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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
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Operations ResearchAdvances in
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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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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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