887nabla119890

119899minus1 nabla119890119899minus1

)ℎ]

le 119862[Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

ℎ4

Δ119905

(int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω

+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω)] + 120576Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(41)

Multiplying (41) by 2Δ119905 and summing over in time from 119894 = 1

to 119894 = 119899 yield

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+ 120576(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+

119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

(42)Secondly we takeΔ119905 rarr 0 andΔ119905must approach zero in sucha way that Δ119905 and ℎ satisfy

Δ119905 = 119874 (ℎ2) (43)

and by inverse inequality we have

(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

le 119862Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

(44)

At the same time using Lemma 2 we obtain119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

= (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899) +

119899minus1

sum

119894=1

(

119890119894minus1

minus 119890119894minus (119890119894minus1

minus 119890119894)

Δ119905

119890119894)

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

100381710038171003817100381711989011989910038171003817100381710038171ℎ

+

119899minus1

sum

119894=1

10038171003817100381710038171003817119890119894minus 119890119894minus110038171003817100381710038171003817

10038171003817100381710038171003817119890119894100381710038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(45)From (42)ndash(45) taking suitable small 120576 such that 1 minus 120576119862 gt 0we have

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(46)

Journal of Applied Mathematics 7

Finally applying discrete Gronwallrsquos lemma yields

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎle 119862[(Δ119905)

2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ ℎ2(1199062

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(47)

In order to derive (27) set wℎ= 120585119899 in (24b) and employ

Lemma 1 and assumption (A3) to give

10038171003817100381710038171205851198991003817100381710038171003817

2

= minus(119887nabla119890119899 120585119899)ℎminus (120578119899 120585119899) minus (119887nabla120588

119899 120585119899)ℎ

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4100381710038171003817100381712059011989910038171003817100381710038172Ω

)

minus ((119887 minus 119887) nabla120588119899 120585119899)ℎ+

1

4

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4(100381710038171003817100381712059011989910038171003817100381710038172Ω

+100381710038171003817100381711990611989910038171003817100381710038172Ω

)) +

1

2

10038171003817100381710038171205851198991003817100381710038171003817

2

(48)

Combining (47) with (48) yields

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ2(|119906|2

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(49)

By using of interpolation theory and the triangle inequality(37) (47) and (49) lead to (25) (26) and (27) respectivelywhich are the desired results

Remark 2 From (37) we have

Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ= Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817

(1205871

ℎ119906 minus 119906ℎ)

119894100381710038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(50)

This byproduct can be regarded as the superclose resultbetween 1205871

ℎ119906 and 119906

ℎin mean broken1198671-norm It seems that

both (25) and (50) have never been seen in the existing stud-ies At the same time by employing the new characteristicnonconforming MFE scheme we can also obtain the sameerror estimate of (27) as traditional characteristicMFEM[10]

Remark 3 From the analysis of Theorem 2 in this paperwe may see that Lemma 1 is the key result leading to the

Table 1 Numerical results of 119906 minus 119906ℎ1ℎ

119898 times 119899 119905 = 02 120572 119905 = 03 120572 119905 = 04 120572

8 times 8 075277 075017 066433 16 times 16 042984 081 041849 084 035474 09132 times 32 021758 099 021412 097 017552 102119898 times 119899 119905 = 05 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 055291 042211 040937 16 times 16 029234 092 023117 087 021120 09632 times 32 014466 102 010807 110 009343 118

Table 2 Numerical results of 119906 minus 119906ℎ

119898 times 119899 119905 = 04 120572 119905 = 05 120572 119905 = 07 120572

8 times 8 00298190 00276370 00223240 16 times 16 00073087 203 00062445 215 00048038 22232 times 32 00020769 182 00017926 180 00013309 185119898 times 119899 119905 = 08 120572 119905 = 09 120572 119905 = 10 120572

8 times 8 00198730 00175900 00154090 16 times 16 00044472 216 00041982 207 00039150 19832 times 32 00011894 190 00010738 197 00009466 205

successful optimal order error estimations If we want toget higher order accuracy similar to Lemma 1 the non-conforming finite elements for approximating 119906 should alsopossess a very special property that is the consistency errorestimates with 119874(ℎ

2) order and satisfy (18) For the famous

nonconformingWilson element [32] whose shape function isspan1 119909 119910 1199092 1199102 by a counter-example it has been provenin [32] that its consistency error estimate is of119874(ℎ) order andcannot be improved any more For the rotated bilinear 119876

1

element [33] whose shape function is span1 119909 119910 1199092 minus 1199102

although its consistency error with 119874(ℎ2) order and (nabla(119906 minus

1205871

ℎ119906) nablaV

ℎ)ℎ= 0 on squaremeshes is satisfied the second term

of (18) is not valid Thus when they are applied to (1) on newcharacteristic mixed finite element scheme up to now theoptimal order error estimates of (25) (26) and (27) cannotbe obtained directly

5 Numerical Example

In order to verify our theoretical analysis in previous sectionswe consider the convection-dominated diffusion problem (1)as follows

119906119905+ 119906119909+ 119906119910minus 10minus4(119906119909119909+ 119906119910119910)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(51)

withΩ = [0 1] times [0 1] a(119909 119910) = (1 1) and 119887(119909 119910) = 10minus4

The right hand term 119891(119909 119910 119905) is taken such that 119906 =

119890minus119905 sin(120587119909) sin(2120587119910) 120590 = minus10

minus4119890minus119905(120587 cos(120587119909) sin(2120587119910)

2120587 sin(120587119909) cos(2120587119910)) are the exact solutions

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

The interpolation operators on 119890 are defined as follows

Π1 V isin 119867

1(119890) 997888rarr Π

1V isin 1

int

119894

(Π1V minus V) 119889119904 = 0 (119894 = 1 2 3 4)

int

119890

(Π1V minus V) 119889119909 119889119910 = 0

Π2 119901 isin 119867

1(119890) 997888rarr Π

2119901 isin

2

int

2119894

(Π2119901 minus 119901) 119889119904 = 0 (119894 = 1 2)

Π3 119902 isin 119867

1(119890) 997888rarr Π

3119902 isin 3

int

2119894minus1

(Π3119902 minus 119902) 119889119904 = 0 (119894 = 1 2)

(7)

Then the associated nonconforming 119864119876rot1

element space119872ℎ

[29] and lowest order Raviart-Thomas element space Vℎ[25

27] are defined as

119872ℎ= Vℎ Vℎ|119890= V ∘ 119865

minus1

119890 V isin

1

int

119865

[Vℎ] 119889119904 = 0 119865 sub 120597119890

Vℎ= wℎ= (119908ℎ1 119908ℎ2)

wℎ|119890= (1199081∘ 119865minus1

119890 1199082∘ 119865minus1

119890)

w = (1199081 1199082) isin 2times 3

(8)

respectively where [120593] represents the jump value of 120593 acrossthe boundary 119865 and [120593] = 120593 if 119865 sub 120597Ω

Similarly the interpolation operators 1205871

ℎand 120587

2

ℎare

defined as

1205871

ℎ 1198671(Ω) 997888rarr 119872

ℎ 120587

1

ℎ

10038161003816100381610038161003816119890= 1205871

119890

1205871

119890V = (Π

1V) ∘ 119865

minus1

119890 forallV isin 119867

1(Ω)

1205872

ℎ (1198671(Ω))

2

997888rarr Vℎ 1205872

ℎ|119890= 1205872

119890

1205872

119890w = ((Π

21199081) ∘ 119865minus1

119890 (Π31199082) ∘ 119865minus1

119890)

forallw = (1199081 1199082) isin (119867

1(Ω))

2

(9)

3 New Characteristic Nonconforming MFEScheme and Two Lemmas

Let 120595(119909 119910) = (1 + |a(119909 119910)|2)12 and 120591 = 120591(119909 119910) be the chara-cteristic direction associated with 119906

119905+ a(119909 119910) sdot nabla119906 such that

120597

120597120591

=

1

120595 (119909 119910)

120597

120597119905

+

a (119909 119910)120595 (119909 119910)

sdot nabla (10)

Then (1) can be put in the following system

120595 (119909 119910)

120597119906

120597120591

minus nabla sdot (119887 (119909 119910) nabla119906) = 119891 (119909 119910 119905)

forall (119909 119910 119905) isin Ω times (0 119879]

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879]

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(11)

By introducing 120590 = minus119887(119909 119910)nabla119906 and using Greenrsquosformula we obtain the new characteristic mixed form of (11)Find (119906 120590) (0 119879] rarr 119867

1

0(Ω) times (119871

2(Ω))

2 such that

(120595 (119909 119910)

120597119906

120597120591

V) minus (120590 nablaV) = (119891 (119909 119910 119905) V) forallV isin 1198671

0(Ω)

(120590w) + (119887 (119909 119910) nabla119906w) = 0 forallw isin (1198712(Ω))

2

(12)

Let Δ119905 gt 0119873 = 119879Δ119905 isin Z 119905119899 = 119899Δ119905 and 120601119899 = 120601(119909 119910 119905119899)

When solving 119906119899+1

ℎ we would like to make the scheme as

implicit as possible by using of the characteristic vector 120591Denote119883 = (119909 119910) isin Ω and

119883 = 119883 minus a (119909 119910) Δ119905 (13)

similar to [1 3] and then we have the following approxima-tion

120595 (119909 119910)

120597119906

120597120591

10038161003816100381610038161003816100381610038161003816119905119899

asymp 120595 (119909 119910)

119906 (119883 119905119899) minus 119906 (119883 119905

119899minus1)

radic(119883 minus 119883)

2

+ (Δ119905)2

=

119906 (119883 119905119899) minus 119906 (119883 119905

119899minus1)

Δ119905

=

119906119899minus 119906119899minus1

Δ119905

(14)

This leads to the following characteristic nonconformingMFE scheme Find (119906

ℎ 120590ℎ) 1199050 1199051 119905119873 rarr 119872

ℎtimesVℎ such

that

(

119906119899

ℎminus 119906119899minus1

ℎ

Δ119905

Vℎ) minus (120590

119899

ℎ nablaVℎ)ℎ= (119891119899 Vℎ) forallV

ℎisin 119872ℎ

(15a)

(120590119899

ℎwℎ) + (119887nabla119906

119899

ℎwℎ)ℎ= 0 forallw

ℎisin Vℎ (15b)

1199060

ℎ= 1205871

ℎ1199060(119909 119910) 120590

0

ℎ= 1205872

ℎ(119887nabla1199060(119909 119910)) forall (119909 119910) isin Ω

(15c)

where 119906119899ℎ= 119906ℎ(119883 119905119899) (119906 V)

ℎ= sum119890isin119879ℎ

int119890119906V119889119909119889119910 Generally

speaking 119906119899minus1ℎ

(119899 = 2 119873) are not node values and shouldbe derived by interpolation formulas on 119906119899minus1

ℎ

Remark 1 In [1] the expanded characteristic MFE schemewas presented by introducing two new auxiliary variableswhich avoided the inversion of the coefficient 119887 when 119887 issmallThe newmixed schemes (15a) (15b) and (15c) not onlykeep the advantage of expanded characteristic MFE schemebut also donot need to solve three variables

4 Journal of Applied Mathematics

Now we prove the existence and uniqueness of the solu-tion of (15a) (15b) and (15c)

Theorem 1 Under assumption (A3) there exists a uniquesolution (119906

ℎ 120590ℎ) isin 119872

ℎtimes Vℎto the schemes (15a) (15b) and

(15c)

Proof The linear system generated by (15a) (15b) and (15c)is square so the existence of the solution is implied by its uni-queness From (15a) (15b) and (15c) we have

(

119906119899

ℎ

Δ119905

Vℎ) minus (120590

119899

ℎ nablaVℎ)ℎ= (

119906119899minus1

ℎ

Δ119905

Vℎ) + (119891

119899 Vℎ) forallV

ℎisin 119872ℎ

(120590119899

ℎwℎ) + (119887nabla119906

119899

ℎwℎ)ℎ= 0 forallw

ℎisin Vℎ

(16)

Let 119906119899ℎand 119891 be zero and thus 119906119899

ℎis zero too taking V

ℎ=

119906119899

ℎ wℎ= (1119887)120590

119899

ℎin (16) and adding them together we have

1

Δ119905

1003817100381710038171003817119906119899

ℎ

1003817100381710038171003817

2

+ (

1

119887

120590119899

ℎ 120590119899

ℎ) = 0 (17)

Thus assumption (A3) implies that 119906119899ℎ= 120590119899

ℎ= 0 The proof is

complete

To get error estimates we state the following two impor-tant lemmas

Lemma 1 (see [27 29 31]) Assume that 119906 isin 1198671(Ω) p isin

(1198672(Ω))

2 for all Vℎisin 119872ℎ wℎisin Vℎ and then there hold

(nabla (119906 minus 1205871

ℎ119906) nablaV

ℎ)ℎ= 0 (nabla (119906 minus 120587

1

ℎ119906) wℎ)ℎ= 0

(18)

(p minus 1205872ℎpwℎ) le 119862ℎ

2|p|2Ω

1003817100381710038171003817wℎ

1003817100381710038171003817 (19)

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119890isin119879ℎ

int

120597119890

pVℎsdot n 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862ℎ2|p|2Ω

1003817100381710038171003817Vℎ

10038171003817100381710038171ℎ

(20)

where sdot 1ℎ

= (sum119890isin119879ℎ

| sdot |1119890)12 is a norm on 119872

ℎ and n

denotes the outward unit normal vector on 120597119890

Lemma 2 (see [1 3]) Let 120593 isin 1198712(Ω) and 120593 = 120593(119883minus119892(119883)Δ119905)

where function 119892 and its gradient nabla119892 are bounded then1003817100381710038171003817120593 minus 120593

1003817100381710038171003817minus1

le 11986210038171003817100381710038171205931003817100381710038171003817Δ119905 (21)

where 120593minus1= sup

120601isin1198671(Ω)((120593 120601)120601

1Ω)

4 Convergence Analysis and Optimal OrderError Estimates

In this section we aim to analyze the convergence analysisand error estimates of characteristic nonconforming MFEMIn order to do this let

119906ℎminus 119906 = 119906

ℎminus 1205871

ℎ119906 + 1205871

ℎ119906 minus 119906 = 119890 + 120588

120590ℎminus 120590 = 120590

ℎminus 1205872

ℎ120590 + 1205872

ℎ120590 minus 120590 = 120585 + 120578

(22)

Taking 119905 = 119905119899 in (12) yields

(120595

120597119906119899

120597120591

Vℎ) minus (120590

119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n119889119904 = (119891

119899 Vℎ)

forallVℎisin 119872ℎ

(23a)

(120590119899wℎ) + (119887nabla119906

119899wℎ)ℎ= 0 forallw

ℎisin Vℎ (23b)

From (23a) (23b) (15a) (15b) and (15c) we get

(

119890119899minus 119890119899minus1

Δ119905

Vℎ) minus (120585

119899 nablaVℎ)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

Vℎ) minus (

120588119899minus 120588119899minus1

Δ119905

Vℎ)

+ (120578119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n 119889119904 forallV

ℎisin 119872ℎ

(24a)

(120585119899wℎ) + (119887nabla119890

119899wℎ)ℎ= minus (120578

119899wℎ) minus (119887nabla120588

119899wℎ)ℎ

forallwℎisin Vℎ

(24b)

We are now in a position to prove the optimal order errorestimates

Theorem 2 Let (119906 120590) and (119906119899

ℎ 120590119899

ℎ) be the solutions of (12)

(15a) (15b) and (15c) respectively (12059721199061205971205912) isin 1198712(0 119879

1198712(Ω)) 119906

119905isin 1198712(0 119879119867

2(Ω)) 119906 isin 119871

infin(0 119879119867

2(Ω)) 120590 isin

119871infin(0 119879119867

2(Ω)) and assume that Δ119905 = 119874(ℎ

2) Then under

assumption (A3) we have

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)10038171003817100381710038171ℎ

le 119862 (Δ119905 + ℎ) (25)

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ

2) (26)

max0le119899le119873

1003817100381710038171003817(120590ℎminus 120590) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ) (27)

Proof Taking Vℎ= 119890119899 in (24a) and w

ℎ= nabla119890119899 in (24b) and

adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899) minus (

120588119899minus 120588119899minus1

Δ119905

119890119899)

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899)

+ sum

119890isin119879ℎ

int

120597119890

120590119899119890119899sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899)

ℎ

=

5

sum

119894=1

(Err)119894

(28)

Journal of Applied Mathematics 5

On the one hand we consider the right hand of (28)Using the method similar to [3] we have

(Err)1le 119862

100381710038171003817100381710038171003817100381710038171003817

120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(29)

(Err)2can be estimated as

1003816100381610038161003816(Err)2

1003816100381610038161003816le

1

Δ119905

(int

Ω

(int

119905119899

119905119899minus1

120588119905119889119904)

2

119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

1

radicΔ119905

(int

Ω

int

119905119899

119905119899minus1

1205882

119905119889119904 119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

119862

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817120588119905

1003817100381710038171003817

2

119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(30)

By Lemma 2 we obtain

1003816100381610038161003816(Err)3

1003816100381610038161003816le

1

Δ119905

10038171003817100381710038171003817120588119899minus1

minus 120588119899minus110038171003817

100381710038171003817minus1

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817120588119899minus110038171003817

100381710038171003817

2

+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862ℎ410038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(31)

It follows from Lemma 1 that

1003816100381610038161003816(Err)4

1003816100381610038161003816le 119862ℎ410038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ (32)

Let 119887 = (1|119890|) int119890119887(119909 119910)119889119909 119889119910 By Lemma 1 we have

1003816100381610038161003816(Err)5

1003816100381610038161003816=

10038161003816100381610038161003816minus((119887 minus 119887) nabla120588

119899 nabla119890119899)ℎ

10038161003816100381610038161003816

le 119862ℎ|119887|1198821infin(Ω)

100381710038171003817100381712058811989910038171003817100381710038171ℎ

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862ℎ410038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(33)

On the other hand the left hand of (28) can be bounded by

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

ge

1

2Δ119905

((119890119899 119890119899) minus (119890

119899minus1 119890119899minus1

)) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

ge

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus (1 + 119862Δ119905)

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(34)

where the inequality 119890119899minus12 le (1+119862Δ119905)119890119899minus1

2 proved in [3]is used in the last step

Combining (29)ndash(34) with (28) gives

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862(Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus1 119905119899 1198712(Ω))

+

ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+ℎ4(

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω))

+ 1205761

10038171003817100381710038171198901198991003817100381710038171003817

2

+ 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

100381710038171003817100381711989011989910038171003817100381710038171ℎ

(35)

Taking 1 minus 2Δ1199051205761gt 0 multiplying (35) by 2Δ119905 summing over

from 119894 = 1 to 119894 = 119899 and noticing that 1198900 = 0 we obtain

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4int

119905119899

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+Δ119905ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)) + 119862

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

(36)

It follows from discrete Gronwallrsquos lemma that

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(37)

From (37) we get the optimal order error estimate of 119890119899rather than 119890

1198991ℎ So we start to reestimate 119890119899

1ℎin the

following manner and derive the estimation of 120585119899 simul-taneously

Firstly choosing Vℎ= ((119890119899minus 119890119899minus1

)Δ119905) in (24a) and wℎ=

nabla((119890119899minus 119890119899minus1

)Δ119905) in (24b) and adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

minus (

120588119899minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

6 Journal of Applied Mathematics

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

+ sum

119890isin119879ℎ

int

120597119890

120590119899 119890119899minus 119890119899minus1

Δ119905

sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899minus 119890119899minus1

Δ119905

)

ℎ

=

5

sum

119894=1

(Err)1015840119894

(38)The left hand can be estimated as

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

ge

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899) minus (119887nabla119890

119899minus1 nabla119890119899minus1

)]

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(39)

and (Err)1015840119894 (119894 = 1 2 3 4 5) can be bounded by

10038161003816100381610038161003816(Err)10158401

10038161003816100381610038161003816le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158402

10038161003816100381610038161003816le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158403

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158404

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158405

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

(40)From (38)ndash(40) we get

1

2

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899)ℎminus (119887nabla119890

119899minus1 nabla119890119899minus1

)ℎ]

le 119862[Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

ℎ4

Δ119905

(int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω

+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω)] + 120576Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(41)

Multiplying (41) by 2Δ119905 and summing over in time from 119894 = 1

to 119894 = 119899 yield

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+ 120576(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+

119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

(42)Secondly we takeΔ119905 rarr 0 andΔ119905must approach zero in sucha way that Δ119905 and ℎ satisfy

Δ119905 = 119874 (ℎ2) (43)

and by inverse inequality we have

(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

le 119862Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

(44)

At the same time using Lemma 2 we obtain119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

= (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899) +

119899minus1

sum

119894=1

(

119890119894minus1

minus 119890119894minus (119890119894minus1

minus 119890119894)

Δ119905

119890119894)

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

100381710038171003817100381711989011989910038171003817100381710038171ℎ

+

119899minus1

sum

119894=1

10038171003817100381710038171003817119890119894minus 119890119894minus110038171003817100381710038171003817

10038171003817100381710038171003817119890119894100381710038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(45)From (42)ndash(45) taking suitable small 120576 such that 1 minus 120576119862 gt 0we have

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(46)

Journal of Applied Mathematics 7

Finally applying discrete Gronwallrsquos lemma yields

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎle 119862[(Δ119905)

2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ ℎ2(1199062

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(47)

In order to derive (27) set wℎ= 120585119899 in (24b) and employ

Lemma 1 and assumption (A3) to give

10038171003817100381710038171205851198991003817100381710038171003817

2

= minus(119887nabla119890119899 120585119899)ℎminus (120578119899 120585119899) minus (119887nabla120588

119899 120585119899)ℎ

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4100381710038171003817100381712059011989910038171003817100381710038172Ω

)

minus ((119887 minus 119887) nabla120588119899 120585119899)ℎ+

1

4

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4(100381710038171003817100381712059011989910038171003817100381710038172Ω

+100381710038171003817100381711990611989910038171003817100381710038172Ω

)) +

1

2

10038171003817100381710038171205851198991003817100381710038171003817

2

(48)

Combining (47) with (48) yields

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ2(|119906|2

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(49)

By using of interpolation theory and the triangle inequality(37) (47) and (49) lead to (25) (26) and (27) respectivelywhich are the desired results

Remark 2 From (37) we have

Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ= Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817

(1205871

ℎ119906 minus 119906ℎ)

119894100381710038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(50)

This byproduct can be regarded as the superclose resultbetween 1205871

ℎ119906 and 119906

ℎin mean broken1198671-norm It seems that

both (25) and (50) have never been seen in the existing stud-ies At the same time by employing the new characteristicnonconforming MFE scheme we can also obtain the sameerror estimate of (27) as traditional characteristicMFEM[10]

Remark 3 From the analysis of Theorem 2 in this paperwe may see that Lemma 1 is the key result leading to the

Table 1 Numerical results of 119906 minus 119906ℎ1ℎ

119898 times 119899 119905 = 02 120572 119905 = 03 120572 119905 = 04 120572

8 times 8 075277 075017 066433 16 times 16 042984 081 041849 084 035474 09132 times 32 021758 099 021412 097 017552 102119898 times 119899 119905 = 05 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 055291 042211 040937 16 times 16 029234 092 023117 087 021120 09632 times 32 014466 102 010807 110 009343 118

Table 2 Numerical results of 119906 minus 119906ℎ

119898 times 119899 119905 = 04 120572 119905 = 05 120572 119905 = 07 120572

8 times 8 00298190 00276370 00223240 16 times 16 00073087 203 00062445 215 00048038 22232 times 32 00020769 182 00017926 180 00013309 185119898 times 119899 119905 = 08 120572 119905 = 09 120572 119905 = 10 120572

8 times 8 00198730 00175900 00154090 16 times 16 00044472 216 00041982 207 00039150 19832 times 32 00011894 190 00010738 197 00009466 205

successful optimal order error estimations If we want toget higher order accuracy similar to Lemma 1 the non-conforming finite elements for approximating 119906 should alsopossess a very special property that is the consistency errorestimates with 119874(ℎ

2) order and satisfy (18) For the famous

nonconformingWilson element [32] whose shape function isspan1 119909 119910 1199092 1199102 by a counter-example it has been provenin [32] that its consistency error estimate is of119874(ℎ) order andcannot be improved any more For the rotated bilinear 119876

1

element [33] whose shape function is span1 119909 119910 1199092 minus 1199102

although its consistency error with 119874(ℎ2) order and (nabla(119906 minus

1205871

ℎ119906) nablaV

ℎ)ℎ= 0 on squaremeshes is satisfied the second term

of (18) is not valid Thus when they are applied to (1) on newcharacteristic mixed finite element scheme up to now theoptimal order error estimates of (25) (26) and (27) cannotbe obtained directly

5 Numerical Example

In order to verify our theoretical analysis in previous sectionswe consider the convection-dominated diffusion problem (1)as follows

119906119905+ 119906119909+ 119906119910minus 10minus4(119906119909119909+ 119906119910119910)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(51)

withΩ = [0 1] times [0 1] a(119909 119910) = (1 1) and 119887(119909 119910) = 10minus4

The right hand term 119891(119909 119910 119905) is taken such that 119906 =

119890minus119905 sin(120587119909) sin(2120587119910) 120590 = minus10

minus4119890minus119905(120587 cos(120587119909) sin(2120587119910)

2120587 sin(120587119909) cos(2120587119910)) are the exact solutions

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Now we prove the existence and uniqueness of the solu-tion of (15a) (15b) and (15c)

Theorem 1 Under assumption (A3) there exists a uniquesolution (119906

ℎ 120590ℎ) isin 119872

ℎtimes Vℎto the schemes (15a) (15b) and

(15c)

Proof The linear system generated by (15a) (15b) and (15c)is square so the existence of the solution is implied by its uni-queness From (15a) (15b) and (15c) we have

(

119906119899

ℎ

Δ119905

Vℎ) minus (120590

119899

ℎ nablaVℎ)ℎ= (

119906119899minus1

ℎ

Δ119905

Vℎ) + (119891

119899 Vℎ) forallV

ℎisin 119872ℎ

(120590119899

ℎwℎ) + (119887nabla119906

119899

ℎwℎ)ℎ= 0 forallw

ℎisin Vℎ

(16)

Let 119906119899ℎand 119891 be zero and thus 119906119899

ℎis zero too taking V

ℎ=

119906119899

ℎ wℎ= (1119887)120590

119899

ℎin (16) and adding them together we have

1

Δ119905

1003817100381710038171003817119906119899

ℎ

1003817100381710038171003817

2

+ (

1

119887

120590119899

ℎ 120590119899

ℎ) = 0 (17)

Thus assumption (A3) implies that 119906119899ℎ= 120590119899

ℎ= 0 The proof is

complete

To get error estimates we state the following two impor-tant lemmas

Lemma 1 (see [27 29 31]) Assume that 119906 isin 1198671(Ω) p isin

(1198672(Ω))

2 for all Vℎisin 119872ℎ wℎisin Vℎ and then there hold

(nabla (119906 minus 1205871

ℎ119906) nablaV

ℎ)ℎ= 0 (nabla (119906 minus 120587

1

ℎ119906) wℎ)ℎ= 0

(18)

(p minus 1205872ℎpwℎ) le 119862ℎ

2|p|2Ω

1003817100381710038171003817wℎ

1003817100381710038171003817 (19)

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119890isin119879ℎ

int

120597119890

pVℎsdot n 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862ℎ2|p|2Ω

1003817100381710038171003817Vℎ

10038171003817100381710038171ℎ

(20)

where sdot 1ℎ

= (sum119890isin119879ℎ

| sdot |1119890)12 is a norm on 119872

ℎ and n

denotes the outward unit normal vector on 120597119890

Lemma 2 (see [1 3]) Let 120593 isin 1198712(Ω) and 120593 = 120593(119883minus119892(119883)Δ119905)

where function 119892 and its gradient nabla119892 are bounded then1003817100381710038171003817120593 minus 120593

1003817100381710038171003817minus1

le 11986210038171003817100381710038171205931003817100381710038171003817Δ119905 (21)

where 120593minus1= sup

120601isin1198671(Ω)((120593 120601)120601

1Ω)

4 Convergence Analysis and Optimal OrderError Estimates

In this section we aim to analyze the convergence analysisand error estimates of characteristic nonconforming MFEMIn order to do this let

119906ℎminus 119906 = 119906

ℎminus 1205871

ℎ119906 + 1205871

ℎ119906 minus 119906 = 119890 + 120588

120590ℎminus 120590 = 120590

ℎminus 1205872

ℎ120590 + 1205872

ℎ120590 minus 120590 = 120585 + 120578

(22)

Taking 119905 = 119905119899 in (12) yields

(120595

120597119906119899

120597120591

Vℎ) minus (120590

119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n119889119904 = (119891

119899 Vℎ)

forallVℎisin 119872ℎ

(23a)

(120590119899wℎ) + (119887nabla119906

119899wℎ)ℎ= 0 forallw

ℎisin Vℎ (23b)

From (23a) (23b) (15a) (15b) and (15c) we get

(

119890119899minus 119890119899minus1

Δ119905

Vℎ) minus (120585

119899 nablaVℎ)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

Vℎ) minus (

120588119899minus 120588119899minus1

Δ119905

Vℎ)

+ (120578119899 nablaVℎ)ℎ+ sum

119890isin119879ℎ

int

120597119890

120590119899Vℎsdot n 119889119904 forallV

ℎisin 119872ℎ

(24a)

(120585119899wℎ) + (119887nabla119890

119899wℎ)ℎ= minus (120578

119899wℎ) minus (119887nabla120588

119899wℎ)ℎ

forallwℎisin Vℎ

(24b)

We are now in a position to prove the optimal order errorestimates

Theorem 2 Let (119906 120590) and (119906119899

ℎ 120590119899

ℎ) be the solutions of (12)

(15a) (15b) and (15c) respectively (12059721199061205971205912) isin 1198712(0 119879

1198712(Ω)) 119906

119905isin 1198712(0 119879119867

2(Ω)) 119906 isin 119871

infin(0 119879119867

2(Ω)) 120590 isin

119871infin(0 119879119867

2(Ω)) and assume that Δ119905 = 119874(ℎ

2) Then under

assumption (A3) we have

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)10038171003817100381710038171ℎ

le 119862 (Δ119905 + ℎ) (25)

max0le119899le119873

1003817100381710038171003817(119906ℎminus 119906) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ

2) (26)

max0le119899le119873

1003817100381710038171003817(120590ℎminus 120590) (119905

119899)1003817100381710038171003817le 119862 (Δ119905 + ℎ) (27)

Proof Taking Vℎ= 119890119899 in (24a) and w

ℎ= nabla119890119899 in (24b) and

adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899) minus (

120588119899minus 120588119899minus1

Δ119905

119890119899)

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899)

+ sum

119890isin119879ℎ

int

120597119890

120590119899119890119899sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899)

ℎ

=

5

sum

119894=1

(Err)119894

(28)

Journal of Applied Mathematics 5

On the one hand we consider the right hand of (28)Using the method similar to [3] we have

(Err)1le 119862

100381710038171003817100381710038171003817100381710038171003817

120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(29)

(Err)2can be estimated as

1003816100381610038161003816(Err)2

1003816100381610038161003816le

1

Δ119905

(int

Ω

(int

119905119899

119905119899minus1

120588119905119889119904)

2

119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

1

radicΔ119905

(int

Ω

int

119905119899

119905119899minus1

1205882

119905119889119904 119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

119862

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817120588119905

1003817100381710038171003817

2

119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(30)

By Lemma 2 we obtain

1003816100381610038161003816(Err)3

1003816100381610038161003816le

1

Δ119905

10038171003817100381710038171003817120588119899minus1

minus 120588119899minus110038171003817

100381710038171003817minus1

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817120588119899minus110038171003817

100381710038171003817

2

+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862ℎ410038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(31)

It follows from Lemma 1 that

1003816100381610038161003816(Err)4

1003816100381610038161003816le 119862ℎ410038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ (32)

Let 119887 = (1|119890|) int119890119887(119909 119910)119889119909 119889119910 By Lemma 1 we have

1003816100381610038161003816(Err)5

1003816100381610038161003816=

10038161003816100381610038161003816minus((119887 minus 119887) nabla120588

119899 nabla119890119899)ℎ

10038161003816100381610038161003816

le 119862ℎ|119887|1198821infin(Ω)

100381710038171003817100381712058811989910038171003817100381710038171ℎ

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862ℎ410038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(33)

On the other hand the left hand of (28) can be bounded by

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

ge

1

2Δ119905

((119890119899 119890119899) minus (119890

119899minus1 119890119899minus1

)) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

ge

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus (1 + 119862Δ119905)

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(34)

where the inequality 119890119899minus12 le (1+119862Δ119905)119890119899minus1

2 proved in [3]is used in the last step

Combining (29)ndash(34) with (28) gives

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862(Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus1 119905119899 1198712(Ω))

+

ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+ℎ4(

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω))

+ 1205761

10038171003817100381710038171198901198991003817100381710038171003817

2

+ 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

100381710038171003817100381711989011989910038171003817100381710038171ℎ

(35)

Taking 1 minus 2Δ1199051205761gt 0 multiplying (35) by 2Δ119905 summing over

from 119894 = 1 to 119894 = 119899 and noticing that 1198900 = 0 we obtain

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4int

119905119899

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+Δ119905ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)) + 119862

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

(36)

It follows from discrete Gronwallrsquos lemma that

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(37)

From (37) we get the optimal order error estimate of 119890119899rather than 119890

1198991ℎ So we start to reestimate 119890119899

1ℎin the

following manner and derive the estimation of 120585119899 simul-taneously

Firstly choosing Vℎ= ((119890119899minus 119890119899minus1

)Δ119905) in (24a) and wℎ=

nabla((119890119899minus 119890119899minus1

)Δ119905) in (24b) and adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

minus (

120588119899minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

6 Journal of Applied Mathematics

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

+ sum

119890isin119879ℎ

int

120597119890

120590119899 119890119899minus 119890119899minus1

Δ119905

sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899minus 119890119899minus1

Δ119905

)

ℎ

=

5

sum

119894=1

(Err)1015840119894

(38)The left hand can be estimated as

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

ge

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899) minus (119887nabla119890

119899minus1 nabla119890119899minus1

)]

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(39)

and (Err)1015840119894 (119894 = 1 2 3 4 5) can be bounded by

10038161003816100381610038161003816(Err)10158401

10038161003816100381610038161003816le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158402

10038161003816100381610038161003816le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158403

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158404

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158405

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

(40)From (38)ndash(40) we get

1

2

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899)ℎminus (119887nabla119890

119899minus1 nabla119890119899minus1

)ℎ]

le 119862[Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

ℎ4

Δ119905

(int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω

+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω)] + 120576Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(41)

Multiplying (41) by 2Δ119905 and summing over in time from 119894 = 1

to 119894 = 119899 yield

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+ 120576(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+

119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

(42)Secondly we takeΔ119905 rarr 0 andΔ119905must approach zero in sucha way that Δ119905 and ℎ satisfy

Δ119905 = 119874 (ℎ2) (43)

and by inverse inequality we have

(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

le 119862Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

(44)

At the same time using Lemma 2 we obtain119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

= (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899) +

119899minus1

sum

119894=1

(

119890119894minus1

minus 119890119894minus (119890119894minus1

minus 119890119894)

Δ119905

119890119894)

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

100381710038171003817100381711989011989910038171003817100381710038171ℎ

+

119899minus1

sum

119894=1

10038171003817100381710038171003817119890119894minus 119890119894minus110038171003817100381710038171003817

10038171003817100381710038171003817119890119894100381710038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(45)From (42)ndash(45) taking suitable small 120576 such that 1 minus 120576119862 gt 0we have

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(46)

Journal of Applied Mathematics 7

Finally applying discrete Gronwallrsquos lemma yields

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎle 119862[(Δ119905)

2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ ℎ2(1199062

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(47)

In order to derive (27) set wℎ= 120585119899 in (24b) and employ

Lemma 1 and assumption (A3) to give

10038171003817100381710038171205851198991003817100381710038171003817

2

= minus(119887nabla119890119899 120585119899)ℎminus (120578119899 120585119899) minus (119887nabla120588

119899 120585119899)ℎ

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4100381710038171003817100381712059011989910038171003817100381710038172Ω

)

minus ((119887 minus 119887) nabla120588119899 120585119899)ℎ+

1

4

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4(100381710038171003817100381712059011989910038171003817100381710038172Ω

+100381710038171003817100381711990611989910038171003817100381710038172Ω

)) +

1

2

10038171003817100381710038171205851198991003817100381710038171003817

2

(48)

Combining (47) with (48) yields

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ2(|119906|2

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(49)

By using of interpolation theory and the triangle inequality(37) (47) and (49) lead to (25) (26) and (27) respectivelywhich are the desired results

Remark 2 From (37) we have

Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ= Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817

(1205871

ℎ119906 minus 119906ℎ)

119894100381710038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(50)

This byproduct can be regarded as the superclose resultbetween 1205871

ℎ119906 and 119906

ℎin mean broken1198671-norm It seems that

both (25) and (50) have never been seen in the existing stud-ies At the same time by employing the new characteristicnonconforming MFE scheme we can also obtain the sameerror estimate of (27) as traditional characteristicMFEM[10]

Remark 3 From the analysis of Theorem 2 in this paperwe may see that Lemma 1 is the key result leading to the

Table 1 Numerical results of 119906 minus 119906ℎ1ℎ

119898 times 119899 119905 = 02 120572 119905 = 03 120572 119905 = 04 120572

8 times 8 075277 075017 066433 16 times 16 042984 081 041849 084 035474 09132 times 32 021758 099 021412 097 017552 102119898 times 119899 119905 = 05 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 055291 042211 040937 16 times 16 029234 092 023117 087 021120 09632 times 32 014466 102 010807 110 009343 118

Table 2 Numerical results of 119906 minus 119906ℎ

119898 times 119899 119905 = 04 120572 119905 = 05 120572 119905 = 07 120572

8 times 8 00298190 00276370 00223240 16 times 16 00073087 203 00062445 215 00048038 22232 times 32 00020769 182 00017926 180 00013309 185119898 times 119899 119905 = 08 120572 119905 = 09 120572 119905 = 10 120572

8 times 8 00198730 00175900 00154090 16 times 16 00044472 216 00041982 207 00039150 19832 times 32 00011894 190 00010738 197 00009466 205

successful optimal order error estimations If we want toget higher order accuracy similar to Lemma 1 the non-conforming finite elements for approximating 119906 should alsopossess a very special property that is the consistency errorestimates with 119874(ℎ

2) order and satisfy (18) For the famous

nonconformingWilson element [32] whose shape function isspan1 119909 119910 1199092 1199102 by a counter-example it has been provenin [32] that its consistency error estimate is of119874(ℎ) order andcannot be improved any more For the rotated bilinear 119876

1

element [33] whose shape function is span1 119909 119910 1199092 minus 1199102

although its consistency error with 119874(ℎ2) order and (nabla(119906 minus

1205871

ℎ119906) nablaV

ℎ)ℎ= 0 on squaremeshes is satisfied the second term

of (18) is not valid Thus when they are applied to (1) on newcharacteristic mixed finite element scheme up to now theoptimal order error estimates of (25) (26) and (27) cannotbe obtained directly

5 Numerical Example

In order to verify our theoretical analysis in previous sectionswe consider the convection-dominated diffusion problem (1)as follows

119906119905+ 119906119909+ 119906119910minus 10minus4(119906119909119909+ 119906119910119910)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(51)

withΩ = [0 1] times [0 1] a(119909 119910) = (1 1) and 119887(119909 119910) = 10minus4

The right hand term 119891(119909 119910 119905) is taken such that 119906 =

119890minus119905 sin(120587119909) sin(2120587119910) 120590 = minus10

minus4119890minus119905(120587 cos(120587119909) sin(2120587119910)

2120587 sin(120587119909) cos(2120587119910)) are the exact solutions

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

On the one hand we consider the right hand of (28)Using the method similar to [3] we have

(Err)1le 119862

100381710038171003817100381710038171003817100381710038171003817

120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(29)

(Err)2can be estimated as

1003816100381610038161003816(Err)2

1003816100381610038161003816le

1

Δ119905

(int

Ω

(int

119905119899

119905119899minus1

120588119905119889119904)

2

119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

1

radicΔ119905

(int

Ω

int

119905119899

119905119899minus1

1205882

119905119889119904 119889119909 119889119910)

12

10038171003817100381710038171198901198991003817100381710038171003817

le

119862

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817120588119905

1003817100381710038171003817

2

119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1205761

2

10038171003817100381710038171198901198991003817100381710038171003817

2

(30)

By Lemma 2 we obtain

1003816100381610038161003816(Err)3

1003816100381610038161003816le

1

Δ119905

10038171003817100381710038171003817120588119899minus1

minus 120588119899minus110038171003817

100381710038171003817minus1

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817120588119899minus110038171003817

100381710038171003817

2

+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862ℎ410038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(31)

It follows from Lemma 1 that

1003816100381610038161003816(Err)4

1003816100381610038161003816le 119862ℎ410038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ (32)

Let 119887 = (1|119890|) int119890119887(119909 119910)119889119909 119889119910 By Lemma 1 we have

1003816100381610038161003816(Err)5

1003816100381610038161003816=

10038161003816100381610038161003816minus((119887 minus 119887) nabla120588

119899 nabla119890119899)ℎ

10038161003816100381610038161003816

le 119862ℎ|119887|1198821infin(Ω)

100381710038171003817100381712058811989910038171003817100381710038171ℎ

100381710038171003817100381711989011989910038171003817100381710038171ℎ

le 119862ℎ410038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

1198871

6

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(33)

On the other hand the left hand of (28) can be bounded by

(

119890119899minus 119890119899minus1

Δ119905

119890119899) + (119887nabla119890

119899 nabla119890119899)ℎ

ge

1

2Δ119905

((119890119899 119890119899) minus (119890

119899minus1 119890119899minus1

)) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

ge

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus (1 + 119862Δ119905)

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

(34)

where the inequality 119890119899minus12 le (1+119862Δ119905)119890119899minus1

2 proved in [3]is used in the last step

Combining (29)ndash(34) with (28) gives

1

2Δ119905

(10038171003817100381710038171198901198991003817100381710038171003817

2

minus

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

) + 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862(Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus1 119905119899 1198712(Ω))

+

ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+ℎ4(

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω))

+ 1205761

10038171003817100381710038171198901198991003817100381710038171003817

2

+ 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

100381710038171003817100381711989011989910038171003817100381710038171ℎ

(35)

Taking 1 minus 2Δ1199051205761gt 0 multiplying (35) by 2Δ119905 summing over

from 119894 = 1 to 119894 = 119899 and noticing that 1198900 = 0 we obtain

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4int

119905119899

0

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904

+Δ119905ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)) + 119862

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

(36)

It follows from discrete Gronwallrsquos lemma that

10038171003817100381710038171198901198991003817100381710038171003817

2

+ Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(37)

From (37) we get the optimal order error estimate of 119890119899rather than 119890

1198991ℎ So we start to reestimate 119890119899

1ℎin the

following manner and derive the estimation of 120585119899 simul-taneously

Firstly choosing Vℎ= ((119890119899minus 119890119899minus1

)Δ119905) in (24a) and wℎ=

nabla((119890119899minus 119890119899minus1

)Δ119905) in (24b) and adding them we have

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

= (120595

120597119906119899

120597120591

minus

119906119899minus 119906119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

minus (

120588119899minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

6 Journal of Applied Mathematics

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

+ sum

119890isin119879ℎ

int

120597119890

120590119899 119890119899minus 119890119899minus1

Δ119905

sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899minus 119890119899minus1

Δ119905

)

ℎ

=

5

sum

119894=1

(Err)1015840119894

(38)The left hand can be estimated as

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

ge

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899) minus (119887nabla119890

119899minus1 nabla119890119899minus1

)]

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(39)

and (Err)1015840119894 (119894 = 1 2 3 4 5) can be bounded by

10038161003816100381610038161003816(Err)10158401

10038161003816100381610038161003816le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158402

10038161003816100381610038161003816le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158403

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158404

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158405

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

(40)From (38)ndash(40) we get

1

2

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899)ℎminus (119887nabla119890

119899minus1 nabla119890119899minus1

)ℎ]

le 119862[Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

ℎ4

Δ119905

(int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω

+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω)] + 120576Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(41)

Multiplying (41) by 2Δ119905 and summing over in time from 119894 = 1

to 119894 = 119899 yield

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+ 120576(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+

119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

(42)Secondly we takeΔ119905 rarr 0 andΔ119905must approach zero in sucha way that Δ119905 and ℎ satisfy

Δ119905 = 119874 (ℎ2) (43)

and by inverse inequality we have

(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

le 119862Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

(44)

At the same time using Lemma 2 we obtain119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

= (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899) +

119899minus1

sum

119894=1

(

119890119894minus1

minus 119890119894minus (119890119894minus1

minus 119890119894)

Δ119905

119890119894)

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

100381710038171003817100381711989011989910038171003817100381710038171ℎ

+

119899minus1

sum

119894=1

10038171003817100381710038171003817119890119894minus 119890119894minus110038171003817100381710038171003817

10038171003817100381710038171003817119890119894100381710038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(45)From (42)ndash(45) taking suitable small 120576 such that 1 minus 120576119862 gt 0we have

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(46)

Journal of Applied Mathematics 7

Finally applying discrete Gronwallrsquos lemma yields

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎle 119862[(Δ119905)

2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ ℎ2(1199062

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(47)

In order to derive (27) set wℎ= 120585119899 in (24b) and employ

Lemma 1 and assumption (A3) to give

10038171003817100381710038171205851198991003817100381710038171003817

2

= minus(119887nabla119890119899 120585119899)ℎminus (120578119899 120585119899) minus (119887nabla120588

119899 120585119899)ℎ

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4100381710038171003817100381712059011989910038171003817100381710038172Ω

)

minus ((119887 minus 119887) nabla120588119899 120585119899)ℎ+

1

4

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4(100381710038171003817100381712059011989910038171003817100381710038172Ω

+100381710038171003817100381711990611989910038171003817100381710038172Ω

)) +

1

2

10038171003817100381710038171205851198991003817100381710038171003817

2

(48)

Combining (47) with (48) yields

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ2(|119906|2

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(49)

By using of interpolation theory and the triangle inequality(37) (47) and (49) lead to (25) (26) and (27) respectivelywhich are the desired results

Remark 2 From (37) we have

Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ= Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817

(1205871

ℎ119906 minus 119906ℎ)

119894100381710038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(50)

This byproduct can be regarded as the superclose resultbetween 1205871

ℎ119906 and 119906

ℎin mean broken1198671-norm It seems that

both (25) and (50) have never been seen in the existing stud-ies At the same time by employing the new characteristicnonconforming MFE scheme we can also obtain the sameerror estimate of (27) as traditional characteristicMFEM[10]

Remark 3 From the analysis of Theorem 2 in this paperwe may see that Lemma 1 is the key result leading to the

Table 1 Numerical results of 119906 minus 119906ℎ1ℎ

119898 times 119899 119905 = 02 120572 119905 = 03 120572 119905 = 04 120572

8 times 8 075277 075017 066433 16 times 16 042984 081 041849 084 035474 09132 times 32 021758 099 021412 097 017552 102119898 times 119899 119905 = 05 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 055291 042211 040937 16 times 16 029234 092 023117 087 021120 09632 times 32 014466 102 010807 110 009343 118

Table 2 Numerical results of 119906 minus 119906ℎ

119898 times 119899 119905 = 04 120572 119905 = 05 120572 119905 = 07 120572

8 times 8 00298190 00276370 00223240 16 times 16 00073087 203 00062445 215 00048038 22232 times 32 00020769 182 00017926 180 00013309 185119898 times 119899 119905 = 08 120572 119905 = 09 120572 119905 = 10 120572

8 times 8 00198730 00175900 00154090 16 times 16 00044472 216 00041982 207 00039150 19832 times 32 00011894 190 00010738 197 00009466 205

successful optimal order error estimations If we want toget higher order accuracy similar to Lemma 1 the non-conforming finite elements for approximating 119906 should alsopossess a very special property that is the consistency errorestimates with 119874(ℎ

2) order and satisfy (18) For the famous

nonconformingWilson element [32] whose shape function isspan1 119909 119910 1199092 1199102 by a counter-example it has been provenin [32] that its consistency error estimate is of119874(ℎ) order andcannot be improved any more For the rotated bilinear 119876

1

element [33] whose shape function is span1 119909 119910 1199092 minus 1199102

although its consistency error with 119874(ℎ2) order and (nabla(119906 minus

1205871

ℎ119906) nablaV

ℎ)ℎ= 0 on squaremeshes is satisfied the second term

of (18) is not valid Thus when they are applied to (1) on newcharacteristic mixed finite element scheme up to now theoptimal order error estimates of (25) (26) and (27) cannotbe obtained directly

5 Numerical Example

In order to verify our theoretical analysis in previous sectionswe consider the convection-dominated diffusion problem (1)as follows

119906119905+ 119906119909+ 119906119910minus 10minus4(119906119909119909+ 119906119910119910)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(51)

withΩ = [0 1] times [0 1] a(119909 119910) = (1 1) and 119887(119909 119910) = 10minus4

The right hand term 119891(119909 119910 119905) is taken such that 119906 =

119890minus119905 sin(120587119909) sin(2120587119910) 120590 = minus10

minus4119890minus119905(120587 cos(120587119909) sin(2120587119910)

2120587 sin(120587119909) cos(2120587119910)) are the exact solutions

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

minus (

120588119899minus1

minus 120588119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

+ sum

119890isin119879ℎ

int

120597119890

120590119899 119890119899minus 119890119899minus1

Δ119905

sdot n 119889119904 minus (119887nabla120588119899 nabla119890119899minus 119890119899minus1

Δ119905

)

ℎ

=

5

sum

119894=1

(Err)1015840119894

(38)The left hand can be estimated as

(

119890119899minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

) + (119887nabla119890119899 nabla

119890119899minus 119890119899minus1

Δ119905

)

ℎ

ge

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899) minus (119887nabla119890

119899minus1 nabla119890119899minus1

)]

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(39)

and (Err)1015840119894 (119894 = 1 2 3 4 5) can be bounded by

10038161003816100381610038161003816(Err)10158401

10038161003816100381610038161003816le 119862Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158402

10038161003816100381610038161003816le

119862ℎ4

Δ119905

int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

1

4

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

10038161003816100381610038161003816(Err)10158403

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158404

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

10038161003816100381610038161003816(Err)10158405

10038161003816100381610038161003816le

119862ℎ4

Δ119905

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

120576

3

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

(40)From (38)ndash(40) we get

1

2

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+

1

2Δ119905

[(119887nabla119890119899 nabla119890119899)ℎminus (119887nabla119890

119899minus1 nabla119890119899minus1

)ℎ]

le 119862[Δ119905

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(119905119899minus11199051198991198712(Ω))

+

ℎ4

Δ119905

(int

119905119899

119905119899minus1

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2Ω119889119904 +

10038171003817100381710038171199061198991003817100381710038171003817

2

2Ω+

10038171003817100381710038171003817119906119899minus110038171003817

100381710038171003817

2

2Ω

+10038171003817100381710038171205901198991003817100381710038171003817

2

2Ω)] + 120576Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+ (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899minus 119890119899minus1

Δ119905

)

(41)

Multiplying (41) by 2Δ119905 and summing over in time from 119894 = 1

to 119894 = 119899 yield

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 1198871

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+ 120576(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

+

119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

(42)Secondly we takeΔ119905 rarr 0 andΔ119905must approach zero in sucha way that Δ119905 and ℎ satisfy

Δ119905 = 119874 (ℎ2) (43)

and by inverse inequality we have

(Δ119905)2

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

1ℎ

le 119862Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

(44)

At the same time using Lemma 2 we obtain119899

sum

119894=1

(

119890119894minus1

minus 119890119894minus1

Δ119905

119890119894minus 119890119894minus1)

= (

119890119899minus1

minus 119890119899minus1

Δ119905

119890119899) +

119899minus1

sum

119894=1

(

119890119894minus1

minus 119890119894minus (119890119894minus1

minus 119890119894)

Δ119905

119890119894)

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

100381710038171003817100381711989011989910038171003817100381710038171ℎ

+

119899minus1

sum

119894=1

10038171003817100381710038171003817119890119894minus 119890119894minus110038171003817100381710038171003817

10038171003817100381710038171003817119890119894100381710038171003817100381710038171ℎ

le 119862

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+

1198871

2

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(45)From (42)ndash(45) taking suitable small 120576 such that 1 minus 120576119862 gt 0we have

Δ119905

100381710038171003817100381710038171003817100381710038171003817

119890119899minus 119890119899minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ4

119899

sum

119894=1

(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

2Ω+

1003817100381710038171003817100381712059011989410038171003817100381710038171003817

2

2Ω)]

+

10038171003817100381710038171003817119890119899minus110038171003817

100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

100381710038171003817100381710038171003817100381710038171003817

119890119894minus 119890119894minus1

Δ119905

100381710038171003817100381710038171003817100381710038171003817

2

+ 119862Δ119905

119899minus1

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ

(46)

Journal of Applied Mathematics 7

Finally applying discrete Gronwallrsquos lemma yields

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎle 119862[(Δ119905)

2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ ℎ2(1199062

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(47)

In order to derive (27) set wℎ= 120585119899 in (24b) and employ

Lemma 1 and assumption (A3) to give

10038171003817100381710038171205851198991003817100381710038171003817

2

= minus(119887nabla119890119899 120585119899)ℎminus (120578119899 120585119899) minus (119887nabla120588

119899 120585119899)ℎ

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4100381710038171003817100381712059011989910038171003817100381710038172Ω

)

minus ((119887 minus 119887) nabla120588119899 120585119899)ℎ+

1

4

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4(100381710038171003817100381712059011989910038171003817100381710038172Ω

+100381710038171003817100381711990611989910038171003817100381710038172Ω

)) +

1

2

10038171003817100381710038171205851198991003817100381710038171003817

2

(48)

Combining (47) with (48) yields

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ2(|119906|2

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(49)

By using of interpolation theory and the triangle inequality(37) (47) and (49) lead to (25) (26) and (27) respectivelywhich are the desired results

Remark 2 From (37) we have

Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ= Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817

(1205871

ℎ119906 minus 119906ℎ)

119894100381710038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(50)

This byproduct can be regarded as the superclose resultbetween 1205871

ℎ119906 and 119906

ℎin mean broken1198671-norm It seems that

both (25) and (50) have never been seen in the existing stud-ies At the same time by employing the new characteristicnonconforming MFE scheme we can also obtain the sameerror estimate of (27) as traditional characteristicMFEM[10]

Remark 3 From the analysis of Theorem 2 in this paperwe may see that Lemma 1 is the key result leading to the

Table 1 Numerical results of 119906 minus 119906ℎ1ℎ

119898 times 119899 119905 = 02 120572 119905 = 03 120572 119905 = 04 120572

8 times 8 075277 075017 066433 16 times 16 042984 081 041849 084 035474 09132 times 32 021758 099 021412 097 017552 102119898 times 119899 119905 = 05 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 055291 042211 040937 16 times 16 029234 092 023117 087 021120 09632 times 32 014466 102 010807 110 009343 118

Table 2 Numerical results of 119906 minus 119906ℎ

119898 times 119899 119905 = 04 120572 119905 = 05 120572 119905 = 07 120572

8 times 8 00298190 00276370 00223240 16 times 16 00073087 203 00062445 215 00048038 22232 times 32 00020769 182 00017926 180 00013309 185119898 times 119899 119905 = 08 120572 119905 = 09 120572 119905 = 10 120572

8 times 8 00198730 00175900 00154090 16 times 16 00044472 216 00041982 207 00039150 19832 times 32 00011894 190 00010738 197 00009466 205

successful optimal order error estimations If we want toget higher order accuracy similar to Lemma 1 the non-conforming finite elements for approximating 119906 should alsopossess a very special property that is the consistency errorestimates with 119874(ℎ

2) order and satisfy (18) For the famous

nonconformingWilson element [32] whose shape function isspan1 119909 119910 1199092 1199102 by a counter-example it has been provenin [32] that its consistency error estimate is of119874(ℎ) order andcannot be improved any more For the rotated bilinear 119876

1

element [33] whose shape function is span1 119909 119910 1199092 minus 1199102

although its consistency error with 119874(ℎ2) order and (nabla(119906 minus

1205871

ℎ119906) nablaV

ℎ)ℎ= 0 on squaremeshes is satisfied the second term

of (18) is not valid Thus when they are applied to (1) on newcharacteristic mixed finite element scheme up to now theoptimal order error estimates of (25) (26) and (27) cannotbe obtained directly

5 Numerical Example

In order to verify our theoretical analysis in previous sectionswe consider the convection-dominated diffusion problem (1)as follows

119906119905+ 119906119909+ 119906119910minus 10minus4(119906119909119909+ 119906119910119910)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(51)

withΩ = [0 1] times [0 1] a(119909 119910) = (1 1) and 119887(119909 119910) = 10minus4

The right hand term 119891(119909 119910 119905) is taken such that 119906 =

119890minus119905 sin(120587119909) sin(2120587119910) 120590 = minus10

minus4119890minus119905(120587 cos(120587119909) sin(2120587119910)

2120587 sin(120587119909) cos(2120587119910)) are the exact solutions

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Finally applying discrete Gronwallrsquos lemma yields

10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎle 119862[(Δ119905)

2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ ℎ2(1199062

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(47)

In order to derive (27) set wℎ= 120585119899 in (24b) and employ

Lemma 1 and assumption (A3) to give

10038171003817100381710038171205851198991003817100381710038171003817

2

= minus(119887nabla119890119899 120585119899)ℎminus (120578119899 120585119899) minus (119887nabla120588

119899 120585119899)ℎ

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4100381710038171003817100381712059011989910038171003817100381710038172Ω

)

minus ((119887 minus 119887) nabla120588119899 120585119899)ℎ+

1

4

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862 (10038171003817100381710038171198901198991003817100381710038171003817

2

1ℎ+ ℎ4(100381710038171003817100381712059011989910038171003817100381710038172Ω

+100381710038171003817100381711990611989910038171003817100381710038172Ω

)) +

1

2

10038171003817100381710038171205851198991003817100381710038171003817

2

(48)

Combining (47) with (48) yields

10038171003817100381710038171205851198991003817100381710038171003817

2

le 119862[(Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ41003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ℎ2(|119906|2

119871infin(01199051198991198672(Ω))

+ 1205902

119871infin(01199051198991198672(Ω))

) ]

(49)

By using of interpolation theory and the triangle inequality(37) (47) and (49) lead to (25) (26) and (27) respectivelywhich are the desired results

Remark 2 From (37) we have

Δ119905

119899

sum

119894=1

1003817100381710038171003817100381711989011989410038171003817100381710038171003817

2

1ℎ= Δ119905

119899

sum

119894=1

100381710038171003817100381710038171003817

(1205871

ℎ119906 minus 119906ℎ)

119894100381710038171003817100381710038171003817

2

1ℎ

le 119862((Δ119905)2

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971205912

100381710038171003817100381710038171003817100381710038171003817

2

1198712(01199051198991198712(Ω))

+ ℎ4(1003817100381710038171003817119906119905

1003817100381710038171003817

2

1198712(01199051198991198672(Ω))

+ 1199062

119871infin(01199051198991198672(Ω))

+1205902

119871infin(0119905119899(1198672(Ω))2

)))

(50)

This byproduct can be regarded as the superclose resultbetween 1205871

ℎ119906 and 119906

ℎin mean broken1198671-norm It seems that

both (25) and (50) have never been seen in the existing stud-ies At the same time by employing the new characteristicnonconforming MFE scheme we can also obtain the sameerror estimate of (27) as traditional characteristicMFEM[10]

Remark 3 From the analysis of Theorem 2 in this paperwe may see that Lemma 1 is the key result leading to the

Table 1 Numerical results of 119906 minus 119906ℎ1ℎ

119898 times 119899 119905 = 02 120572 119905 = 03 120572 119905 = 04 120572

8 times 8 075277 075017 066433 16 times 16 042984 081 041849 084 035474 09132 times 32 021758 099 021412 097 017552 102119898 times 119899 119905 = 05 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 055291 042211 040937 16 times 16 029234 092 023117 087 021120 09632 times 32 014466 102 010807 110 009343 118

Table 2 Numerical results of 119906 minus 119906ℎ

119898 times 119899 119905 = 04 120572 119905 = 05 120572 119905 = 07 120572

8 times 8 00298190 00276370 00223240 16 times 16 00073087 203 00062445 215 00048038 22232 times 32 00020769 182 00017926 180 00013309 185119898 times 119899 119905 = 08 120572 119905 = 09 120572 119905 = 10 120572

8 times 8 00198730 00175900 00154090 16 times 16 00044472 216 00041982 207 00039150 19832 times 32 00011894 190 00010738 197 00009466 205

successful optimal order error estimations If we want toget higher order accuracy similar to Lemma 1 the non-conforming finite elements for approximating 119906 should alsopossess a very special property that is the consistency errorestimates with 119874(ℎ

2) order and satisfy (18) For the famous

nonconformingWilson element [32] whose shape function isspan1 119909 119910 1199092 1199102 by a counter-example it has been provenin [32] that its consistency error estimate is of119874(ℎ) order andcannot be improved any more For the rotated bilinear 119876

1

element [33] whose shape function is span1 119909 119910 1199092 minus 1199102

although its consistency error with 119874(ℎ2) order and (nabla(119906 minus

1205871

ℎ119906) nablaV

ℎ)ℎ= 0 on squaremeshes is satisfied the second term

of (18) is not valid Thus when they are applied to (1) on newcharacteristic mixed finite element scheme up to now theoptimal order error estimates of (25) (26) and (27) cannotbe obtained directly

5 Numerical Example

In order to verify our theoretical analysis in previous sectionswe consider the convection-dominated diffusion problem (1)as follows

119906119905+ 119906119909+ 119906119910minus 10minus4(119906119909119909+ 119906119910119910)

= 119891 (119909 119910 119905) (119909 119910 119905) isin Ω times (0 119879)

119906 (119909 119910 119905) = 0 (119909 119910 119905) isin 120597Ω times (0 119879)

119906 (119909 119910 0) = 1199060(119909 119910) (119909 119910) isin Ω

(51)

withΩ = [0 1] times [0 1] a(119909 119910) = (1 1) and 119887(119909 119910) = 10minus4

The right hand term 119891(119909 119910 119905) is taken such that 119906 =

119890minus119905 sin(120587119909) sin(2120587119910) 120590 = minus10

minus4119890minus119905(120587 cos(120587119909) sin(2120587119910)

2120587 sin(120587119909) cos(2120587119910)) are the exact solutions

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Table 3 Numerical results of 120590 minus 120590ℎ

119898 times 119899 119905 = 01 120572 119905 = 04 120572 119905 = 05 120572

8 times 8 49528119890 minus 005 42661119890 minus 005 38292119890 minus 005 16 times 16 23945119890 minus 005 105 18843119890 minus 005 118 16806119890 minus 005 11932 times 32 11749119890 minus 005 103 90029119890 minus 006 107 80521119890 minus 006 106119898 times 119899 119905 = 07 120572 119905 = 08 120572 119905 = 09 120572

8 times 8 30714119890 minus 005 27735119890 minus 005 2524119890 minus 005 16 times 16 13326119890 minus 005 120 1224119890 minus 005 118 11443119890 minus 005 11432 times 32 6455119890 minus 006 105 58353119890 minus 006 107 53751119890 minus 006 109

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 04

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 1 Errors at 119905 = 04

0

minus2

minus4

minus6

minus8

minus10

minus12minus35 minus3 minus25 minus2

119905 = 05

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

Figure 2 Errors at 119905 = 05

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14minus3minus35 minus2

119905 = 08

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(error)

log(ℎ)

119906 minus 119906ℎ1ℎ

minus25

Figure 3 Errors at 119905 = 08

minus3minus35 minus25 minus2

119905 = 09

ℎ

ℎ2

119906 minus 119906ℎ

120590 minus 120590ℎ

log(ℎ)

0

minus2

minus4

minus6

minus8

minus10

minus12

minus14

log(error)

119906 minus 119906ℎ1ℎ

Figure 4 Errors at 119905 = 09

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

We first divide the domainΩ into119898 and 119899 equal intervalsalong 119909-axis and 119910-axis and the numerical results at differenttimes are listed in Tables 1 2 and 3 and pictured in Figures1 2 3 and 4 respectively (119906

ℎ pℎ) denotes the characteristic

nonconformingMFE solution of the problem (15a) (15b) and(15c) Δ119905 represents the time step and the experiment is donewith Δ119905 = ℎ

2 120572 stands for the convergence orderIt can be seen from the above Tables 1 2 and 3 that

119906 minus 119906ℎ1ℎ

and 120590minus120590ℎ are convergent at optimal rate of119874(ℎ)

and 119906 minus 119906ℎ is convergent at optimal rate of 119874(ℎ2) respec-

tively which coincide with our theoretical investigation inSection 4

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (Grant nos 10971203 11101384 and11271340) and the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant no 20094101110006)The author would like to thank the referees for their helpfulsuggestions

References

[1] L Guo and H Z Chen ldquoAn expanded characteristic-mixedfinite element method for a convection-dominated transportproblemrdquo Journal of Computational Mathematics vol 23 no 5pp 479ndash490 2005

[2] Z W Jiang Q Yang and A Q Li ldquoA characteristics-finitevolume element method for a convection-dominated diffusionequationrdquo Journal of Systems Science andMathematical Sciencesvol 31 no 1 pp 80ndash91 2011

[3] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[4] L Z Qian X L Feng and Y N He ldquoThe characteristicfinite difference streamline diffusion method for convection-dominated diffusion problemsrdquo Applied Mathematical Mod-elling vol 36 no 2 pp 561ndash572 2012

[5] P Hansbo ldquoThe characteristic streamline diffusion method forconvection-diffusion problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 96 no 2 pp 239ndash253 1992

[6] M A Celia T F Russell I Herrera and R E Ewing ldquoAnEulerian-Lagrangian localized adjoint method for the advec-tion-diffusion equationrdquo Advances in Water Resources vol 13no 4 pp 186ndash205 1990

[7] H Wang R E Ewing and T F Russell ldquoEulerian-Lagrangianlocalized adjoint methods for convection-diffusion equationsand their convergence analysisrdquo IMA Journal of NumericalAnalysis vol 15 no 3 pp 405ndash459 1995

[8] H X Rui ldquoA conservative characteristic finite volume elementmethod for solution of the advection-diffusion equationrdquo Com-puter Methods in Applied Mechanics and Engineering vol 197no 45ndash48 pp 3862ndash3869 2008

[9] F Z Gao and Y R Yuan ldquoThe characteristic finite volumeelementmethod for the nonlinear convection-dominated diffu-sion problemrdquoComputersampMathematics withApplications vol56 no 1 pp 71ndash81 2008

[10] H T Che and Z W Jiang ldquoA characteristics-mixed covolumemethod for a convection-dominated transport problemrdquo Jour-nal of Computational and Applied Mathematics vol 231 no 2pp 760ndash770 2009

[11] Z X Chen S H Chou and D Y Kwak ldquoCharacteristic-mixedcovolume methods for advection-dominated diffusion prob-lemsrdquo Numerical Linear Algebra with Applications vol 13 no9 pp 677ndash697 2006

[12] C N Dawson T F Russell andM FWheeler ldquoSome improvederror estimates for the modified method of characteristicsrdquoSIAM Journal on Numerical Analysis vol 26 no 6 pp 1487ndash1512 1989

[13] Z X Chen ldquoCharacteristic-nonconforming finite-elementmethods for advection-dominated diffusion problemsrdquo Com-puters amp Mathematics with Applications vol 48 no 7-8 pp1087ndash1100 2004

[14] D Y Shi and X L Wang ldquoA low order anisotropic noncon-forming characteristic finite element method for a convection-dominated transport problemrdquo Applied Mathematics and Com-putation vol 213 no 2 pp 411ndash418 2009

[15] D Y Shi and X L Wang ldquoTwo low order characteristic finiteelement methods for a convection-dominated transport prob-lemrdquo Computers amp Mathematics with Applications vol 59 no12 pp 3630ndash3639 2010

[16] Z J Zhou F X Chen and H Z Chen ldquoCharacteristic mixedfinite element approximation of transient convection diffusionoptimal control problemsrdquo Mathematics and Computers inSimulation vol 82 no 11 pp 2109ndash2128 2012

[17] Z Y Liu andH Z Chen ldquoModified characteristics-mixed finiteelement method with adjusted advection for linear convection-dominated diffusion problemsrdquo Chinese Journal of EngineeringMathematics vol 26 no 2 pp 200ndash208 2009

[18] T Arbogast and M F Wheeler ldquoA characteristics-mixed finiteelement method for advection-dominated transport problemsrdquoSIAM Journal onNumerical Analysis vol 32 no 2 pp 404ndash4241995

[19] T J Sun and Y R Yuan ldquoAn approximation of incompressiblemiscible displacement in porous media by mixed finite elementmethod and characteristics-mixed finite elementmethodrdquo Jour-nal of Computational and Applied Mathematics vol 228 no 1pp 391ndash411 2009

[20] F X Chen andH Z Chen ldquoAn expanded characteristics-mixedfinite element method for quasilinear convection-dominateddiffusion equationsrdquo Journal of Systems Science and Mathemat-ical Sciences vol 29 no 5 pp 585ndash597 2009

[21] Z X Chen ldquoCharacteristic mixed discontinuous finite elementmethods for advection-dominated diffusion problemsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 191no 23-24 pp 2509ndash2538 2002

[22] D Q Yang ldquoA characteristic mixed method with dynamicfinite-element space for convection-dominated diffusion prob-lemsrdquo Journal of Computational and Applied Mathematics vol43 no 3 pp 343ndash353 1992

[23] H Z Chen Z J Zhou H Wang and H Y Man ldquoAn optimal-order error estimate for a family of characteristic-mixed meth-ods to transient convection-diffusion problemsrdquo Discrete andContinuous Dynamical Systems vol 15 no 2 pp 325ndash341 2011

[24] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[25] P A Raviart and J MThomas ldquoAmixed finite element methodfor 2nd order elliptic problemsrdquo in Mathematical Aspects of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

Finite Element Methods vol 606 of Lecture Notes in Mathemat-ics pp 292ndash315 Springer Berlin Germany 1977

[26] S C Chen and H R Chen ldquoNew mixed element schemes for asecond-order elliptic problemrdquo Mathematica Numerica Sinicavol 32 no 2 pp 213ndash218 2010

[27] Q Lin and N N Yan The Construction and Analysis of HighAccurate Finite ElementMethods Hebei University Press Baod-ing China 1996

[28] S Larsson and V Thomee Partial Differential Equations withNumerical Methods vol 45 of Texts in Applied MathematicsSpringer Berlin Germany 2003

[29] D Y Shi and Y D Zhang ldquoHigh accuracy analysis of a newnonconforming mixed finite element scheme for Sobolev equa-tionsrdquoAppliedMathematics and Computation vol 218 no 7 pp3176ndash3186 2011

[30] P G Ciarlet The Finite Element Method for Elliptic Problemsvol 4 North-Holland Publishing Amsterdam The Nether-lands 1978 Studies in Mathematics and its Applications

[31] D Y Shi P L Xie and S C Chen ldquoNonconforming finite ele-ment approximation to hyperbolic integrodifferential equationson anisotropic meshesrdquo Acta Mathematicae Applicatae Sinicavol 30 no 4 pp 654ndash666 2007

[32] Z C Shi ldquoA remark on the optimal order of convergenceof Wilsonrsquos nonconforming elementrdquo Mathematica NumericaSinica vol 8 no 2 pp 159ndash163 1986

[33] R Rannacher and S Turek ldquoSimple nonconforming quadrilat-eral Stokes elementrdquo Numerical Methods for Partial DifferentialEquations vol 8 no 2 pp 97ndash111 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of