Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2013 Article ID 189045 9 pageshttpdxdoiorg1011552013189045

Research ArticleMixed Finite Element Methods for the Poisson EquationUsing Biorthogonal and Quasi-Biorthogonal Systems

Bishnu P Lamichhane

School of Mathematical and Physical Sciences University of Newcastle Callaghan NSW 2308 Australia

Correspondence should be addressed to Bishnu P Lamichhane blamichhagmailcom

Received 10 October 2012 Revised 20 February 2013 Accepted 25 February 2013

Academic Editor Norbert Heuer

Copyright copy 2013 Bishnu P Lamichhane This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce two three-field mixed formulations for the Poisson equation and propose finite element methods for theirapproximation Both mixed formulations are obtained by introducing a weak equation for the gradient of the solution by means ofa Lagrange multiplier space Two efficient numerical schemes are proposed based on using a pair of bases for the gradient of thesolution and the Lagrange multiplier space forming biorthogonal and quasi-biorthogonal systems respectively We also establishan optimal a priori error estimate for both finite element approximations

1 Introduction

In many practical situations it is important to compute dualvariables of partial differential equationsmore accurately Forexample the gradient of the solution is the dual variable incase of the Poisson equation whereas the stress or pressurevariable is the dual variable in case of elasticity equationWorking with the standard finite element approach thesevariables should be obtained a posteriori by differentiationwhich will result in a loss of accuracy In these situationsa mixed method is often preferred as these variables can bedirectly computed using a mixed method

In this paper we introduce two mixed finite elementmethods for the Poisson equation using biorthogonal orquasi-biorthogonal systems Both formulations are obtainedby introducing the gradient of the solution of Poisson equa-tion as a new unknown and writing an additional variationalequation in terms of a Lagrange multiplier This gives rise totwo additional vector unknowns the gradient of the solutionand the Lagrange multiplier In order to obtain an efficientnumerical scheme we carefully choose a pair of bases forthe space of the gradient of the solution and the Lagrangemultiplier space in the discrete setting Choosing the pairof bases forming a biorthogonal or quasi-biorthogonal sys-tem for these two spaces we can eliminate the degrees offreedom associated with the gradient of the solution and

the Lagrange multiplier and arrive at a positive definiteformulation The positive definite formulation involves onlythe degrees of freedom associated with the solution of thePoisson equation Hence a reduced system is obtained whichis easy to solve The first formulation is discretized by usinga quasi-biorthogonal system whereas the second one whichis a stabilized version of the first one is discretized using abiorthogonal system

There are many mixed finite element methods for thePoisson equation [1ndash8] However all of them are based onthe two-field formulation of the Poisson equation and henceare not amenable to the application of the biorthogonaland quasi-biorthogonal systems We need a three-field for-mulation to apply the biorthogonal and quasi-biorthogonalsystems which leads to a symmetric formulation (see [9]for a three-field formulation in linear elasticity) The useof biorthogonal and quasi-biorthogonal systems allows aneasy static condensation of the auxiliary variables (gradi-ent of the solution and Lagrange multiplier) leading to areduced linear system These variables can be recoveredjust by inverting a diagonal matrix Therefore in thispaper we present three-field formulations of the Poissonequation to apply the biorthogonal and quasi-biorthogonalsystems

The structure of the rest of the paper is as follows InSection 2 we introduce our three-field formulations and

2 Advances in Numerical Analysis

show their well-posedness We propose finite element meth-ods for both formulations and prove a priori error estimatesin Section 3 Finally a short conclusion is drawn in Section 4

2 Two Three-Field Formulationsof the Poisson Equation

In this section we introduce two three-field mixed formu-lations of the Poisson problem Let Ω sub R119889 119889 isin 2 3be a bounded convex domain with polygonal or polyhedralboundary 120597Ω with the outward pointing normal n on 120597ΩThere are many mixed formulations of the Poisson equation

minusΔ119906 = 119891 in Ω (1)

with Dirichlet boundary condition

119906 = 0 on 120597Ω (2)

Neumann boundary condition

120597119906

120597n= 0 on 120597Ω (3)

or a mixture of these two (see [2ndash6 8 10])We start with the followingminimization problem for the

Poisson problem

119869 (119906) = infVisin11986710(Ω)

119869 (V) (4)

with

119869 (V) =1

2intΩ

|nablaV|2119889119909 minus ℓ (V) (5)

where for simplicity we have assumed the Dirichlet bound-ary condition 119891 isin 119871

2(Ω) and

ℓ (V) = intΩ

119891V 119889119909 (6)

Let 119881 = 1198671

0(Ω) and 119877 = [119871

2(Ω)]119889 and for two vector-valued

functions 120572 Ω rarr R119889 and 120573 Ω rarr R119889 the Sobolevinner product on the Sobolev space [119867119896(Ω)]119889 is defined as

⟨120572120573⟩119896Ω

=

119889

sum

119894=1

⟨120572119894 120573119894⟩119896Ω (7)

where (120572)119894 = 120572119894 and (120573)119894 = 120573119894 with 120572119894 120573119894 isin 119867119896(Ω) for

119894 = 1 119889 and the norm sdot 119896Ω is induced from this innerproduct where 119896 isin N cup 0 Note that ⟨120572119894 120573119894⟩119896Ω is thestandard inner product on the Sobolev space119867119896(Ω)

Our new formulation is obtained by introducing an aux-iliary variable 120590 = nabla119906 such that the minimization problem(4) is rewritten as the following constrained minimizationproblem

arg min(119906120590)isin[119881times119877]

120590=nabla119906

(1

21205902

0Ωminus ℓ (119906)) (8)

where the natural norm for the product space 119881 times 119877 isradic1199062

1Ω+ 1205902

0Ωfor (119906120590) isin 119881 times 119877 We write a weak

variational equation for equation 120590 = nabla119906 in terms ofthe Lagrange multiplier space 119877 to obtain the saddle-pointproblem of the minimization problem (8) The saddle-pointformulation is to find (119906120590120601) isin 119881 times 119877 times 119877 such that

119886 ((119906120590) (V 120591)) + 119887 ((V 120591) 120601) = ℓ (V) (V 120591) isin 119881 times 119877

119887 ((119906120590) 120595) = 0 120595 isin 119877

(9)

where

119886 ((119906120590) (V 120591)) = intΩ

120590 sdot 120591 119889119909

119887 ((119906120590) 120595) = intΩ

(120590 minus nabla119906) sdot 120595 119889119909

(10)

A similar three-field formulationmdashcalled the Hu-Washizuformulationmdashis very popular in designing finite elementmethods to alleviate locking effect in linear elasticity [11ndash13]

In order to show that the saddle-point problem (9) hasa unique solution we want to apply a standard saddle-point theory [3 4 10] To this end we need to show thefollowing three conditions of well-posedness

(1) The linear form ℓ(sdot) the bilinear forms 119886(sdot sdot) and119887(sdot sdot) are continuous on the spaces on which they aredefined

(2) Thebilinear form 119886(sdot sdot) is coercive on the kernel space119870 defined as

119870 = (V 120591) isin 119881 times 119877 119887 ((V 120591) 120595) = 0 120595 isin 119877 (11)

(3) Thebilinear form 119887(sdot sdot) satisfies the inf-sup condition

sup(V120591)isin119881times119877

119887 ((V 120591) 120595)

radicV21Ω

+ 1205912

0Ω

ge 120573100381710038171003817100381712059510038171003817100381710038170Ω

120595 isin 119877 (12)

for a constant 120573 gt 0The linear form ℓ(sdot) and the two bilinear forms 119886(sdot sdot) and

119887(sdot sdot) are continuous by the Cauchy-Schwarz inequality Thecoercivity of the bilinear form 119886(sdot sdot) on the kernel space 119870follows from Poincare inequality

119886 ((119906120590) (119906120590)) = 1205902

0Ωge1

2(1205902

0Ω+ 1199062

1Ω) (13)

It remains to show that the bilinear form 119887(sdot sdot) satisfies theinf-sup condition

Lemma 1 The bilinear form 119887(sdot sdot) satisfies the inf-sup condi-tion that is

sup(V120591)isin119881times119877

119887 ((V 120591) 120601)

radic |V 21Ω

+ 1205912

0Ω

ge100381710038171003817100381712060110038171003817100381710038170Ω

120601 isin 119877 (14)

Advances in Numerical Analysis 3

Proof Using V = 0 in the expression on the left above we have

sup(V120591)isin119881times119877

119887 ((V 120591) 120601)

radic |V 21Ω

+ 1205912

0Ω

ge sup120591isin119877

intΩ120591 sdot 120601 119889119909

1205912

0Ω

=100381710038171003817100381712060110038171003817100381710038170Ω

(15)

To summarize we have proved the following theorem

Theorem 2 The saddle-point problem (9) has a unique solu-tion (119906120590120601) isin 119881 times 119877 times 119877 and

1199061Ω + 1205900Ω +100381710038171003817100381712060110038171003817100381710038170Ω

le ℓ0Ω (16)

Themain difficulty in this formulation is that the bilinearform 119886(sdot sdot) is not coercive on the whole space 119881 times 119877 Itis only coercive on the subspace 119870 of 119881 times 119877 One has toconsider this difficulty in developing a finite element methodfor this problem One approach to make the bilinear form119886(sdot sdot) coercive on the whole space 119881 times 119877 is to stabilize itThere are many approaches to stabilize this formulation (see[2 5 6]) Here we modify the bilinear form 119886(sdot sdot) to get aconsistent stabilization as proposed in [14] for the Mindlin-Reissner plate so that it is now coercive on the whole space119881 times 119877 This is obtained by adding the term

intΩ

(120590 minus nabla119906) sdot (120591 minus nablaV) 119889119909 (17)

to the bilinear form 119886(sdot sdot) Thus our problem is to find(119906120590120601) isin 119881 times 119877 times 119877 such that

119860 ((119906120590) (V 120591)) + 119861 ((V 120591) 120601) = ℓ (V) (V 120591) isin 119881 times 119877

119861 ((119906120590) 120595) = 0 120595 isin 119877

(18)

where

119860 ((119906120590) (V 120591)) = intΩ

120590120591 119889119909

+ intΩ

(120590 minus nabla119906) sdot (120591 minus nablaV) 119889119909

119861 ((119906120590) 120595) = intΩ

(120590 minus nabla119906) sdot 120595 119889119909

(19)

Theorem 3 The saddle-point problems (9) and (18) yield thesame solution

Proof By construction the solution to (18) is also the solutionto (9)Moreover as the second equation of (18) yields120590 = nabla119906we can substitute this into the first equation and set V = 0 toget 120601 = minusnabla119906 This proves that the solution to (18) is also thesolution to (9)

Here the bilinear form 119860(sdot sdot) is coercive on the wholespace 119881 times 119877 from the triangle and Poincare inequalities

Lemma 4 The bilinear form 119860(sdot sdot) is coercive on 119881 times 119877 Thatis

119860 ((119906120590) (119906120590)) ge 119862 (1199062

1Ω+ 1205902

0Ω) (119906120590) isin 119881 times 119877

(20)

Proof We start with the triangle inequality

nabla1199062

0Ωle nabla119906 minus 120590

2

0Ω+ 1205902

0Ω (21)

From Poincare inequality there exists a constant 119862 gt 0 suchthat

1198621199062

1Ωle nabla119906

2

0Ω(22)

and hence we have

1198621199062

1Ωle (nabla119906 minus 120590

2

0Ω+ 1205902

0Ω) = 119860 ((119906120590) (119906120590))

(23)

Moreover the other conditions of well-posedness for thisnew saddle-point formulation (18) can also be shown exactlyas for (9)

In the following we present finite element approxima-tions for both proposed three-field formulations In the firststep we propose a finite element method for the nonstabi-lized formulation (9) which is based on quasi-biorthogonalsystems In the second step we present a finite elementmethod for the stabilized formulation (18) which is basedon biorthogonal systems The first method works only forsimplicial elements whereas the second method works formeshes of parallelotopes and simplices

3 Finite Element Approximationand a Priori Error Estimate

Let Tℎ be a quasi-uniform partition of the domain Ω insimplices or parallelotopes having the mesh size ℎ Let bea reference simplex or 119889-cube where the reference simplex isdefined as

= x isin R119889 119909119894 gt 0 119894 = 1 119889

119889

sum

119894=1

119909119894 lt 1 (24)

and a 119889-cube = (minus1 1)119889 Note that for 119889 = 2 a simplex is a

triangle and for 119889 = 2 it is a tetrahedron Similarly a 2-cubeis a square whereas 3-cube is a cube

The finite element space is defined by affinemaps 119865119879 froma reference element to a physical element 119879 isin Tℎ LetQ()be the space of bilineartrilinear polynomials in if is areference squarecube or the space of linear polynomials in if is a reference simplex Then the finite element spacebased on the mesh Tℎ is defined as the space of continuousfunctions whose restrictions to an element 119879 are obtainedby maps of bilinear trilinear or linear functions from thereference element that is

119878ℎ = Vℎ isin 1198671(Ω) Vℎ|119879

= Vℎ ∘ 119865minus1

119879 Vℎ isin Q () 119879 isin Tℎ

119881ℎ = 119878ℎ cap 1198671

0(Ω)

(25)

4 Advances in Numerical Analysis

(See [3 10 15]) We start with some abstract assumptions fortwo discrete spaces [119871ℎ]

119889sub 119877 and [119872ℎ]

119889sub 119877 to discretize the

gradient of the solution and the Lagrange multiplier spaceswhere 119871ℎ and119872ℎ both are piecewise polynomial spaces withrespect to themeshTℎ and are both subsets of119871

2(Ω) Explicit

construction of local basis functions for these spaces for themixed finite elementmethod (9) is given in Section 31 and forthe mixed finite element method (18) is given in Section 32

We assume that 119871ℎ and119872ℎ satisfy the following assump-tions (see also [16 17])

Assumption 1 (i) dim119872ℎ = dim 119871ℎ(ii)There is a constant 120573 gt 0 independent of themeshTℎ

such that

1003817100381710038171003817120601ℎ10038171003817100381710038171198712(Ω)

le 120573 sup120583ℎisin119872ℎ0

intΩ120583ℎ120601ℎ119889119909

1003817100381710038171003817120583ℎ10038171003817100381710038171198712(Ω)

120601ℎ isin 119871ℎ (26)

(iii) The space119872ℎ has the approximation property

inf120582ℎisin119872ℎ

1003817100381710038171003817120601 minus 120582ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (27)

(iv) The space 119871ℎ has the approximation property

inf120601ℎisin119871ℎ

1003817100381710038171003817120601 minus 120601ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (28)

Now we define biorthogonality and quasi-biorthogonali-ty (see [16 18 19])

Definition 5 The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called biorthogonal if the resulting Gram matrix Gis diagonal The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called quasi-biorthogonal and the resulting Grammatrix G is called quasidiagonal if G is of the form

G = [D1 0

R D2] or G = [

D1 R0 D2

] (29)

where D1 and D2 are diagonal matrices and R is a sparserectangular matrix

We recall that a Gram matrix G of two sets of basisfunctions 1205831198941le119894le119899 and 1205931198941le119894le119899 is the matrix G whose 119894119895thentry is

intΩ

120583119894120593119895119889119909 (30)

It is worth noting that the quasi-diagonalmatrix is inverted byinverting two diagonal matrices and scaling the rectangularmatrix

31 Finite Element Approximation for the Nonstabilized For-mulation (9) We now turn our attention to a finite elementapproximation for the nonstabilized formulation (9) Thisapproximation works only for simplicial meshes and is basedon a quasi-biorthogonal system as in [19] Let

119861ℎ = 119887ℎ | 119887ℎ|119879

= 119862

119889+1

prod

119894=1

120582119879

119894 119879 isin Tℎ (31)

be the space of bubble functions where 119862 is a constant and120582119879

119894119889+1

119894=1is the set of barycentric coordinates on 119879 Let 119871ℎ =

119878ℎoplus119861ℎ Explicit construction of basis functions of 119871ℎ and119872ℎon a reference element is provided below

The finite element problem is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times

[119871ℎ]119889times [119872ℎ]

119889 such that

119886 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119887 ((Vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(32)

We note that the standard finite element space 119878ℎ is enrichedwith elementwise defined bubble functions to obtain thespace 119871ℎ which is done to obtain the coercivity of the bilinearform 119886(sdot sdot) on the kernel space 119870ℎ defined as (see Lemma 7)

119870ℎ = (Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((Vℎ 120591ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(33)

To simplify the numerical analysis of the finite elementproblem we introduce a quasiprojection operator 119876ℎ

1198712(Ω) rarr 119871ℎ which is defined as 120595

ℎ

intΩ

119876ℎV120583ℎ119889119909 = intΩ

V120583ℎ119889119909 V isin 1198712(Ω) 120583ℎ isin 119872ℎ (34)

This type of operator was introduced in [9] to obtain the finiteelement interpolation of nonsmooth functions satisfyingboundary conditions and is used in [20] in the context ofmortar finite elementsThe definition of119876ℎ allows us to writethe weak gradient as

120590ℎ = 119876ℎ (nabla119906ℎ) (35)

where operator 119876ℎ is applied to the vector nabla119906ℎ component-wise We see that 119876ℎ is well defined due to Assumption 1(i)and (ii) Furthermore the restriction of 119876ℎ to 119871ℎ is theidentity Hence 119876ℎ is a projection onto the space 119871ℎ Wenote that 119876ℎ is not the orthogonal projection onto 119871ℎ butan oblique projection onto 119871ℎ see [21 22] The stability andapproximation properties of 119876ℎ in 119871

2 and 1198671-norm can be

shown as in [16 23] In the following we will use a genericconstant 119862 which will take different values at different placesbut will be always independent of the mesh size ℎ

Lemma 6 Under Assumption 1(i)ndash(iii)

1003817100381710038171003817119876ℎV10038171003817100381710038171198712(Ω)

le 119862V1198712(Ω) 119891119900119903 V isin 1198712(Ω)

1003816100381610038161003816119876ℎ11990810038161003816100381610038161198671(Ω)

le 119862|119908|1198671(Ω) 119891119900119903 119908 isin 1198671(Ω)

(36)

and for 0 lt 119904 le 1 and V isin 1198671+119904(Ω)1003817100381710038171003817V minus 119876ℎV

10038171003817100381710038171198712(Ω) le 119862ℎ1+119904|V|119867119904+1(Ω)

1003817100381710038171003817V minus 119876ℎV10038171003817100381710038171198671(Ω)

le 119862ℎ119904|V|119867119904+1(Ω)

(37)

Advances in Numerical Analysis 5

We need to construct another finite element space119872ℎ sub1198712(Ω) that satisfies Assumption 1(i)ndash(iii) and also leads to an

efficient numerical scheme The main goal is to choose thebasis functions for119872ℎ and 119871ℎ so that the matrix D associatedwith the bilinear form int

Ω120590ℎ 120595ℎ119889119909 is easy to invert To this

end we consider the algebraic form of the weak equation forthe gradient of the solution which is given by

minusB + D = 0 (38)

where and are the solution vectors for 119906ℎ and 120590ℎ and D

is the Gram matrix between the bases of [119872ℎ]119889 and [119871ℎ]

119889Hence if the matrix D is diagonal or quasi-diagonal we caneasily eliminate the degrees of freedom corresponding to 120590ℎand 120601

ℎ This then leads to a formulation involving only one

unknown 119906ℎ Statically condensing out variables 120590ℎ and 120601ℎwe arrive at the variational formulation of the reduced systemgiven by (68) (see Section 33) Note that the matrix D is theGram matrix between the bases of [119872ℎ]

119889 and [119871ℎ]119889

We cannot use a biorthogonal system in this case as thestability will be lost (see the Appendix) This motivates ourinterest in a quasi-biorthogonal system We now show theconstruction of the local basis functions for119871ℎ and119872ℎ so thatthey form a quasi-biorthogonal system The construction isshown on the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 in the two-dimensional case and on the referencetetrahedron = (119909 119910 119911) 0 le 119909 0 le 119910 0 le 119911 119909 +119910+ 119911 le 1

in the three-dimensional caseWe now start with the reference triangle in the two-

dimensional case Let 1205931 1205934 defined as

1205931 = 1 minus 119909 minus 119910 1205932 = 119909 1205933 = 119910

1205934 = 27119909119910 (1 minus 119909 minus 119910)

(39)

be the local basis functions of 119871ℎ on the reference triangleassociated with three vertices (0 0) and (1 0) (0 1) and onebarycenter (13 13) Note that 1205934 is the standard bubblefunction used for example in the minielement for the Stokesequations [24]

Then if we define the local basis functions 1205831 1205834 of119872ℎ as

1205831 =34

9minus 4119909 minus 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205832 = minus2

9+ 4119909 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205833 = minus2

9+ 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205834 = 27119909119910 (1 minus 119909 minus 119910)

(40)

the two sets of basis functions 1205931 1205934 and 1205831 1205834

form a quasi-biorthogonal system on the reference triangleThese four basis functions 1205831 1205834 of 119872ℎ are associatedwith three vertices (0 0) and (1 0) (0 1) and the barycenter(13 13) of the reference triangle

Similarly on the reference tetrahedron the local basisfunctions 1205931 1205935 of 119871ℎ defined as

1205931 = 1 minus 119909 minus 119910 minus 119911 1205932 = 119909 1205933 = 119910

1205934 = 119911 1205935 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(41)

and the local basis functions 1205831 1205835 of119872ℎ defined as

1205831 =401

92minus 5119909 minus 5119910 minus 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205832 = minus59

92+ 5119909 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205833 = minus59

92+ 5119910 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205834 = minus59

92+ 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205835 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(42)

are quasi-biorthogonalThese five basis functions 1205931 1205935of 119871ℎ or 1205831 1205835 of 119872ℎ are associated with four vertices(0 0 0) (1 0 0) (0 1 0) and (0 0 1) and the barycenter(14 14 and 14) of the reference tetrahedron Using thefinite element basis functions for 119871ℎ the finite element basisfunctions for119872ℎ are constructed in such a way that the Grammatrix on the reference element is given by

D=

[[[[[[[[[

[

1

60 0 0

01

60 0

0 01

60

3

40

3

40

3

40

81

560

]]]]]]]]]

]

(43)

for the two-dimensional case and

D=

[[[[[[[[[[[[[

[

1

240 0 0 0

01

240 0 0

0 01

240 0

0 0 01

240

4

315

4

315

4

315

4

315

4096

155925

]]]]]]]]]]]]]

]

(44)

for the three-dimensional case After ordering the degreesof freedom in 120590ℎ and 120601ℎ so that the degrees of freedomassociated with the barycenters of elements come after thedegrees of freedom associated with the vertices of elementsthe global Gram matrix D is quasi-diagonal

Thus in the two-dimensional case the local basis func-tions 1205831 1205832 and 1205833 are associated with the vertices of thereference triangle and the function 1205834 is associated withthe barycenter of the triangle and defined elementwise Thatmeans 1205834 is supported only on the reference element Thesituation is similar in the three-dimensional case Hence

6 Advances in Numerical Analysis

using a proper ordering the support of a global basis function120593119894 of119871ℎ is the same as that of the global basis function120583119894 of119872ℎfor 119894 = 1 119899 However the global basis functions 1205831 120583119899of119872ℎ are not continuous on interelement boundaries

Now we show that119872ℎ satisfies Assumption 1(i)ndash(iii) Asthe first assumption is satisfied by construction we considerthe second assumption Let 120601ℎ = sum

119899

119896=1119886119896120601119896 isin 119878ℎ and set 120583ℎ =

sum119899

119896=1119886119896120583119896 isin 119872ℎ By using the quasibiorthogonality relation

(1) and the quasiuniformity assumption we get

intΩ

120601ℎ120583ℎ119889x =119899

sum

119894119895=1

119886119894119886119895 intΩ

120601119894120583119895119889x

=

119899

sum

119894=1

1198862

119894119888119894 ge 119862

119899

sum

119894=1

1198862

119894ℎ119889

119894ge 119862

1003817100381710038171003817120601ℎ1003817100381710038171003817

2

0Ω

(45)

where ℎ119894 denotes the meshsize at 119894th vertex Taking intoaccount the fact that 120601ℎ

2

0Ωequiv 120583ℎ

2

0Ωequiv sum119899

119894=11198862

119894ℎ119889

119894 we

find that Assumption 1(ii) is satisfied Since the sum of thelocal basis functions of 119872ℎ is one Assumption 1(iii) can beproved as in [23 25] Assumption 1(iv) follows by standardarguments

There are (119889 + 2) local basis functions of 119871ℎ where (119889 +1) of them are associated with vertices and one is associatedwith the element barycenter Then the local basis functionsof119872ℎ are also divided into two parts (119889 + 2) basis functionsassociated with the vertices and 1 basis function associatedwith the element Let 120583119879 be the local basis function associatedwith the element barycenter of 119879 isin Tℎ for119872ℎ Let 119884ℎ be thespace of piecewise constant functions with respect toTℎ and119875ℎ 119871

2(Ω) rarr 119884ℎ is defined as

119875ℎ119908|119879 =

int119879119908120583119879119889119909

int119879120583119879119889119909

(46)

Then 119875ℎ is a projection operator onto 119884ℎ Moreover

nablaVℎ = 119875ℎ119876ℎnablaVℎ (47)

and this implies that1003817100381710038171003817nablaVℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnablaVℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnablaVℎ

10038171003817100381710038170Ω (48)

This yields the following coercivity result

Lemma 7 The bilinear form 119886(sdot sdot) is coercive on the kernelspace 119870ℎ In fact

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ)) ge1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119906ℎ

1003817100381710038171003817

2

0Ω)

(119906ℎ120590ℎ) isin 119870ℎ

(49)

Proof Since 120590ℎ = 119876ℎnabla119906ℎ on the kernel space 119870ℎ we have

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ))

=1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817119876ℎnabla119906ℎ

1003817100381710038171003817

2

0Ω)

(50)

The proof follows from the result1003817100381710038171003817nabla119906ℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnabla119906ℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnabla119906ℎ

10038171003817100381710038170Ω (51)

Remark 8 Using a quadrilateral or hexahedral meshes wecannot have nablaVℎ = 119875ℎ119876ℎnablaVℎ and hence the coercivity failsOne needs to modify the finite element method for thesemeshes

The inf-sup condition follows exactly as in the continuoussetting using Assumption 1(ii)

Lemma 9 There exists a constant 120573 gt 0 such that

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge 1205731003817100381710038171003817120601ℎ

10038171003817100381710038170Ω 120601ℎisin 119877ℎ

(52)

Proof The proof follows exactly as in the continuous settingWe use Vℎ = 0 in the expression

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

(53)

to get

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge sup120591ℎisin119877ℎ

intΩ120591ℎ sdot 120601ℎ119889119909

1003817100381710038171003817120591ℎ1003817100381710038171003817

2

0Ω

(54)

Assumption 1(ii) then yields the final result

Hence we have the following stability and approximationresult

Theorem 10 The discrete saddle-point problem (32) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and

1003817100381710038171003817119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (55)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (32)

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(56)

Owing to Assumption 1(iii)-(iv) the discrete solutionconverges optimally to the continuous solution

32 Finite Element Approximation for the Stabilized Formula-tion (18) After stabilizing formulation (9) the bilinear form119860(sdot sdot) (18) is coercive on the whole product space 119881 times 119877Therefore we can use the standard finite element space 119878ℎto discretize the gradient of the solution for the saddle-pointproblem (18) so that 119871ℎ = 119878ℎ Thus our discrete saddle-point

Advances in Numerical Analysis 7

formulation of (18) is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎtimes[119871ℎ]119889times[119872ℎ]

119889

so that

119860 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119861 ((vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119861 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(57)

We can now verify the three conditions of well-posedness forthe discrete saddle point problem The continuity of 119860(sdot sdot)119861(sdot sdot) and ℓ(sdot) follows as in the continuous setting andcoercivity is immediate from Lemma 4 as119881ℎ times[119871ℎ]

119889sub 119881times119877

Explicit construction of basis functions of119872ℎ on a referenceelement will be given below

Assumption 1(ii) guarantees that the bilinear form 119887(sdot sdot)

satisfies the inf-sup condition as in the case of the previousfinite element method

We have thus proved the following theorem from thestandard theory of saddle-point problem (see [4])

Theorem 11 The discrete saddle-point problem (57) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and1003817100381710038171003817119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (58)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (9)1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(59)

The optimal convergence of the discrete solution is thenguaranteed by the standard approximation property of119881ℎ and119871ℎ and Assumption 1(iii)

In the following we give these basis functions for lin-ear simplicial finite elements in two and three dimensions[18] The parallelotope case is handled by using the tensorproduct construction of the one-dimensional basis functionspresented in [20 26] These basis functions are very useful inthe mortar finite element method and extensively studied inone- and two-dimensional cases [16 23 26]

Note that we have imposed the condition dim119872ℎ =

dim 119871ℎ to get that our Grammatrix is a square matrixThere-fore the local basis functions for linearbilineartrilinearfinite spaces are associated with the vertices of trianglestetrahedra quadrilaterals and hexahedra For example forthe reference triangle = (119909 119910) 0 lt 119909 0 lt 119910 119909 + 119910 lt 1we have

1205831 = 3 minus 4119909 minus 4119910 1205832 = 4119909 minus 1

1205833 = 4119910 minus 1

(60)

where the basis functions 1205831 1205832 and 1205833 are associated withthree vertices (0 0) (1 0) and (0 1) of the reference triangle

For the reference tetrahedron = (119909 119910 119911) 0 lt 119909 0 lt

119910 0 lt 119911 119909 + 119910 + 119911 lt 1 we have

1205831 = 4 minus 5119909 minus 5119910 minus 5119911 1205832 = 5119909 minus 1

1205833 = 5119910 minus 1 1205834 = 5119911 minus 1

(61)

where the basis functions 1205831 1205832 1205833 and 1205834 associated withfour vertices (0 0 0) (1 0 0) (0 1 0) and (0 0 1) of thereference tetrahedron (see also [17 18]) Note that the basisfunctions are constructed in such a way that the biorthogo-nality condition is satisfied on the reference element

The global basis functions for the test space are con-structed by gluing the local basis functions together followingexactly the same procedure of constructing global finiteelement basis functions from the local ones These globalbasis functions then satisfy the condition of biorthogonalitywith global finite element basis functions of 119878ℎ These basisfunctions also satisfy Assumption 1(i)ndash(iv) (see [17 18])

33 Static Condensation Here we want to eliminate 120590ℎ andLagrange multiplier 120601

ℎfrom the saddle-point problem (57)

We make use of the operator 119876ℎ defined previously Usingthe biorthogonality relation between the basis functions of 119871ℎand 119872ℎ the action of operator 119876ℎ on a function V isin 119871

2(Ω)

can be written as

119876ℎV =119899

sum

119894=1

intΩ120583119894V 119889119909

119888119894

120593119894 (62)

which tells that the operator119876ℎ is local in the sense to be givenbelow (see also [27])

Using the property of operator 119876ℎ we can eliminate thedegrees of freedom corresponding to 120590ℎ and 120601ℎ so that ourproblem is to find 119906ℎ isin 119878ℎ such that

119869 (119906ℎ) = minVℎisin119881ℎ

119869 (Vℎ) (63)

where

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ) (64)

for the nonstabilized formulation (9) and

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ω

+1003817100381710038171003817119876ℎ (nablaVℎ) minus nablaVℎ

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ)

(65)

for the stabilized formulation (18)Let the bilinear form 119860(sdot sdot) be defined as

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909 (66)

for the non-stabilized formulation (9) and

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909

+ intΩ

(120590ℎ minus nabla119906ℎ) sdot (120591ℎ minus nablaVℎ) 119889119909

(67)

8 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Numerical Analysis

show their well-posedness We propose finite element meth-ods for both formulations and prove a priori error estimatesin Section 3 Finally a short conclusion is drawn in Section 4

2 Two Three-Field Formulationsof the Poisson Equation

In this section we introduce two three-field mixed formu-lations of the Poisson problem Let Ω sub R119889 119889 isin 2 3be a bounded convex domain with polygonal or polyhedralboundary 120597Ω with the outward pointing normal n on 120597ΩThere are many mixed formulations of the Poisson equation

minusΔ119906 = 119891 in Ω (1)

with Dirichlet boundary condition

119906 = 0 on 120597Ω (2)

Neumann boundary condition

120597119906

120597n= 0 on 120597Ω (3)

or a mixture of these two (see [2ndash6 8 10])We start with the followingminimization problem for the

Poisson problem

119869 (119906) = infVisin11986710(Ω)

119869 (V) (4)

with

119869 (V) =1

2intΩ

|nablaV|2119889119909 minus ℓ (V) (5)

where for simplicity we have assumed the Dirichlet bound-ary condition 119891 isin 119871

2(Ω) and

ℓ (V) = intΩ

119891V 119889119909 (6)

Let 119881 = 1198671

0(Ω) and 119877 = [119871

2(Ω)]119889 and for two vector-valued

functions 120572 Ω rarr R119889 and 120573 Ω rarr R119889 the Sobolevinner product on the Sobolev space [119867119896(Ω)]119889 is defined as

⟨120572120573⟩119896Ω

=

119889

sum

119894=1

⟨120572119894 120573119894⟩119896Ω (7)

where (120572)119894 = 120572119894 and (120573)119894 = 120573119894 with 120572119894 120573119894 isin 119867119896(Ω) for

119894 = 1 119889 and the norm sdot 119896Ω is induced from this innerproduct where 119896 isin N cup 0 Note that ⟨120572119894 120573119894⟩119896Ω is thestandard inner product on the Sobolev space119867119896(Ω)

Our new formulation is obtained by introducing an aux-iliary variable 120590 = nabla119906 such that the minimization problem(4) is rewritten as the following constrained minimizationproblem

arg min(119906120590)isin[119881times119877]

120590=nabla119906

(1

21205902

0Ωminus ℓ (119906)) (8)

where the natural norm for the product space 119881 times 119877 isradic1199062

1Ω+ 1205902

0Ωfor (119906120590) isin 119881 times 119877 We write a weak

variational equation for equation 120590 = nabla119906 in terms ofthe Lagrange multiplier space 119877 to obtain the saddle-pointproblem of the minimization problem (8) The saddle-pointformulation is to find (119906120590120601) isin 119881 times 119877 times 119877 such that

119886 ((119906120590) (V 120591)) + 119887 ((V 120591) 120601) = ℓ (V) (V 120591) isin 119881 times 119877

119887 ((119906120590) 120595) = 0 120595 isin 119877

(9)

where

119886 ((119906120590) (V 120591)) = intΩ

120590 sdot 120591 119889119909

119887 ((119906120590) 120595) = intΩ

(120590 minus nabla119906) sdot 120595 119889119909

(10)

A similar three-field formulationmdashcalled the Hu-Washizuformulationmdashis very popular in designing finite elementmethods to alleviate locking effect in linear elasticity [11ndash13]

In order to show that the saddle-point problem (9) hasa unique solution we want to apply a standard saddle-point theory [3 4 10] To this end we need to show thefollowing three conditions of well-posedness

(1) The linear form ℓ(sdot) the bilinear forms 119886(sdot sdot) and119887(sdot sdot) are continuous on the spaces on which they aredefined

(2) Thebilinear form 119886(sdot sdot) is coercive on the kernel space119870 defined as

119870 = (V 120591) isin 119881 times 119877 119887 ((V 120591) 120595) = 0 120595 isin 119877 (11)

(3) Thebilinear form 119887(sdot sdot) satisfies the inf-sup condition

sup(V120591)isin119881times119877

119887 ((V 120591) 120595)

radicV21Ω

+ 1205912

0Ω

ge 120573100381710038171003817100381712059510038171003817100381710038170Ω

120595 isin 119877 (12)

for a constant 120573 gt 0The linear form ℓ(sdot) and the two bilinear forms 119886(sdot sdot) and

119887(sdot sdot) are continuous by the Cauchy-Schwarz inequality Thecoercivity of the bilinear form 119886(sdot sdot) on the kernel space 119870follows from Poincare inequality

119886 ((119906120590) (119906120590)) = 1205902

0Ωge1

2(1205902

0Ω+ 1199062

1Ω) (13)

It remains to show that the bilinear form 119887(sdot sdot) satisfies theinf-sup condition

Lemma 1 The bilinear form 119887(sdot sdot) satisfies the inf-sup condi-tion that is

sup(V120591)isin119881times119877

119887 ((V 120591) 120601)

radic |V 21Ω

+ 1205912

0Ω

ge100381710038171003817100381712060110038171003817100381710038170Ω

120601 isin 119877 (14)

Advances in Numerical Analysis 3

Proof Using V = 0 in the expression on the left above we have

sup(V120591)isin119881times119877

119887 ((V 120591) 120601)

radic |V 21Ω

+ 1205912

0Ω

ge sup120591isin119877

intΩ120591 sdot 120601 119889119909

1205912

0Ω

=100381710038171003817100381712060110038171003817100381710038170Ω

(15)

To summarize we have proved the following theorem

Theorem 2 The saddle-point problem (9) has a unique solu-tion (119906120590120601) isin 119881 times 119877 times 119877 and

1199061Ω + 1205900Ω +100381710038171003817100381712060110038171003817100381710038170Ω

le ℓ0Ω (16)

Themain difficulty in this formulation is that the bilinearform 119886(sdot sdot) is not coercive on the whole space 119881 times 119877 Itis only coercive on the subspace 119870 of 119881 times 119877 One has toconsider this difficulty in developing a finite element methodfor this problem One approach to make the bilinear form119886(sdot sdot) coercive on the whole space 119881 times 119877 is to stabilize itThere are many approaches to stabilize this formulation (see[2 5 6]) Here we modify the bilinear form 119886(sdot sdot) to get aconsistent stabilization as proposed in [14] for the Mindlin-Reissner plate so that it is now coercive on the whole space119881 times 119877 This is obtained by adding the term

intΩ

(120590 minus nabla119906) sdot (120591 minus nablaV) 119889119909 (17)

to the bilinear form 119886(sdot sdot) Thus our problem is to find(119906120590120601) isin 119881 times 119877 times 119877 such that

119860 ((119906120590) (V 120591)) + 119861 ((V 120591) 120601) = ℓ (V) (V 120591) isin 119881 times 119877

119861 ((119906120590) 120595) = 0 120595 isin 119877

(18)

where

119860 ((119906120590) (V 120591)) = intΩ

120590120591 119889119909

+ intΩ

(120590 minus nabla119906) sdot (120591 minus nablaV) 119889119909

119861 ((119906120590) 120595) = intΩ

(120590 minus nabla119906) sdot 120595 119889119909

(19)

Theorem 3 The saddle-point problems (9) and (18) yield thesame solution

Proof By construction the solution to (18) is also the solutionto (9)Moreover as the second equation of (18) yields120590 = nabla119906we can substitute this into the first equation and set V = 0 toget 120601 = minusnabla119906 This proves that the solution to (18) is also thesolution to (9)

Here the bilinear form 119860(sdot sdot) is coercive on the wholespace 119881 times 119877 from the triangle and Poincare inequalities

Lemma 4 The bilinear form 119860(sdot sdot) is coercive on 119881 times 119877 Thatis

119860 ((119906120590) (119906120590)) ge 119862 (1199062

1Ω+ 1205902

0Ω) (119906120590) isin 119881 times 119877

(20)

Proof We start with the triangle inequality

nabla1199062

0Ωle nabla119906 minus 120590

2

0Ω+ 1205902

0Ω (21)

From Poincare inequality there exists a constant 119862 gt 0 suchthat

1198621199062

1Ωle nabla119906

2

0Ω(22)

and hence we have

1198621199062

1Ωle (nabla119906 minus 120590

2

0Ω+ 1205902

0Ω) = 119860 ((119906120590) (119906120590))

(23)

Moreover the other conditions of well-posedness for thisnew saddle-point formulation (18) can also be shown exactlyas for (9)

In the following we present finite element approxima-tions for both proposed three-field formulations In the firststep we propose a finite element method for the nonstabi-lized formulation (9) which is based on quasi-biorthogonalsystems In the second step we present a finite elementmethod for the stabilized formulation (18) which is basedon biorthogonal systems The first method works only forsimplicial elements whereas the second method works formeshes of parallelotopes and simplices

3 Finite Element Approximationand a Priori Error Estimate

Let Tℎ be a quasi-uniform partition of the domain Ω insimplices or parallelotopes having the mesh size ℎ Let bea reference simplex or 119889-cube where the reference simplex isdefined as

= x isin R119889 119909119894 gt 0 119894 = 1 119889

119889

sum

119894=1

119909119894 lt 1 (24)

and a 119889-cube = (minus1 1)119889 Note that for 119889 = 2 a simplex is a

triangle and for 119889 = 2 it is a tetrahedron Similarly a 2-cubeis a square whereas 3-cube is a cube

The finite element space is defined by affinemaps 119865119879 froma reference element to a physical element 119879 isin Tℎ LetQ()be the space of bilineartrilinear polynomials in if is areference squarecube or the space of linear polynomials in if is a reference simplex Then the finite element spacebased on the mesh Tℎ is defined as the space of continuousfunctions whose restrictions to an element 119879 are obtainedby maps of bilinear trilinear or linear functions from thereference element that is

119878ℎ = Vℎ isin 1198671(Ω) Vℎ|119879

= Vℎ ∘ 119865minus1

119879 Vℎ isin Q () 119879 isin Tℎ

119881ℎ = 119878ℎ cap 1198671

0(Ω)

(25)

4 Advances in Numerical Analysis

(See [3 10 15]) We start with some abstract assumptions fortwo discrete spaces [119871ℎ]

119889sub 119877 and [119872ℎ]

119889sub 119877 to discretize the

gradient of the solution and the Lagrange multiplier spaceswhere 119871ℎ and119872ℎ both are piecewise polynomial spaces withrespect to themeshTℎ and are both subsets of119871

2(Ω) Explicit

construction of local basis functions for these spaces for themixed finite elementmethod (9) is given in Section 31 and forthe mixed finite element method (18) is given in Section 32

We assume that 119871ℎ and119872ℎ satisfy the following assump-tions (see also [16 17])

Assumption 1 (i) dim119872ℎ = dim 119871ℎ(ii)There is a constant 120573 gt 0 independent of themeshTℎ

such that

1003817100381710038171003817120601ℎ10038171003817100381710038171198712(Ω)

le 120573 sup120583ℎisin119872ℎ0

intΩ120583ℎ120601ℎ119889119909

1003817100381710038171003817120583ℎ10038171003817100381710038171198712(Ω)

120601ℎ isin 119871ℎ (26)

(iii) The space119872ℎ has the approximation property

inf120582ℎisin119872ℎ

1003817100381710038171003817120601 minus 120582ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (27)

(iv) The space 119871ℎ has the approximation property

inf120601ℎisin119871ℎ

1003817100381710038171003817120601 minus 120601ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (28)

Now we define biorthogonality and quasi-biorthogonali-ty (see [16 18 19])

Definition 5 The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called biorthogonal if the resulting Gram matrix Gis diagonal The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called quasi-biorthogonal and the resulting Grammatrix G is called quasidiagonal if G is of the form

G = [D1 0

R D2] or G = [

D1 R0 D2

] (29)

where D1 and D2 are diagonal matrices and R is a sparserectangular matrix

We recall that a Gram matrix G of two sets of basisfunctions 1205831198941le119894le119899 and 1205931198941le119894le119899 is the matrix G whose 119894119895thentry is

intΩ

120583119894120593119895119889119909 (30)

It is worth noting that the quasi-diagonalmatrix is inverted byinverting two diagonal matrices and scaling the rectangularmatrix

31 Finite Element Approximation for the Nonstabilized For-mulation (9) We now turn our attention to a finite elementapproximation for the nonstabilized formulation (9) Thisapproximation works only for simplicial meshes and is basedon a quasi-biorthogonal system as in [19] Let

119861ℎ = 119887ℎ | 119887ℎ|119879

= 119862

119889+1

prod

119894=1

120582119879

119894 119879 isin Tℎ (31)

be the space of bubble functions where 119862 is a constant and120582119879

119894119889+1

119894=1is the set of barycentric coordinates on 119879 Let 119871ℎ =

119878ℎoplus119861ℎ Explicit construction of basis functions of 119871ℎ and119872ℎon a reference element is provided below

The finite element problem is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times

[119871ℎ]119889times [119872ℎ]

119889 such that

119886 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119887 ((Vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(32)

We note that the standard finite element space 119878ℎ is enrichedwith elementwise defined bubble functions to obtain thespace 119871ℎ which is done to obtain the coercivity of the bilinearform 119886(sdot sdot) on the kernel space 119870ℎ defined as (see Lemma 7)

119870ℎ = (Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((Vℎ 120591ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(33)

To simplify the numerical analysis of the finite elementproblem we introduce a quasiprojection operator 119876ℎ

1198712(Ω) rarr 119871ℎ which is defined as 120595

ℎ

intΩ

119876ℎV120583ℎ119889119909 = intΩ

V120583ℎ119889119909 V isin 1198712(Ω) 120583ℎ isin 119872ℎ (34)

This type of operator was introduced in [9] to obtain the finiteelement interpolation of nonsmooth functions satisfyingboundary conditions and is used in [20] in the context ofmortar finite elementsThe definition of119876ℎ allows us to writethe weak gradient as

120590ℎ = 119876ℎ (nabla119906ℎ) (35)

where operator 119876ℎ is applied to the vector nabla119906ℎ component-wise We see that 119876ℎ is well defined due to Assumption 1(i)and (ii) Furthermore the restriction of 119876ℎ to 119871ℎ is theidentity Hence 119876ℎ is a projection onto the space 119871ℎ Wenote that 119876ℎ is not the orthogonal projection onto 119871ℎ butan oblique projection onto 119871ℎ see [21 22] The stability andapproximation properties of 119876ℎ in 119871

2 and 1198671-norm can be

shown as in [16 23] In the following we will use a genericconstant 119862 which will take different values at different placesbut will be always independent of the mesh size ℎ

Lemma 6 Under Assumption 1(i)ndash(iii)

1003817100381710038171003817119876ℎV10038171003817100381710038171198712(Ω)

le 119862V1198712(Ω) 119891119900119903 V isin 1198712(Ω)

1003816100381610038161003816119876ℎ11990810038161003816100381610038161198671(Ω)

le 119862|119908|1198671(Ω) 119891119900119903 119908 isin 1198671(Ω)

(36)

and for 0 lt 119904 le 1 and V isin 1198671+119904(Ω)1003817100381710038171003817V minus 119876ℎV

10038171003817100381710038171198712(Ω) le 119862ℎ1+119904|V|119867119904+1(Ω)

1003817100381710038171003817V minus 119876ℎV10038171003817100381710038171198671(Ω)

le 119862ℎ119904|V|119867119904+1(Ω)

(37)

Advances in Numerical Analysis 5

We need to construct another finite element space119872ℎ sub1198712(Ω) that satisfies Assumption 1(i)ndash(iii) and also leads to an

efficient numerical scheme The main goal is to choose thebasis functions for119872ℎ and 119871ℎ so that the matrix D associatedwith the bilinear form int

Ω120590ℎ 120595ℎ119889119909 is easy to invert To this

end we consider the algebraic form of the weak equation forthe gradient of the solution which is given by

minusB + D = 0 (38)

where and are the solution vectors for 119906ℎ and 120590ℎ and D

is the Gram matrix between the bases of [119872ℎ]119889 and [119871ℎ]

119889Hence if the matrix D is diagonal or quasi-diagonal we caneasily eliminate the degrees of freedom corresponding to 120590ℎand 120601

ℎ This then leads to a formulation involving only one

unknown 119906ℎ Statically condensing out variables 120590ℎ and 120601ℎwe arrive at the variational formulation of the reduced systemgiven by (68) (see Section 33) Note that the matrix D is theGram matrix between the bases of [119872ℎ]

119889 and [119871ℎ]119889

We cannot use a biorthogonal system in this case as thestability will be lost (see the Appendix) This motivates ourinterest in a quasi-biorthogonal system We now show theconstruction of the local basis functions for119871ℎ and119872ℎ so thatthey form a quasi-biorthogonal system The construction isshown on the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 in the two-dimensional case and on the referencetetrahedron = (119909 119910 119911) 0 le 119909 0 le 119910 0 le 119911 119909 +119910+ 119911 le 1

in the three-dimensional caseWe now start with the reference triangle in the two-

dimensional case Let 1205931 1205934 defined as

1205931 = 1 minus 119909 minus 119910 1205932 = 119909 1205933 = 119910

1205934 = 27119909119910 (1 minus 119909 minus 119910)

(39)

be the local basis functions of 119871ℎ on the reference triangleassociated with three vertices (0 0) and (1 0) (0 1) and onebarycenter (13 13) Note that 1205934 is the standard bubblefunction used for example in the minielement for the Stokesequations [24]

Then if we define the local basis functions 1205831 1205834 of119872ℎ as

1205831 =34

9minus 4119909 minus 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205832 = minus2

9+ 4119909 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205833 = minus2

9+ 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205834 = 27119909119910 (1 minus 119909 minus 119910)

(40)

the two sets of basis functions 1205931 1205934 and 1205831 1205834

form a quasi-biorthogonal system on the reference triangleThese four basis functions 1205831 1205834 of 119872ℎ are associatedwith three vertices (0 0) and (1 0) (0 1) and the barycenter(13 13) of the reference triangle

Similarly on the reference tetrahedron the local basisfunctions 1205931 1205935 of 119871ℎ defined as

1205931 = 1 minus 119909 minus 119910 minus 119911 1205932 = 119909 1205933 = 119910

1205934 = 119911 1205935 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(41)

and the local basis functions 1205831 1205835 of119872ℎ defined as

1205831 =401

92minus 5119909 minus 5119910 minus 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205832 = minus59

92+ 5119909 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205833 = minus59

92+ 5119910 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205834 = minus59

92+ 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205835 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(42)

are quasi-biorthogonalThese five basis functions 1205931 1205935of 119871ℎ or 1205831 1205835 of 119872ℎ are associated with four vertices(0 0 0) (1 0 0) (0 1 0) and (0 0 1) and the barycenter(14 14 and 14) of the reference tetrahedron Using thefinite element basis functions for 119871ℎ the finite element basisfunctions for119872ℎ are constructed in such a way that the Grammatrix on the reference element is given by

D=

[[[[[[[[[

[

1

60 0 0

01

60 0

0 01

60

3

40

3

40

3

40

81

560

]]]]]]]]]

]

(43)

for the two-dimensional case and

D=

[[[[[[[[[[[[[

[

1

240 0 0 0

01

240 0 0

0 01

240 0

0 0 01

240

4

315

4

315

4

315

4

315

4096

155925

]]]]]]]]]]]]]

]

(44)

for the three-dimensional case After ordering the degreesof freedom in 120590ℎ and 120601ℎ so that the degrees of freedomassociated with the barycenters of elements come after thedegrees of freedom associated with the vertices of elementsthe global Gram matrix D is quasi-diagonal

Thus in the two-dimensional case the local basis func-tions 1205831 1205832 and 1205833 are associated with the vertices of thereference triangle and the function 1205834 is associated withthe barycenter of the triangle and defined elementwise Thatmeans 1205834 is supported only on the reference element Thesituation is similar in the three-dimensional case Hence

6 Advances in Numerical Analysis

using a proper ordering the support of a global basis function120593119894 of119871ℎ is the same as that of the global basis function120583119894 of119872ℎfor 119894 = 1 119899 However the global basis functions 1205831 120583119899of119872ℎ are not continuous on interelement boundaries

Now we show that119872ℎ satisfies Assumption 1(i)ndash(iii) Asthe first assumption is satisfied by construction we considerthe second assumption Let 120601ℎ = sum

119899

119896=1119886119896120601119896 isin 119878ℎ and set 120583ℎ =

sum119899

119896=1119886119896120583119896 isin 119872ℎ By using the quasibiorthogonality relation

(1) and the quasiuniformity assumption we get

intΩ

120601ℎ120583ℎ119889x =119899

sum

119894119895=1

119886119894119886119895 intΩ

120601119894120583119895119889x

=

119899

sum

119894=1

1198862

119894119888119894 ge 119862

119899

sum

119894=1

1198862

119894ℎ119889

119894ge 119862

1003817100381710038171003817120601ℎ1003817100381710038171003817

2

0Ω

(45)

where ℎ119894 denotes the meshsize at 119894th vertex Taking intoaccount the fact that 120601ℎ

2

0Ωequiv 120583ℎ

2

0Ωequiv sum119899

119894=11198862

119894ℎ119889

119894 we

find that Assumption 1(ii) is satisfied Since the sum of thelocal basis functions of 119872ℎ is one Assumption 1(iii) can beproved as in [23 25] Assumption 1(iv) follows by standardarguments

There are (119889 + 2) local basis functions of 119871ℎ where (119889 +1) of them are associated with vertices and one is associatedwith the element barycenter Then the local basis functionsof119872ℎ are also divided into two parts (119889 + 2) basis functionsassociated with the vertices and 1 basis function associatedwith the element Let 120583119879 be the local basis function associatedwith the element barycenter of 119879 isin Tℎ for119872ℎ Let 119884ℎ be thespace of piecewise constant functions with respect toTℎ and119875ℎ 119871

2(Ω) rarr 119884ℎ is defined as

119875ℎ119908|119879 =

int119879119908120583119879119889119909

int119879120583119879119889119909

(46)

Then 119875ℎ is a projection operator onto 119884ℎ Moreover

nablaVℎ = 119875ℎ119876ℎnablaVℎ (47)

and this implies that1003817100381710038171003817nablaVℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnablaVℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnablaVℎ

10038171003817100381710038170Ω (48)

This yields the following coercivity result

Lemma 7 The bilinear form 119886(sdot sdot) is coercive on the kernelspace 119870ℎ In fact

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ)) ge1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119906ℎ

1003817100381710038171003817

2

0Ω)

(119906ℎ120590ℎ) isin 119870ℎ

(49)

Proof Since 120590ℎ = 119876ℎnabla119906ℎ on the kernel space 119870ℎ we have

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ))

=1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817119876ℎnabla119906ℎ

1003817100381710038171003817

2

0Ω)

(50)

The proof follows from the result1003817100381710038171003817nabla119906ℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnabla119906ℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnabla119906ℎ

10038171003817100381710038170Ω (51)

Remark 8 Using a quadrilateral or hexahedral meshes wecannot have nablaVℎ = 119875ℎ119876ℎnablaVℎ and hence the coercivity failsOne needs to modify the finite element method for thesemeshes

The inf-sup condition follows exactly as in the continuoussetting using Assumption 1(ii)

Lemma 9 There exists a constant 120573 gt 0 such that

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge 1205731003817100381710038171003817120601ℎ

10038171003817100381710038170Ω 120601ℎisin 119877ℎ

(52)

Proof The proof follows exactly as in the continuous settingWe use Vℎ = 0 in the expression

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

(53)

to get

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge sup120591ℎisin119877ℎ

intΩ120591ℎ sdot 120601ℎ119889119909

1003817100381710038171003817120591ℎ1003817100381710038171003817

2

0Ω

(54)

Assumption 1(ii) then yields the final result

Hence we have the following stability and approximationresult

Theorem 10 The discrete saddle-point problem (32) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and

1003817100381710038171003817119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (55)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (32)

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(56)

Owing to Assumption 1(iii)-(iv) the discrete solutionconverges optimally to the continuous solution

32 Finite Element Approximation for the Stabilized Formula-tion (18) After stabilizing formulation (9) the bilinear form119860(sdot sdot) (18) is coercive on the whole product space 119881 times 119877Therefore we can use the standard finite element space 119878ℎto discretize the gradient of the solution for the saddle-pointproblem (18) so that 119871ℎ = 119878ℎ Thus our discrete saddle-point

Advances in Numerical Analysis 7

formulation of (18) is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎtimes[119871ℎ]119889times[119872ℎ]

119889

so that

119860 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119861 ((vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119861 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(57)

We can now verify the three conditions of well-posedness forthe discrete saddle point problem The continuity of 119860(sdot sdot)119861(sdot sdot) and ℓ(sdot) follows as in the continuous setting andcoercivity is immediate from Lemma 4 as119881ℎ times[119871ℎ]

119889sub 119881times119877

Explicit construction of basis functions of119872ℎ on a referenceelement will be given below

Assumption 1(ii) guarantees that the bilinear form 119887(sdot sdot)

satisfies the inf-sup condition as in the case of the previousfinite element method

We have thus proved the following theorem from thestandard theory of saddle-point problem (see [4])

Theorem 11 The discrete saddle-point problem (57) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and1003817100381710038171003817119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (58)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (9)1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(59)

The optimal convergence of the discrete solution is thenguaranteed by the standard approximation property of119881ℎ and119871ℎ and Assumption 1(iii)

In the following we give these basis functions for lin-ear simplicial finite elements in two and three dimensions[18] The parallelotope case is handled by using the tensorproduct construction of the one-dimensional basis functionspresented in [20 26] These basis functions are very useful inthe mortar finite element method and extensively studied inone- and two-dimensional cases [16 23 26]

Note that we have imposed the condition dim119872ℎ =

dim 119871ℎ to get that our Grammatrix is a square matrixThere-fore the local basis functions for linearbilineartrilinearfinite spaces are associated with the vertices of trianglestetrahedra quadrilaterals and hexahedra For example forthe reference triangle = (119909 119910) 0 lt 119909 0 lt 119910 119909 + 119910 lt 1we have

1205831 = 3 minus 4119909 minus 4119910 1205832 = 4119909 minus 1

1205833 = 4119910 minus 1

(60)

where the basis functions 1205831 1205832 and 1205833 are associated withthree vertices (0 0) (1 0) and (0 1) of the reference triangle

For the reference tetrahedron = (119909 119910 119911) 0 lt 119909 0 lt

119910 0 lt 119911 119909 + 119910 + 119911 lt 1 we have

1205831 = 4 minus 5119909 minus 5119910 minus 5119911 1205832 = 5119909 minus 1

1205833 = 5119910 minus 1 1205834 = 5119911 minus 1

(61)

where the basis functions 1205831 1205832 1205833 and 1205834 associated withfour vertices (0 0 0) (1 0 0) (0 1 0) and (0 0 1) of thereference tetrahedron (see also [17 18]) Note that the basisfunctions are constructed in such a way that the biorthogo-nality condition is satisfied on the reference element

The global basis functions for the test space are con-structed by gluing the local basis functions together followingexactly the same procedure of constructing global finiteelement basis functions from the local ones These globalbasis functions then satisfy the condition of biorthogonalitywith global finite element basis functions of 119878ℎ These basisfunctions also satisfy Assumption 1(i)ndash(iv) (see [17 18])

33 Static Condensation Here we want to eliminate 120590ℎ andLagrange multiplier 120601

ℎfrom the saddle-point problem (57)

We make use of the operator 119876ℎ defined previously Usingthe biorthogonality relation between the basis functions of 119871ℎand 119872ℎ the action of operator 119876ℎ on a function V isin 119871

2(Ω)

can be written as

119876ℎV =119899

sum

119894=1

intΩ120583119894V 119889119909

119888119894

120593119894 (62)

which tells that the operator119876ℎ is local in the sense to be givenbelow (see also [27])

Using the property of operator 119876ℎ we can eliminate thedegrees of freedom corresponding to 120590ℎ and 120601ℎ so that ourproblem is to find 119906ℎ isin 119878ℎ such that

119869 (119906ℎ) = minVℎisin119881ℎ

119869 (Vℎ) (63)

where

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ) (64)

for the nonstabilized formulation (9) and

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ω

+1003817100381710038171003817119876ℎ (nablaVℎ) minus nablaVℎ

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ)

(65)

for the stabilized formulation (18)Let the bilinear form 119860(sdot sdot) be defined as

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909 (66)

for the non-stabilized formulation (9) and

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909

+ intΩ

(120590ℎ minus nabla119906ℎ) sdot (120591ℎ minus nablaVℎ) 119889119909

(67)

8 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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

Advances in Numerical Analysis 3

Proof Using V = 0 in the expression on the left above we have

sup(V120591)isin119881times119877

119887 ((V 120591) 120601)

radic |V 21Ω

+ 1205912

0Ω

ge sup120591isin119877

intΩ120591 sdot 120601 119889119909

1205912

0Ω

=100381710038171003817100381712060110038171003817100381710038170Ω

(15)

To summarize we have proved the following theorem

Theorem 2 The saddle-point problem (9) has a unique solu-tion (119906120590120601) isin 119881 times 119877 times 119877 and

1199061Ω + 1205900Ω +100381710038171003817100381712060110038171003817100381710038170Ω

le ℓ0Ω (16)

Themain difficulty in this formulation is that the bilinearform 119886(sdot sdot) is not coercive on the whole space 119881 times 119877 Itis only coercive on the subspace 119870 of 119881 times 119877 One has toconsider this difficulty in developing a finite element methodfor this problem One approach to make the bilinear form119886(sdot sdot) coercive on the whole space 119881 times 119877 is to stabilize itThere are many approaches to stabilize this formulation (see[2 5 6]) Here we modify the bilinear form 119886(sdot sdot) to get aconsistent stabilization as proposed in [14] for the Mindlin-Reissner plate so that it is now coercive on the whole space119881 times 119877 This is obtained by adding the term

intΩ

(120590 minus nabla119906) sdot (120591 minus nablaV) 119889119909 (17)

to the bilinear form 119886(sdot sdot) Thus our problem is to find(119906120590120601) isin 119881 times 119877 times 119877 such that

119860 ((119906120590) (V 120591)) + 119861 ((V 120591) 120601) = ℓ (V) (V 120591) isin 119881 times 119877

119861 ((119906120590) 120595) = 0 120595 isin 119877

(18)

where

119860 ((119906120590) (V 120591)) = intΩ

120590120591 119889119909

+ intΩ

(120590 minus nabla119906) sdot (120591 minus nablaV) 119889119909

119861 ((119906120590) 120595) = intΩ

(120590 minus nabla119906) sdot 120595 119889119909

(19)

Theorem 3 The saddle-point problems (9) and (18) yield thesame solution

Proof By construction the solution to (18) is also the solutionto (9)Moreover as the second equation of (18) yields120590 = nabla119906we can substitute this into the first equation and set V = 0 toget 120601 = minusnabla119906 This proves that the solution to (18) is also thesolution to (9)

Here the bilinear form 119860(sdot sdot) is coercive on the wholespace 119881 times 119877 from the triangle and Poincare inequalities

Lemma 4 The bilinear form 119860(sdot sdot) is coercive on 119881 times 119877 Thatis

119860 ((119906120590) (119906120590)) ge 119862 (1199062

1Ω+ 1205902

0Ω) (119906120590) isin 119881 times 119877

(20)

Proof We start with the triangle inequality

nabla1199062

0Ωle nabla119906 minus 120590

2

0Ω+ 1205902

0Ω (21)

From Poincare inequality there exists a constant 119862 gt 0 suchthat

1198621199062

1Ωle nabla119906

2

0Ω(22)

and hence we have

1198621199062

1Ωle (nabla119906 minus 120590

2

0Ω+ 1205902

0Ω) = 119860 ((119906120590) (119906120590))

(23)

Moreover the other conditions of well-posedness for thisnew saddle-point formulation (18) can also be shown exactlyas for (9)

In the following we present finite element approxima-tions for both proposed three-field formulations In the firststep we propose a finite element method for the nonstabi-lized formulation (9) which is based on quasi-biorthogonalsystems In the second step we present a finite elementmethod for the stabilized formulation (18) which is basedon biorthogonal systems The first method works only forsimplicial elements whereas the second method works formeshes of parallelotopes and simplices

3 Finite Element Approximationand a Priori Error Estimate

Let Tℎ be a quasi-uniform partition of the domain Ω insimplices or parallelotopes having the mesh size ℎ Let bea reference simplex or 119889-cube where the reference simplex isdefined as

= x isin R119889 119909119894 gt 0 119894 = 1 119889

119889

sum

119894=1

119909119894 lt 1 (24)

and a 119889-cube = (minus1 1)119889 Note that for 119889 = 2 a simplex is a

triangle and for 119889 = 2 it is a tetrahedron Similarly a 2-cubeis a square whereas 3-cube is a cube

The finite element space is defined by affinemaps 119865119879 froma reference element to a physical element 119879 isin Tℎ LetQ()be the space of bilineartrilinear polynomials in if is areference squarecube or the space of linear polynomials in if is a reference simplex Then the finite element spacebased on the mesh Tℎ is defined as the space of continuousfunctions whose restrictions to an element 119879 are obtainedby maps of bilinear trilinear or linear functions from thereference element that is

119878ℎ = Vℎ isin 1198671(Ω) Vℎ|119879

= Vℎ ∘ 119865minus1

119879 Vℎ isin Q () 119879 isin Tℎ

119881ℎ = 119878ℎ cap 1198671

0(Ω)

(25)

4 Advances in Numerical Analysis

(See [3 10 15]) We start with some abstract assumptions fortwo discrete spaces [119871ℎ]

119889sub 119877 and [119872ℎ]

119889sub 119877 to discretize the

gradient of the solution and the Lagrange multiplier spaceswhere 119871ℎ and119872ℎ both are piecewise polynomial spaces withrespect to themeshTℎ and are both subsets of119871

2(Ω) Explicit

construction of local basis functions for these spaces for themixed finite elementmethod (9) is given in Section 31 and forthe mixed finite element method (18) is given in Section 32

We assume that 119871ℎ and119872ℎ satisfy the following assump-tions (see also [16 17])

Assumption 1 (i) dim119872ℎ = dim 119871ℎ(ii)There is a constant 120573 gt 0 independent of themeshTℎ

such that

1003817100381710038171003817120601ℎ10038171003817100381710038171198712(Ω)

le 120573 sup120583ℎisin119872ℎ0

intΩ120583ℎ120601ℎ119889119909

1003817100381710038171003817120583ℎ10038171003817100381710038171198712(Ω)

120601ℎ isin 119871ℎ (26)

(iii) The space119872ℎ has the approximation property

inf120582ℎisin119872ℎ

1003817100381710038171003817120601 minus 120582ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (27)

(iv) The space 119871ℎ has the approximation property

inf120601ℎisin119871ℎ

1003817100381710038171003817120601 minus 120601ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (28)

Now we define biorthogonality and quasi-biorthogonali-ty (see [16 18 19])

Definition 5 The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called biorthogonal if the resulting Gram matrix Gis diagonal The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called quasi-biorthogonal and the resulting Grammatrix G is called quasidiagonal if G is of the form

G = [D1 0

R D2] or G = [

D1 R0 D2

] (29)

where D1 and D2 are diagonal matrices and R is a sparserectangular matrix

We recall that a Gram matrix G of two sets of basisfunctions 1205831198941le119894le119899 and 1205931198941le119894le119899 is the matrix G whose 119894119895thentry is

intΩ

120583119894120593119895119889119909 (30)

It is worth noting that the quasi-diagonalmatrix is inverted byinverting two diagonal matrices and scaling the rectangularmatrix

31 Finite Element Approximation for the Nonstabilized For-mulation (9) We now turn our attention to a finite elementapproximation for the nonstabilized formulation (9) Thisapproximation works only for simplicial meshes and is basedon a quasi-biorthogonal system as in [19] Let

119861ℎ = 119887ℎ | 119887ℎ|119879

= 119862

119889+1

prod

119894=1

120582119879

119894 119879 isin Tℎ (31)

be the space of bubble functions where 119862 is a constant and120582119879

119894119889+1

119894=1is the set of barycentric coordinates on 119879 Let 119871ℎ =

119878ℎoplus119861ℎ Explicit construction of basis functions of 119871ℎ and119872ℎon a reference element is provided below

The finite element problem is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times

[119871ℎ]119889times [119872ℎ]

119889 such that

119886 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119887 ((Vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(32)

We note that the standard finite element space 119878ℎ is enrichedwith elementwise defined bubble functions to obtain thespace 119871ℎ which is done to obtain the coercivity of the bilinearform 119886(sdot sdot) on the kernel space 119870ℎ defined as (see Lemma 7)

119870ℎ = (Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((Vℎ 120591ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(33)

To simplify the numerical analysis of the finite elementproblem we introduce a quasiprojection operator 119876ℎ

1198712(Ω) rarr 119871ℎ which is defined as 120595

ℎ

intΩ

119876ℎV120583ℎ119889119909 = intΩ

V120583ℎ119889119909 V isin 1198712(Ω) 120583ℎ isin 119872ℎ (34)

This type of operator was introduced in [9] to obtain the finiteelement interpolation of nonsmooth functions satisfyingboundary conditions and is used in [20] in the context ofmortar finite elementsThe definition of119876ℎ allows us to writethe weak gradient as

120590ℎ = 119876ℎ (nabla119906ℎ) (35)

where operator 119876ℎ is applied to the vector nabla119906ℎ component-wise We see that 119876ℎ is well defined due to Assumption 1(i)and (ii) Furthermore the restriction of 119876ℎ to 119871ℎ is theidentity Hence 119876ℎ is a projection onto the space 119871ℎ Wenote that 119876ℎ is not the orthogonal projection onto 119871ℎ butan oblique projection onto 119871ℎ see [21 22] The stability andapproximation properties of 119876ℎ in 119871

2 and 1198671-norm can be

shown as in [16 23] In the following we will use a genericconstant 119862 which will take different values at different placesbut will be always independent of the mesh size ℎ

Lemma 6 Under Assumption 1(i)ndash(iii)

1003817100381710038171003817119876ℎV10038171003817100381710038171198712(Ω)

le 119862V1198712(Ω) 119891119900119903 V isin 1198712(Ω)

1003816100381610038161003816119876ℎ11990810038161003816100381610038161198671(Ω)

le 119862|119908|1198671(Ω) 119891119900119903 119908 isin 1198671(Ω)

(36)

and for 0 lt 119904 le 1 and V isin 1198671+119904(Ω)1003817100381710038171003817V minus 119876ℎV

10038171003817100381710038171198712(Ω) le 119862ℎ1+119904|V|119867119904+1(Ω)

1003817100381710038171003817V minus 119876ℎV10038171003817100381710038171198671(Ω)

le 119862ℎ119904|V|119867119904+1(Ω)

(37)

Advances in Numerical Analysis 5

We need to construct another finite element space119872ℎ sub1198712(Ω) that satisfies Assumption 1(i)ndash(iii) and also leads to an

efficient numerical scheme The main goal is to choose thebasis functions for119872ℎ and 119871ℎ so that the matrix D associatedwith the bilinear form int

Ω120590ℎ 120595ℎ119889119909 is easy to invert To this

end we consider the algebraic form of the weak equation forthe gradient of the solution which is given by

minusB + D = 0 (38)

where and are the solution vectors for 119906ℎ and 120590ℎ and D

is the Gram matrix between the bases of [119872ℎ]119889 and [119871ℎ]

119889Hence if the matrix D is diagonal or quasi-diagonal we caneasily eliminate the degrees of freedom corresponding to 120590ℎand 120601

ℎ This then leads to a formulation involving only one

unknown 119906ℎ Statically condensing out variables 120590ℎ and 120601ℎwe arrive at the variational formulation of the reduced systemgiven by (68) (see Section 33) Note that the matrix D is theGram matrix between the bases of [119872ℎ]

119889 and [119871ℎ]119889

We cannot use a biorthogonal system in this case as thestability will be lost (see the Appendix) This motivates ourinterest in a quasi-biorthogonal system We now show theconstruction of the local basis functions for119871ℎ and119872ℎ so thatthey form a quasi-biorthogonal system The construction isshown on the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 in the two-dimensional case and on the referencetetrahedron = (119909 119910 119911) 0 le 119909 0 le 119910 0 le 119911 119909 +119910+ 119911 le 1

in the three-dimensional caseWe now start with the reference triangle in the two-

dimensional case Let 1205931 1205934 defined as

1205931 = 1 minus 119909 minus 119910 1205932 = 119909 1205933 = 119910

1205934 = 27119909119910 (1 minus 119909 minus 119910)

(39)

be the local basis functions of 119871ℎ on the reference triangleassociated with three vertices (0 0) and (1 0) (0 1) and onebarycenter (13 13) Note that 1205934 is the standard bubblefunction used for example in the minielement for the Stokesequations [24]

Then if we define the local basis functions 1205831 1205834 of119872ℎ as

1205831 =34

9minus 4119909 minus 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205832 = minus2

9+ 4119909 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205833 = minus2

9+ 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205834 = 27119909119910 (1 minus 119909 minus 119910)

(40)

the two sets of basis functions 1205931 1205934 and 1205831 1205834

form a quasi-biorthogonal system on the reference triangleThese four basis functions 1205831 1205834 of 119872ℎ are associatedwith three vertices (0 0) and (1 0) (0 1) and the barycenter(13 13) of the reference triangle

Similarly on the reference tetrahedron the local basisfunctions 1205931 1205935 of 119871ℎ defined as

1205931 = 1 minus 119909 minus 119910 minus 119911 1205932 = 119909 1205933 = 119910

1205934 = 119911 1205935 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(41)

and the local basis functions 1205831 1205835 of119872ℎ defined as

1205831 =401

92minus 5119909 minus 5119910 minus 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205832 = minus59

92+ 5119909 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205833 = minus59

92+ 5119910 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205834 = minus59

92+ 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205835 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(42)

are quasi-biorthogonalThese five basis functions 1205931 1205935of 119871ℎ or 1205831 1205835 of 119872ℎ are associated with four vertices(0 0 0) (1 0 0) (0 1 0) and (0 0 1) and the barycenter(14 14 and 14) of the reference tetrahedron Using thefinite element basis functions for 119871ℎ the finite element basisfunctions for119872ℎ are constructed in such a way that the Grammatrix on the reference element is given by

D=

[[[[[[[[[

[

1

60 0 0

01

60 0

0 01

60

3

40

3

40

3

40

81

560

]]]]]]]]]

]

(43)

for the two-dimensional case and

D=

[[[[[[[[[[[[[

[

1

240 0 0 0

01

240 0 0

0 01

240 0

0 0 01

240

4

315

4

315

4

315

4

315

4096

155925

]]]]]]]]]]]]]

]

(44)

for the three-dimensional case After ordering the degreesof freedom in 120590ℎ and 120601ℎ so that the degrees of freedomassociated with the barycenters of elements come after thedegrees of freedom associated with the vertices of elementsthe global Gram matrix D is quasi-diagonal

Thus in the two-dimensional case the local basis func-tions 1205831 1205832 and 1205833 are associated with the vertices of thereference triangle and the function 1205834 is associated withthe barycenter of the triangle and defined elementwise Thatmeans 1205834 is supported only on the reference element Thesituation is similar in the three-dimensional case Hence

6 Advances in Numerical Analysis

using a proper ordering the support of a global basis function120593119894 of119871ℎ is the same as that of the global basis function120583119894 of119872ℎfor 119894 = 1 119899 However the global basis functions 1205831 120583119899of119872ℎ are not continuous on interelement boundaries

Now we show that119872ℎ satisfies Assumption 1(i)ndash(iii) Asthe first assumption is satisfied by construction we considerthe second assumption Let 120601ℎ = sum

119899

119896=1119886119896120601119896 isin 119878ℎ and set 120583ℎ =

sum119899

119896=1119886119896120583119896 isin 119872ℎ By using the quasibiorthogonality relation

(1) and the quasiuniformity assumption we get

intΩ

120601ℎ120583ℎ119889x =119899

sum

119894119895=1

119886119894119886119895 intΩ

120601119894120583119895119889x

=

119899

sum

119894=1

1198862

119894119888119894 ge 119862

119899

sum

119894=1

1198862

119894ℎ119889

119894ge 119862

1003817100381710038171003817120601ℎ1003817100381710038171003817

2

0Ω

(45)

where ℎ119894 denotes the meshsize at 119894th vertex Taking intoaccount the fact that 120601ℎ

2

0Ωequiv 120583ℎ

2

0Ωequiv sum119899

119894=11198862

119894ℎ119889

119894 we

find that Assumption 1(ii) is satisfied Since the sum of thelocal basis functions of 119872ℎ is one Assumption 1(iii) can beproved as in [23 25] Assumption 1(iv) follows by standardarguments

There are (119889 + 2) local basis functions of 119871ℎ where (119889 +1) of them are associated with vertices and one is associatedwith the element barycenter Then the local basis functionsof119872ℎ are also divided into two parts (119889 + 2) basis functionsassociated with the vertices and 1 basis function associatedwith the element Let 120583119879 be the local basis function associatedwith the element barycenter of 119879 isin Tℎ for119872ℎ Let 119884ℎ be thespace of piecewise constant functions with respect toTℎ and119875ℎ 119871

2(Ω) rarr 119884ℎ is defined as

119875ℎ119908|119879 =

int119879119908120583119879119889119909

int119879120583119879119889119909

(46)

Then 119875ℎ is a projection operator onto 119884ℎ Moreover

nablaVℎ = 119875ℎ119876ℎnablaVℎ (47)

and this implies that1003817100381710038171003817nablaVℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnablaVℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnablaVℎ

10038171003817100381710038170Ω (48)

This yields the following coercivity result

Lemma 7 The bilinear form 119886(sdot sdot) is coercive on the kernelspace 119870ℎ In fact

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ)) ge1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119906ℎ

1003817100381710038171003817

2

0Ω)

(119906ℎ120590ℎ) isin 119870ℎ

(49)

Proof Since 120590ℎ = 119876ℎnabla119906ℎ on the kernel space 119870ℎ we have

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ))

=1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817119876ℎnabla119906ℎ

1003817100381710038171003817

2

0Ω)

(50)

The proof follows from the result1003817100381710038171003817nabla119906ℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnabla119906ℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnabla119906ℎ

10038171003817100381710038170Ω (51)

Remark 8 Using a quadrilateral or hexahedral meshes wecannot have nablaVℎ = 119875ℎ119876ℎnablaVℎ and hence the coercivity failsOne needs to modify the finite element method for thesemeshes

The inf-sup condition follows exactly as in the continuoussetting using Assumption 1(ii)

Lemma 9 There exists a constant 120573 gt 0 such that

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge 1205731003817100381710038171003817120601ℎ

10038171003817100381710038170Ω 120601ℎisin 119877ℎ

(52)

Proof The proof follows exactly as in the continuous settingWe use Vℎ = 0 in the expression

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

(53)

to get

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge sup120591ℎisin119877ℎ

intΩ120591ℎ sdot 120601ℎ119889119909

1003817100381710038171003817120591ℎ1003817100381710038171003817

2

0Ω

(54)

Assumption 1(ii) then yields the final result

Hence we have the following stability and approximationresult

Theorem 10 The discrete saddle-point problem (32) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and

1003817100381710038171003817119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (55)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (32)

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(56)

Owing to Assumption 1(iii)-(iv) the discrete solutionconverges optimally to the continuous solution

32 Finite Element Approximation for the Stabilized Formula-tion (18) After stabilizing formulation (9) the bilinear form119860(sdot sdot) (18) is coercive on the whole product space 119881 times 119877Therefore we can use the standard finite element space 119878ℎto discretize the gradient of the solution for the saddle-pointproblem (18) so that 119871ℎ = 119878ℎ Thus our discrete saddle-point

Advances in Numerical Analysis 7

formulation of (18) is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎtimes[119871ℎ]119889times[119872ℎ]

119889

so that

119860 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119861 ((vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119861 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(57)

We can now verify the three conditions of well-posedness forthe discrete saddle point problem The continuity of 119860(sdot sdot)119861(sdot sdot) and ℓ(sdot) follows as in the continuous setting andcoercivity is immediate from Lemma 4 as119881ℎ times[119871ℎ]

119889sub 119881times119877

Explicit construction of basis functions of119872ℎ on a referenceelement will be given below

Assumption 1(ii) guarantees that the bilinear form 119887(sdot sdot)

satisfies the inf-sup condition as in the case of the previousfinite element method

We have thus proved the following theorem from thestandard theory of saddle-point problem (see [4])

Theorem 11 The discrete saddle-point problem (57) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and1003817100381710038171003817119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (58)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (9)1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(59)

The optimal convergence of the discrete solution is thenguaranteed by the standard approximation property of119881ℎ and119871ℎ and Assumption 1(iii)

In the following we give these basis functions for lin-ear simplicial finite elements in two and three dimensions[18] The parallelotope case is handled by using the tensorproduct construction of the one-dimensional basis functionspresented in [20 26] These basis functions are very useful inthe mortar finite element method and extensively studied inone- and two-dimensional cases [16 23 26]

Note that we have imposed the condition dim119872ℎ =

dim 119871ℎ to get that our Grammatrix is a square matrixThere-fore the local basis functions for linearbilineartrilinearfinite spaces are associated with the vertices of trianglestetrahedra quadrilaterals and hexahedra For example forthe reference triangle = (119909 119910) 0 lt 119909 0 lt 119910 119909 + 119910 lt 1we have

1205831 = 3 minus 4119909 minus 4119910 1205832 = 4119909 minus 1

1205833 = 4119910 minus 1

(60)

where the basis functions 1205831 1205832 and 1205833 are associated withthree vertices (0 0) (1 0) and (0 1) of the reference triangle

For the reference tetrahedron = (119909 119910 119911) 0 lt 119909 0 lt

119910 0 lt 119911 119909 + 119910 + 119911 lt 1 we have

1205831 = 4 minus 5119909 minus 5119910 minus 5119911 1205832 = 5119909 minus 1

1205833 = 5119910 minus 1 1205834 = 5119911 minus 1

(61)

where the basis functions 1205831 1205832 1205833 and 1205834 associated withfour vertices (0 0 0) (1 0 0) (0 1 0) and (0 0 1) of thereference tetrahedron (see also [17 18]) Note that the basisfunctions are constructed in such a way that the biorthogo-nality condition is satisfied on the reference element

The global basis functions for the test space are con-structed by gluing the local basis functions together followingexactly the same procedure of constructing global finiteelement basis functions from the local ones These globalbasis functions then satisfy the condition of biorthogonalitywith global finite element basis functions of 119878ℎ These basisfunctions also satisfy Assumption 1(i)ndash(iv) (see [17 18])

33 Static Condensation Here we want to eliminate 120590ℎ andLagrange multiplier 120601

ℎfrom the saddle-point problem (57)

We make use of the operator 119876ℎ defined previously Usingthe biorthogonality relation between the basis functions of 119871ℎand 119872ℎ the action of operator 119876ℎ on a function V isin 119871

2(Ω)

can be written as

119876ℎV =119899

sum

119894=1

intΩ120583119894V 119889119909

119888119894

120593119894 (62)

which tells that the operator119876ℎ is local in the sense to be givenbelow (see also [27])

Using the property of operator 119876ℎ we can eliminate thedegrees of freedom corresponding to 120590ℎ and 120601ℎ so that ourproblem is to find 119906ℎ isin 119878ℎ such that

119869 (119906ℎ) = minVℎisin119881ℎ

119869 (Vℎ) (63)

where

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ) (64)

for the nonstabilized formulation (9) and

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ω

+1003817100381710038171003817119876ℎ (nablaVℎ) minus nablaVℎ

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ)

(65)

for the stabilized formulation (18)Let the bilinear form 119860(sdot sdot) be defined as

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909 (66)

for the non-stabilized formulation (9) and

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909

+ intΩ

(120590ℎ minus nabla119906ℎ) sdot (120591ℎ minus nablaVℎ) 119889119909

(67)

8 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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Stochastic AnalysisInternational Journal of

4 Advances in Numerical Analysis

(See [3 10 15]) We start with some abstract assumptions fortwo discrete spaces [119871ℎ]

119889sub 119877 and [119872ℎ]

119889sub 119877 to discretize the

gradient of the solution and the Lagrange multiplier spaceswhere 119871ℎ and119872ℎ both are piecewise polynomial spaces withrespect to themeshTℎ and are both subsets of119871

2(Ω) Explicit

construction of local basis functions for these spaces for themixed finite elementmethod (9) is given in Section 31 and forthe mixed finite element method (18) is given in Section 32

We assume that 119871ℎ and119872ℎ satisfy the following assump-tions (see also [16 17])

Assumption 1 (i) dim119872ℎ = dim 119871ℎ(ii)There is a constant 120573 gt 0 independent of themeshTℎ

such that

1003817100381710038171003817120601ℎ10038171003817100381710038171198712(Ω)

le 120573 sup120583ℎisin119872ℎ0

intΩ120583ℎ120601ℎ119889119909

1003817100381710038171003817120583ℎ10038171003817100381710038171198712(Ω)

120601ℎ isin 119871ℎ (26)

(iii) The space119872ℎ has the approximation property

inf120582ℎisin119872ℎ

1003817100381710038171003817120601 minus 120582ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (27)

(iv) The space 119871ℎ has the approximation property

inf120601ℎisin119871ℎ

1003817100381710038171003817120601 minus 120601ℎ10038171003817100381710038171198712(Ω)

le 119862ℎ100381610038161003816100381612060110038161003816100381610038161198671(Ω)

120601 isin 1198671(Ω) (28)

Now we define biorthogonality and quasi-biorthogonali-ty (see [16 18 19])

Definition 5 The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called biorthogonal if the resulting Gram matrix Gis diagonal The pair of bases 1205831198941le119894le119899 of 119872ℎ and 1205931198941le119894le119899

of 119871ℎ is called quasi-biorthogonal and the resulting Grammatrix G is called quasidiagonal if G is of the form

G = [D1 0

R D2] or G = [

D1 R0 D2

] (29)

where D1 and D2 are diagonal matrices and R is a sparserectangular matrix

We recall that a Gram matrix G of two sets of basisfunctions 1205831198941le119894le119899 and 1205931198941le119894le119899 is the matrix G whose 119894119895thentry is

intΩ

120583119894120593119895119889119909 (30)

It is worth noting that the quasi-diagonalmatrix is inverted byinverting two diagonal matrices and scaling the rectangularmatrix

31 Finite Element Approximation for the Nonstabilized For-mulation (9) We now turn our attention to a finite elementapproximation for the nonstabilized formulation (9) Thisapproximation works only for simplicial meshes and is basedon a quasi-biorthogonal system as in [19] Let

119861ℎ = 119887ℎ | 119887ℎ|119879

= 119862

119889+1

prod

119894=1

120582119879

119894 119879 isin Tℎ (31)

be the space of bubble functions where 119862 is a constant and120582119879

119894119889+1

119894=1is the set of barycentric coordinates on 119879 Let 119871ℎ =

119878ℎoplus119861ℎ Explicit construction of basis functions of 119871ℎ and119872ℎon a reference element is provided below

The finite element problem is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times

[119871ℎ]119889times [119872ℎ]

119889 such that

119886 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119887 ((Vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(32)

We note that the standard finite element space 119878ℎ is enrichedwith elementwise defined bubble functions to obtain thespace 119871ℎ which is done to obtain the coercivity of the bilinearform 119886(sdot sdot) on the kernel space 119870ℎ defined as (see Lemma 7)

119870ℎ = (Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119887 ((Vℎ 120591ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(33)

To simplify the numerical analysis of the finite elementproblem we introduce a quasiprojection operator 119876ℎ

1198712(Ω) rarr 119871ℎ which is defined as 120595

ℎ

intΩ

119876ℎV120583ℎ119889119909 = intΩ

V120583ℎ119889119909 V isin 1198712(Ω) 120583ℎ isin 119872ℎ (34)

This type of operator was introduced in [9] to obtain the finiteelement interpolation of nonsmooth functions satisfyingboundary conditions and is used in [20] in the context ofmortar finite elementsThe definition of119876ℎ allows us to writethe weak gradient as

120590ℎ = 119876ℎ (nabla119906ℎ) (35)

where operator 119876ℎ is applied to the vector nabla119906ℎ component-wise We see that 119876ℎ is well defined due to Assumption 1(i)and (ii) Furthermore the restriction of 119876ℎ to 119871ℎ is theidentity Hence 119876ℎ is a projection onto the space 119871ℎ Wenote that 119876ℎ is not the orthogonal projection onto 119871ℎ butan oblique projection onto 119871ℎ see [21 22] The stability andapproximation properties of 119876ℎ in 119871

2 and 1198671-norm can be

shown as in [16 23] In the following we will use a genericconstant 119862 which will take different values at different placesbut will be always independent of the mesh size ℎ

Lemma 6 Under Assumption 1(i)ndash(iii)

1003817100381710038171003817119876ℎV10038171003817100381710038171198712(Ω)

le 119862V1198712(Ω) 119891119900119903 V isin 1198712(Ω)

1003816100381610038161003816119876ℎ11990810038161003816100381610038161198671(Ω)

le 119862|119908|1198671(Ω) 119891119900119903 119908 isin 1198671(Ω)

(36)

and for 0 lt 119904 le 1 and V isin 1198671+119904(Ω)1003817100381710038171003817V minus 119876ℎV

10038171003817100381710038171198712(Ω) le 119862ℎ1+119904|V|119867119904+1(Ω)

1003817100381710038171003817V minus 119876ℎV10038171003817100381710038171198671(Ω)

le 119862ℎ119904|V|119867119904+1(Ω)

(37)

Advances in Numerical Analysis 5

We need to construct another finite element space119872ℎ sub1198712(Ω) that satisfies Assumption 1(i)ndash(iii) and also leads to an

efficient numerical scheme The main goal is to choose thebasis functions for119872ℎ and 119871ℎ so that the matrix D associatedwith the bilinear form int

Ω120590ℎ 120595ℎ119889119909 is easy to invert To this

end we consider the algebraic form of the weak equation forthe gradient of the solution which is given by

minusB + D = 0 (38)

where and are the solution vectors for 119906ℎ and 120590ℎ and D

is the Gram matrix between the bases of [119872ℎ]119889 and [119871ℎ]

119889Hence if the matrix D is diagonal or quasi-diagonal we caneasily eliminate the degrees of freedom corresponding to 120590ℎand 120601

ℎ This then leads to a formulation involving only one

unknown 119906ℎ Statically condensing out variables 120590ℎ and 120601ℎwe arrive at the variational formulation of the reduced systemgiven by (68) (see Section 33) Note that the matrix D is theGram matrix between the bases of [119872ℎ]

119889 and [119871ℎ]119889

We cannot use a biorthogonal system in this case as thestability will be lost (see the Appendix) This motivates ourinterest in a quasi-biorthogonal system We now show theconstruction of the local basis functions for119871ℎ and119872ℎ so thatthey form a quasi-biorthogonal system The construction isshown on the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 in the two-dimensional case and on the referencetetrahedron = (119909 119910 119911) 0 le 119909 0 le 119910 0 le 119911 119909 +119910+ 119911 le 1

in the three-dimensional caseWe now start with the reference triangle in the two-

dimensional case Let 1205931 1205934 defined as

1205931 = 1 minus 119909 minus 119910 1205932 = 119909 1205933 = 119910

1205934 = 27119909119910 (1 minus 119909 minus 119910)

(39)

be the local basis functions of 119871ℎ on the reference triangleassociated with three vertices (0 0) and (1 0) (0 1) and onebarycenter (13 13) Note that 1205934 is the standard bubblefunction used for example in the minielement for the Stokesequations [24]

Then if we define the local basis functions 1205831 1205834 of119872ℎ as

1205831 =34

9minus 4119909 minus 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205832 = minus2

9+ 4119909 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205833 = minus2

9+ 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205834 = 27119909119910 (1 minus 119909 minus 119910)

(40)

the two sets of basis functions 1205931 1205934 and 1205831 1205834

form a quasi-biorthogonal system on the reference triangleThese four basis functions 1205831 1205834 of 119872ℎ are associatedwith three vertices (0 0) and (1 0) (0 1) and the barycenter(13 13) of the reference triangle

Similarly on the reference tetrahedron the local basisfunctions 1205931 1205935 of 119871ℎ defined as

1205931 = 1 minus 119909 minus 119910 minus 119911 1205932 = 119909 1205933 = 119910

1205934 = 119911 1205935 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(41)

and the local basis functions 1205831 1205835 of119872ℎ defined as

1205831 =401

92minus 5119909 minus 5119910 minus 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205832 = minus59

92+ 5119909 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205833 = minus59

92+ 5119910 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205834 = minus59

92+ 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205835 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(42)

are quasi-biorthogonalThese five basis functions 1205931 1205935of 119871ℎ or 1205831 1205835 of 119872ℎ are associated with four vertices(0 0 0) (1 0 0) (0 1 0) and (0 0 1) and the barycenter(14 14 and 14) of the reference tetrahedron Using thefinite element basis functions for 119871ℎ the finite element basisfunctions for119872ℎ are constructed in such a way that the Grammatrix on the reference element is given by

D=

[[[[[[[[[

[

1

60 0 0

01

60 0

0 01

60

3

40

3

40

3

40

81

560

]]]]]]]]]

]

(43)

for the two-dimensional case and

D=

[[[[[[[[[[[[[

[

1

240 0 0 0

01

240 0 0

0 01

240 0

0 0 01

240

4

315

4

315

4

315

4

315

4096

155925

]]]]]]]]]]]]]

]

(44)

for the three-dimensional case After ordering the degreesof freedom in 120590ℎ and 120601ℎ so that the degrees of freedomassociated with the barycenters of elements come after thedegrees of freedom associated with the vertices of elementsthe global Gram matrix D is quasi-diagonal

Thus in the two-dimensional case the local basis func-tions 1205831 1205832 and 1205833 are associated with the vertices of thereference triangle and the function 1205834 is associated withthe barycenter of the triangle and defined elementwise Thatmeans 1205834 is supported only on the reference element Thesituation is similar in the three-dimensional case Hence

6 Advances in Numerical Analysis

using a proper ordering the support of a global basis function120593119894 of119871ℎ is the same as that of the global basis function120583119894 of119872ℎfor 119894 = 1 119899 However the global basis functions 1205831 120583119899of119872ℎ are not continuous on interelement boundaries

Now we show that119872ℎ satisfies Assumption 1(i)ndash(iii) Asthe first assumption is satisfied by construction we considerthe second assumption Let 120601ℎ = sum

119899

119896=1119886119896120601119896 isin 119878ℎ and set 120583ℎ =

sum119899

119896=1119886119896120583119896 isin 119872ℎ By using the quasibiorthogonality relation

(1) and the quasiuniformity assumption we get

intΩ

120601ℎ120583ℎ119889x =119899

sum

119894119895=1

119886119894119886119895 intΩ

120601119894120583119895119889x

=

119899

sum

119894=1

1198862

119894119888119894 ge 119862

119899

sum

119894=1

1198862

119894ℎ119889

119894ge 119862

1003817100381710038171003817120601ℎ1003817100381710038171003817

2

0Ω

(45)

where ℎ119894 denotes the meshsize at 119894th vertex Taking intoaccount the fact that 120601ℎ

2

0Ωequiv 120583ℎ

2

0Ωequiv sum119899

119894=11198862

119894ℎ119889

119894 we

find that Assumption 1(ii) is satisfied Since the sum of thelocal basis functions of 119872ℎ is one Assumption 1(iii) can beproved as in [23 25] Assumption 1(iv) follows by standardarguments

There are (119889 + 2) local basis functions of 119871ℎ where (119889 +1) of them are associated with vertices and one is associatedwith the element barycenter Then the local basis functionsof119872ℎ are also divided into two parts (119889 + 2) basis functionsassociated with the vertices and 1 basis function associatedwith the element Let 120583119879 be the local basis function associatedwith the element barycenter of 119879 isin Tℎ for119872ℎ Let 119884ℎ be thespace of piecewise constant functions with respect toTℎ and119875ℎ 119871

2(Ω) rarr 119884ℎ is defined as

119875ℎ119908|119879 =

int119879119908120583119879119889119909

int119879120583119879119889119909

(46)

Then 119875ℎ is a projection operator onto 119884ℎ Moreover

nablaVℎ = 119875ℎ119876ℎnablaVℎ (47)

and this implies that1003817100381710038171003817nablaVℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnablaVℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnablaVℎ

10038171003817100381710038170Ω (48)

This yields the following coercivity result

Lemma 7 The bilinear form 119886(sdot sdot) is coercive on the kernelspace 119870ℎ In fact

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ)) ge1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119906ℎ

1003817100381710038171003817

2

0Ω)

(119906ℎ120590ℎ) isin 119870ℎ

(49)

Proof Since 120590ℎ = 119876ℎnabla119906ℎ on the kernel space 119870ℎ we have

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ))

=1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817119876ℎnabla119906ℎ

1003817100381710038171003817

2

0Ω)

(50)

The proof follows from the result1003817100381710038171003817nabla119906ℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnabla119906ℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnabla119906ℎ

10038171003817100381710038170Ω (51)

Remark 8 Using a quadrilateral or hexahedral meshes wecannot have nablaVℎ = 119875ℎ119876ℎnablaVℎ and hence the coercivity failsOne needs to modify the finite element method for thesemeshes

The inf-sup condition follows exactly as in the continuoussetting using Assumption 1(ii)

Lemma 9 There exists a constant 120573 gt 0 such that

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge 1205731003817100381710038171003817120601ℎ

10038171003817100381710038170Ω 120601ℎisin 119877ℎ

(52)

Proof The proof follows exactly as in the continuous settingWe use Vℎ = 0 in the expression

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

(53)

to get

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge sup120591ℎisin119877ℎ

intΩ120591ℎ sdot 120601ℎ119889119909

1003817100381710038171003817120591ℎ1003817100381710038171003817

2

0Ω

(54)

Assumption 1(ii) then yields the final result

Hence we have the following stability and approximationresult

Theorem 10 The discrete saddle-point problem (32) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and

1003817100381710038171003817119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (55)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (32)

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(56)

Owing to Assumption 1(iii)-(iv) the discrete solutionconverges optimally to the continuous solution

32 Finite Element Approximation for the Stabilized Formula-tion (18) After stabilizing formulation (9) the bilinear form119860(sdot sdot) (18) is coercive on the whole product space 119881 times 119877Therefore we can use the standard finite element space 119878ℎto discretize the gradient of the solution for the saddle-pointproblem (18) so that 119871ℎ = 119878ℎ Thus our discrete saddle-point

Advances in Numerical Analysis 7

formulation of (18) is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎtimes[119871ℎ]119889times[119872ℎ]

119889

so that

119860 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119861 ((vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119861 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(57)

We can now verify the three conditions of well-posedness forthe discrete saddle point problem The continuity of 119860(sdot sdot)119861(sdot sdot) and ℓ(sdot) follows as in the continuous setting andcoercivity is immediate from Lemma 4 as119881ℎ times[119871ℎ]

119889sub 119881times119877

Explicit construction of basis functions of119872ℎ on a referenceelement will be given below

Assumption 1(ii) guarantees that the bilinear form 119887(sdot sdot)

satisfies the inf-sup condition as in the case of the previousfinite element method

We have thus proved the following theorem from thestandard theory of saddle-point problem (see [4])

Theorem 11 The discrete saddle-point problem (57) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and1003817100381710038171003817119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (58)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (9)1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(59)

The optimal convergence of the discrete solution is thenguaranteed by the standard approximation property of119881ℎ and119871ℎ and Assumption 1(iii)

In the following we give these basis functions for lin-ear simplicial finite elements in two and three dimensions[18] The parallelotope case is handled by using the tensorproduct construction of the one-dimensional basis functionspresented in [20 26] These basis functions are very useful inthe mortar finite element method and extensively studied inone- and two-dimensional cases [16 23 26]

Note that we have imposed the condition dim119872ℎ =

dim 119871ℎ to get that our Grammatrix is a square matrixThere-fore the local basis functions for linearbilineartrilinearfinite spaces are associated with the vertices of trianglestetrahedra quadrilaterals and hexahedra For example forthe reference triangle = (119909 119910) 0 lt 119909 0 lt 119910 119909 + 119910 lt 1we have

1205831 = 3 minus 4119909 minus 4119910 1205832 = 4119909 minus 1

1205833 = 4119910 minus 1

(60)

where the basis functions 1205831 1205832 and 1205833 are associated withthree vertices (0 0) (1 0) and (0 1) of the reference triangle

For the reference tetrahedron = (119909 119910 119911) 0 lt 119909 0 lt

119910 0 lt 119911 119909 + 119910 + 119911 lt 1 we have

1205831 = 4 minus 5119909 minus 5119910 minus 5119911 1205832 = 5119909 minus 1

1205833 = 5119910 minus 1 1205834 = 5119911 minus 1

(61)

where the basis functions 1205831 1205832 1205833 and 1205834 associated withfour vertices (0 0 0) (1 0 0) (0 1 0) and (0 0 1) of thereference tetrahedron (see also [17 18]) Note that the basisfunctions are constructed in such a way that the biorthogo-nality condition is satisfied on the reference element

The global basis functions for the test space are con-structed by gluing the local basis functions together followingexactly the same procedure of constructing global finiteelement basis functions from the local ones These globalbasis functions then satisfy the condition of biorthogonalitywith global finite element basis functions of 119878ℎ These basisfunctions also satisfy Assumption 1(i)ndash(iv) (see [17 18])

33 Static Condensation Here we want to eliminate 120590ℎ andLagrange multiplier 120601

ℎfrom the saddle-point problem (57)

We make use of the operator 119876ℎ defined previously Usingthe biorthogonality relation between the basis functions of 119871ℎand 119872ℎ the action of operator 119876ℎ on a function V isin 119871

2(Ω)

can be written as

119876ℎV =119899

sum

119894=1

intΩ120583119894V 119889119909

119888119894

120593119894 (62)

which tells that the operator119876ℎ is local in the sense to be givenbelow (see also [27])

Using the property of operator 119876ℎ we can eliminate thedegrees of freedom corresponding to 120590ℎ and 120601ℎ so that ourproblem is to find 119906ℎ isin 119878ℎ such that

119869 (119906ℎ) = minVℎisin119881ℎ

119869 (Vℎ) (63)

where

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ) (64)

for the nonstabilized formulation (9) and

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ω

+1003817100381710038171003817119876ℎ (nablaVℎ) minus nablaVℎ

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ)

(65)

for the stabilized formulation (18)Let the bilinear form 119860(sdot sdot) be defined as

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909 (66)

for the non-stabilized formulation (9) and

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909

+ intΩ

(120590ℎ minus nabla119906ℎ) sdot (120591ℎ minus nablaVℎ) 119889119909

(67)

8 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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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

Advances in Numerical Analysis 5

We need to construct another finite element space119872ℎ sub1198712(Ω) that satisfies Assumption 1(i)ndash(iii) and also leads to an

efficient numerical scheme The main goal is to choose thebasis functions for119872ℎ and 119871ℎ so that the matrix D associatedwith the bilinear form int

Ω120590ℎ 120595ℎ119889119909 is easy to invert To this

end we consider the algebraic form of the weak equation forthe gradient of the solution which is given by

minusB + D = 0 (38)

where and are the solution vectors for 119906ℎ and 120590ℎ and D

is the Gram matrix between the bases of [119872ℎ]119889 and [119871ℎ]

119889Hence if the matrix D is diagonal or quasi-diagonal we caneasily eliminate the degrees of freedom corresponding to 120590ℎand 120601

ℎ This then leads to a formulation involving only one

unknown 119906ℎ Statically condensing out variables 120590ℎ and 120601ℎwe arrive at the variational formulation of the reduced systemgiven by (68) (see Section 33) Note that the matrix D is theGram matrix between the bases of [119872ℎ]

119889 and [119871ℎ]119889

We cannot use a biorthogonal system in this case as thestability will be lost (see the Appendix) This motivates ourinterest in a quasi-biorthogonal system We now show theconstruction of the local basis functions for119871ℎ and119872ℎ so thatthey form a quasi-biorthogonal system The construction isshown on the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 in the two-dimensional case and on the referencetetrahedron = (119909 119910 119911) 0 le 119909 0 le 119910 0 le 119911 119909 +119910+ 119911 le 1

in the three-dimensional caseWe now start with the reference triangle in the two-

dimensional case Let 1205931 1205934 defined as

1205931 = 1 minus 119909 minus 119910 1205932 = 119909 1205933 = 119910

1205934 = 27119909119910 (1 minus 119909 minus 119910)

(39)

be the local basis functions of 119871ℎ on the reference triangleassociated with three vertices (0 0) and (1 0) (0 1) and onebarycenter (13 13) Note that 1205934 is the standard bubblefunction used for example in the minielement for the Stokesequations [24]

Then if we define the local basis functions 1205831 1205834 of119872ℎ as

1205831 =34

9minus 4119909 minus 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205832 = minus2

9+ 4119909 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205833 = minus2

9+ 4119910 minus

140

3119909119910 (1 minus 119909 minus 119910)

1205834 = 27119909119910 (1 minus 119909 minus 119910)

(40)

the two sets of basis functions 1205931 1205934 and 1205831 1205834

form a quasi-biorthogonal system on the reference triangleThese four basis functions 1205831 1205834 of 119872ℎ are associatedwith three vertices (0 0) and (1 0) (0 1) and the barycenter(13 13) of the reference triangle

Similarly on the reference tetrahedron the local basisfunctions 1205931 1205935 of 119871ℎ defined as

1205931 = 1 minus 119909 minus 119910 minus 119911 1205932 = 119909 1205933 = 119910

1205934 = 119911 1205935 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(41)

and the local basis functions 1205831 1205835 of119872ℎ defined as

1205831 =401

92minus 5119909 minus 5119910 minus 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205832 = minus59

92+ 5119909 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205833 = minus59

92+ 5119910 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205834 = minus59

92+ 5119911 minus

6930

23119909119910119911 (1 minus 119909 minus 119910 minus 119911)

1205835 = 256119909119910119911 (1 minus 119909 minus 119910 minus 119911)

(42)

are quasi-biorthogonalThese five basis functions 1205931 1205935of 119871ℎ or 1205831 1205835 of 119872ℎ are associated with four vertices(0 0 0) (1 0 0) (0 1 0) and (0 0 1) and the barycenter(14 14 and 14) of the reference tetrahedron Using thefinite element basis functions for 119871ℎ the finite element basisfunctions for119872ℎ are constructed in such a way that the Grammatrix on the reference element is given by

D=

[[[[[[[[[

[

1

60 0 0

01

60 0

0 01

60

3

40

3

40

3

40

81

560

]]]]]]]]]

]

(43)

for the two-dimensional case and

D=

[[[[[[[[[[[[[

[

1

240 0 0 0

01

240 0 0

0 01

240 0

0 0 01

240

4

315

4

315

4

315

4

315

4096

155925

]]]]]]]]]]]]]

]

(44)

for the three-dimensional case After ordering the degreesof freedom in 120590ℎ and 120601ℎ so that the degrees of freedomassociated with the barycenters of elements come after thedegrees of freedom associated with the vertices of elementsthe global Gram matrix D is quasi-diagonal

Thus in the two-dimensional case the local basis func-tions 1205831 1205832 and 1205833 are associated with the vertices of thereference triangle and the function 1205834 is associated withthe barycenter of the triangle and defined elementwise Thatmeans 1205834 is supported only on the reference element Thesituation is similar in the three-dimensional case Hence

6 Advances in Numerical Analysis

using a proper ordering the support of a global basis function120593119894 of119871ℎ is the same as that of the global basis function120583119894 of119872ℎfor 119894 = 1 119899 However the global basis functions 1205831 120583119899of119872ℎ are not continuous on interelement boundaries

Now we show that119872ℎ satisfies Assumption 1(i)ndash(iii) Asthe first assumption is satisfied by construction we considerthe second assumption Let 120601ℎ = sum

119899

119896=1119886119896120601119896 isin 119878ℎ and set 120583ℎ =

sum119899

119896=1119886119896120583119896 isin 119872ℎ By using the quasibiorthogonality relation

(1) and the quasiuniformity assumption we get

intΩ

120601ℎ120583ℎ119889x =119899

sum

119894119895=1

119886119894119886119895 intΩ

120601119894120583119895119889x

=

119899

sum

119894=1

1198862

119894119888119894 ge 119862

119899

sum

119894=1

1198862

119894ℎ119889

119894ge 119862

1003817100381710038171003817120601ℎ1003817100381710038171003817

2

0Ω

(45)

where ℎ119894 denotes the meshsize at 119894th vertex Taking intoaccount the fact that 120601ℎ

2

0Ωequiv 120583ℎ

2

0Ωequiv sum119899

119894=11198862

119894ℎ119889

119894 we

find that Assumption 1(ii) is satisfied Since the sum of thelocal basis functions of 119872ℎ is one Assumption 1(iii) can beproved as in [23 25] Assumption 1(iv) follows by standardarguments

There are (119889 + 2) local basis functions of 119871ℎ where (119889 +1) of them are associated with vertices and one is associatedwith the element barycenter Then the local basis functionsof119872ℎ are also divided into two parts (119889 + 2) basis functionsassociated with the vertices and 1 basis function associatedwith the element Let 120583119879 be the local basis function associatedwith the element barycenter of 119879 isin Tℎ for119872ℎ Let 119884ℎ be thespace of piecewise constant functions with respect toTℎ and119875ℎ 119871

2(Ω) rarr 119884ℎ is defined as

119875ℎ119908|119879 =

int119879119908120583119879119889119909

int119879120583119879119889119909

(46)

Then 119875ℎ is a projection operator onto 119884ℎ Moreover

nablaVℎ = 119875ℎ119876ℎnablaVℎ (47)

and this implies that1003817100381710038171003817nablaVℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnablaVℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnablaVℎ

10038171003817100381710038170Ω (48)

This yields the following coercivity result

Lemma 7 The bilinear form 119886(sdot sdot) is coercive on the kernelspace 119870ℎ In fact

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ)) ge1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119906ℎ

1003817100381710038171003817

2

0Ω)

(119906ℎ120590ℎ) isin 119870ℎ

(49)

Proof Since 120590ℎ = 119876ℎnabla119906ℎ on the kernel space 119870ℎ we have

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ))

=1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817119876ℎnabla119906ℎ

1003817100381710038171003817

2

0Ω)

(50)

The proof follows from the result1003817100381710038171003817nabla119906ℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnabla119906ℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnabla119906ℎ

10038171003817100381710038170Ω (51)

Remark 8 Using a quadrilateral or hexahedral meshes wecannot have nablaVℎ = 119875ℎ119876ℎnablaVℎ and hence the coercivity failsOne needs to modify the finite element method for thesemeshes

The inf-sup condition follows exactly as in the continuoussetting using Assumption 1(ii)

Lemma 9 There exists a constant 120573 gt 0 such that

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge 1205731003817100381710038171003817120601ℎ

10038171003817100381710038170Ω 120601ℎisin 119877ℎ

(52)

Proof The proof follows exactly as in the continuous settingWe use Vℎ = 0 in the expression

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

(53)

to get

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge sup120591ℎisin119877ℎ

intΩ120591ℎ sdot 120601ℎ119889119909

1003817100381710038171003817120591ℎ1003817100381710038171003817

2

0Ω

(54)

Assumption 1(ii) then yields the final result

Hence we have the following stability and approximationresult

Theorem 10 The discrete saddle-point problem (32) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and

1003817100381710038171003817119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (55)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (32)

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(56)

Owing to Assumption 1(iii)-(iv) the discrete solutionconverges optimally to the continuous solution

32 Finite Element Approximation for the Stabilized Formula-tion (18) After stabilizing formulation (9) the bilinear form119860(sdot sdot) (18) is coercive on the whole product space 119881 times 119877Therefore we can use the standard finite element space 119878ℎto discretize the gradient of the solution for the saddle-pointproblem (18) so that 119871ℎ = 119878ℎ Thus our discrete saddle-point

Advances in Numerical Analysis 7

formulation of (18) is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎtimes[119871ℎ]119889times[119872ℎ]

119889

so that

119860 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119861 ((vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119861 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(57)

We can now verify the three conditions of well-posedness forthe discrete saddle point problem The continuity of 119860(sdot sdot)119861(sdot sdot) and ℓ(sdot) follows as in the continuous setting andcoercivity is immediate from Lemma 4 as119881ℎ times[119871ℎ]

119889sub 119881times119877

Explicit construction of basis functions of119872ℎ on a referenceelement will be given below

Assumption 1(ii) guarantees that the bilinear form 119887(sdot sdot)

satisfies the inf-sup condition as in the case of the previousfinite element method

We have thus proved the following theorem from thestandard theory of saddle-point problem (see [4])

Theorem 11 The discrete saddle-point problem (57) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and1003817100381710038171003817119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (58)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (9)1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(59)

The optimal convergence of the discrete solution is thenguaranteed by the standard approximation property of119881ℎ and119871ℎ and Assumption 1(iii)

In the following we give these basis functions for lin-ear simplicial finite elements in two and three dimensions[18] The parallelotope case is handled by using the tensorproduct construction of the one-dimensional basis functionspresented in [20 26] These basis functions are very useful inthe mortar finite element method and extensively studied inone- and two-dimensional cases [16 23 26]

Note that we have imposed the condition dim119872ℎ =

dim 119871ℎ to get that our Grammatrix is a square matrixThere-fore the local basis functions for linearbilineartrilinearfinite spaces are associated with the vertices of trianglestetrahedra quadrilaterals and hexahedra For example forthe reference triangle = (119909 119910) 0 lt 119909 0 lt 119910 119909 + 119910 lt 1we have

1205831 = 3 minus 4119909 minus 4119910 1205832 = 4119909 minus 1

1205833 = 4119910 minus 1

(60)

where the basis functions 1205831 1205832 and 1205833 are associated withthree vertices (0 0) (1 0) and (0 1) of the reference triangle

For the reference tetrahedron = (119909 119910 119911) 0 lt 119909 0 lt

119910 0 lt 119911 119909 + 119910 + 119911 lt 1 we have

1205831 = 4 minus 5119909 minus 5119910 minus 5119911 1205832 = 5119909 minus 1

1205833 = 5119910 minus 1 1205834 = 5119911 minus 1

(61)

where the basis functions 1205831 1205832 1205833 and 1205834 associated withfour vertices (0 0 0) (1 0 0) (0 1 0) and (0 0 1) of thereference tetrahedron (see also [17 18]) Note that the basisfunctions are constructed in such a way that the biorthogo-nality condition is satisfied on the reference element

The global basis functions for the test space are con-structed by gluing the local basis functions together followingexactly the same procedure of constructing global finiteelement basis functions from the local ones These globalbasis functions then satisfy the condition of biorthogonalitywith global finite element basis functions of 119878ℎ These basisfunctions also satisfy Assumption 1(i)ndash(iv) (see [17 18])

33 Static Condensation Here we want to eliminate 120590ℎ andLagrange multiplier 120601

ℎfrom the saddle-point problem (57)

We make use of the operator 119876ℎ defined previously Usingthe biorthogonality relation between the basis functions of 119871ℎand 119872ℎ the action of operator 119876ℎ on a function V isin 119871

2(Ω)

can be written as

119876ℎV =119899

sum

119894=1

intΩ120583119894V 119889119909

119888119894

120593119894 (62)

which tells that the operator119876ℎ is local in the sense to be givenbelow (see also [27])

Using the property of operator 119876ℎ we can eliminate thedegrees of freedom corresponding to 120590ℎ and 120601ℎ so that ourproblem is to find 119906ℎ isin 119878ℎ such that

119869 (119906ℎ) = minVℎisin119881ℎ

119869 (Vℎ) (63)

where

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ) (64)

for the nonstabilized formulation (9) and

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ω

+1003817100381710038171003817119876ℎ (nablaVℎ) minus nablaVℎ

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ)

(65)

for the stabilized formulation (18)Let the bilinear form 119860(sdot sdot) be defined as

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909 (66)

for the non-stabilized formulation (9) and

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909

+ intΩ

(120590ℎ minus nabla119906ℎ) sdot (120591ℎ minus nablaVℎ) 119889119909

(67)

8 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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

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Journal of

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Numerical Analysis

using a proper ordering the support of a global basis function120593119894 of119871ℎ is the same as that of the global basis function120583119894 of119872ℎfor 119894 = 1 119899 However the global basis functions 1205831 120583119899of119872ℎ are not continuous on interelement boundaries

Now we show that119872ℎ satisfies Assumption 1(i)ndash(iii) Asthe first assumption is satisfied by construction we considerthe second assumption Let 120601ℎ = sum

119899

119896=1119886119896120601119896 isin 119878ℎ and set 120583ℎ =

sum119899

119896=1119886119896120583119896 isin 119872ℎ By using the quasibiorthogonality relation

(1) and the quasiuniformity assumption we get

intΩ

120601ℎ120583ℎ119889x =119899

sum

119894119895=1

119886119894119886119895 intΩ

120601119894120583119895119889x

=

119899

sum

119894=1

1198862

119894119888119894 ge 119862

119899

sum

119894=1

1198862

119894ℎ119889

119894ge 119862

1003817100381710038171003817120601ℎ1003817100381710038171003817

2

0Ω

(45)

where ℎ119894 denotes the meshsize at 119894th vertex Taking intoaccount the fact that 120601ℎ

2

0Ωequiv 120583ℎ

2

0Ωequiv sum119899

119894=11198862

119894ℎ119889

119894 we

find that Assumption 1(ii) is satisfied Since the sum of thelocal basis functions of 119872ℎ is one Assumption 1(iii) can beproved as in [23 25] Assumption 1(iv) follows by standardarguments

There are (119889 + 2) local basis functions of 119871ℎ where (119889 +1) of them are associated with vertices and one is associatedwith the element barycenter Then the local basis functionsof119872ℎ are also divided into two parts (119889 + 2) basis functionsassociated with the vertices and 1 basis function associatedwith the element Let 120583119879 be the local basis function associatedwith the element barycenter of 119879 isin Tℎ for119872ℎ Let 119884ℎ be thespace of piecewise constant functions with respect toTℎ and119875ℎ 119871

2(Ω) rarr 119884ℎ is defined as

119875ℎ119908|119879 =

int119879119908120583119879119889119909

int119879120583119879119889119909

(46)

Then 119875ℎ is a projection operator onto 119884ℎ Moreover

nablaVℎ = 119875ℎ119876ℎnablaVℎ (47)

and this implies that1003817100381710038171003817nablaVℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnablaVℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnablaVℎ

10038171003817100381710038170Ω (48)

This yields the following coercivity result

Lemma 7 The bilinear form 119886(sdot sdot) is coercive on the kernelspace 119870ℎ In fact

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ)) ge1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119906ℎ

1003817100381710038171003817

2

0Ω)

(119906ℎ120590ℎ) isin 119870ℎ

(49)

Proof Since 120590ℎ = 119876ℎnabla119906ℎ on the kernel space 119870ℎ we have

119886 ((119906ℎ120590ℎ) (119906ℎ120590ℎ))

=1

2(1003817100381710038171003817120590ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817119876ℎnabla119906ℎ

1003817100381710038171003817

2

0Ω)

(50)

The proof follows from the result1003817100381710038171003817nabla119906ℎ

10038171003817100381710038170Ω=1003817100381710038171003817119875ℎ119876ℎnabla119906ℎ

10038171003817100381710038170Ωle1003817100381710038171003817119876ℎnabla119906ℎ

10038171003817100381710038170Ω (51)

Remark 8 Using a quadrilateral or hexahedral meshes wecannot have nablaVℎ = 119875ℎ119876ℎnablaVℎ and hence the coercivity failsOne needs to modify the finite element method for thesemeshes

The inf-sup condition follows exactly as in the continuoussetting using Assumption 1(ii)

Lemma 9 There exists a constant 120573 gt 0 such that

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge 1205731003817100381710038171003817120601ℎ

10038171003817100381710038170Ω 120601ℎisin 119877ℎ

(52)

Proof The proof follows exactly as in the continuous settingWe use Vℎ = 0 in the expression

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

(53)

to get

sup(Vℎ120591ℎ)isin119881ℎtimes119877ℎ

119887 ((Vℎ 120591ℎ) 120601ℎ)

radic1003817100381710038171003817Vℎ

1003817100381710038171003817

2

1Ω+1003817100381710038171003817120591ℎ

1003817100381710038171003817

2

0Ω

ge sup120591ℎisin119877ℎ

intΩ120591ℎ sdot 120601ℎ119889119909

1003817100381710038171003817120591ℎ1003817100381710038171003817

2

0Ω

(54)

Assumption 1(ii) then yields the final result

Hence we have the following stability and approximationresult

Theorem 10 The discrete saddle-point problem (32) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and

1003817100381710038171003817119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (55)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (32)

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(56)

Owing to Assumption 1(iii)-(iv) the discrete solutionconverges optimally to the continuous solution

32 Finite Element Approximation for the Stabilized Formula-tion (18) After stabilizing formulation (9) the bilinear form119860(sdot sdot) (18) is coercive on the whole product space 119881 times 119877Therefore we can use the standard finite element space 119878ℎto discretize the gradient of the solution for the saddle-pointproblem (18) so that 119871ℎ = 119878ℎ Thus our discrete saddle-point

Advances in Numerical Analysis 7

formulation of (18) is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎtimes[119871ℎ]119889times[119872ℎ]

119889

so that

119860 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119861 ((vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119861 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(57)

We can now verify the three conditions of well-posedness forthe discrete saddle point problem The continuity of 119860(sdot sdot)119861(sdot sdot) and ℓ(sdot) follows as in the continuous setting andcoercivity is immediate from Lemma 4 as119881ℎ times[119871ℎ]

119889sub 119881times119877

Explicit construction of basis functions of119872ℎ on a referenceelement will be given below

Assumption 1(ii) guarantees that the bilinear form 119887(sdot sdot)

satisfies the inf-sup condition as in the case of the previousfinite element method

We have thus proved the following theorem from thestandard theory of saddle-point problem (see [4])

Theorem 11 The discrete saddle-point problem (57) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and1003817100381710038171003817119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (58)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (9)1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(59)

The optimal convergence of the discrete solution is thenguaranteed by the standard approximation property of119881ℎ and119871ℎ and Assumption 1(iii)

In the following we give these basis functions for lin-ear simplicial finite elements in two and three dimensions[18] The parallelotope case is handled by using the tensorproduct construction of the one-dimensional basis functionspresented in [20 26] These basis functions are very useful inthe mortar finite element method and extensively studied inone- and two-dimensional cases [16 23 26]

Note that we have imposed the condition dim119872ℎ =

dim 119871ℎ to get that our Grammatrix is a square matrixThere-fore the local basis functions for linearbilineartrilinearfinite spaces are associated with the vertices of trianglestetrahedra quadrilaterals and hexahedra For example forthe reference triangle = (119909 119910) 0 lt 119909 0 lt 119910 119909 + 119910 lt 1we have

1205831 = 3 minus 4119909 minus 4119910 1205832 = 4119909 minus 1

1205833 = 4119910 minus 1

(60)

where the basis functions 1205831 1205832 and 1205833 are associated withthree vertices (0 0) (1 0) and (0 1) of the reference triangle

For the reference tetrahedron = (119909 119910 119911) 0 lt 119909 0 lt

119910 0 lt 119911 119909 + 119910 + 119911 lt 1 we have

1205831 = 4 minus 5119909 minus 5119910 minus 5119911 1205832 = 5119909 minus 1

1205833 = 5119910 minus 1 1205834 = 5119911 minus 1

(61)

where the basis functions 1205831 1205832 1205833 and 1205834 associated withfour vertices (0 0 0) (1 0 0) (0 1 0) and (0 0 1) of thereference tetrahedron (see also [17 18]) Note that the basisfunctions are constructed in such a way that the biorthogo-nality condition is satisfied on the reference element

The global basis functions for the test space are con-structed by gluing the local basis functions together followingexactly the same procedure of constructing global finiteelement basis functions from the local ones These globalbasis functions then satisfy the condition of biorthogonalitywith global finite element basis functions of 119878ℎ These basisfunctions also satisfy Assumption 1(i)ndash(iv) (see [17 18])

33 Static Condensation Here we want to eliminate 120590ℎ andLagrange multiplier 120601

ℎfrom the saddle-point problem (57)

We make use of the operator 119876ℎ defined previously Usingthe biorthogonality relation between the basis functions of 119871ℎand 119872ℎ the action of operator 119876ℎ on a function V isin 119871

2(Ω)

can be written as

119876ℎV =119899

sum

119894=1

intΩ120583119894V 119889119909

119888119894

120593119894 (62)

which tells that the operator119876ℎ is local in the sense to be givenbelow (see also [27])

Using the property of operator 119876ℎ we can eliminate thedegrees of freedom corresponding to 120590ℎ and 120601ℎ so that ourproblem is to find 119906ℎ isin 119878ℎ such that

119869 (119906ℎ) = minVℎisin119881ℎ

119869 (Vℎ) (63)

where

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ) (64)

for the nonstabilized formulation (9) and

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ω

+1003817100381710038171003817119876ℎ (nablaVℎ) minus nablaVℎ

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ)

(65)

for the stabilized formulation (18)Let the bilinear form 119860(sdot sdot) be defined as

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909 (66)

for the non-stabilized formulation (9) and

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909

+ intΩ

(120590ℎ minus nabla119906ℎ) sdot (120591ℎ minus nablaVℎ) 119889119909

(67)

8 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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

Advances in Numerical Analysis 7

formulation of (18) is to find (119906ℎ120590ℎ120601ℎ) isin 119881ℎtimes[119871ℎ]119889times[119872ℎ]

119889

so that

119860 ((119906ℎ120590ℎ) (Vℎ 120591ℎ)) + 119861 ((vℎ 120591ℎ) 120601ℎ) = ℓ (Vℎ)

(Vℎ 120591ℎ) isin 119881ℎ times [119871ℎ]119889

119861 ((119906ℎ120590ℎ) 120595ℎ) = 0 120595ℎisin [119872ℎ]

119889

(57)

We can now verify the three conditions of well-posedness forthe discrete saddle point problem The continuity of 119860(sdot sdot)119861(sdot sdot) and ℓ(sdot) follows as in the continuous setting andcoercivity is immediate from Lemma 4 as119881ℎ times[119871ℎ]

119889sub 119881times119877

Explicit construction of basis functions of119872ℎ on a referenceelement will be given below

Assumption 1(ii) guarantees that the bilinear form 119887(sdot sdot)

satisfies the inf-sup condition as in the case of the previousfinite element method

We have thus proved the following theorem from thestandard theory of saddle-point problem (see [4])

Theorem 11 The discrete saddle-point problem (57) has aunique solution (119906ℎ120590ℎ120601ℎ) isin 119881ℎ times [119871ℎ]

119889times [119872ℎ]

119889 and1003817100381710038171003817119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601ℎ

10038171003817100381710038170Ωle ℓ0Ω (58)

Moreover the following estimate holds for the solution (119906120590

120601) isin 119881 times 119877 times 119877 of (9)1003817100381710038171003817119906 minus 119906ℎ

10038171003817100381710038171Ω+1003817100381710038171003817120590 minus 120590ℎ

10038171003817100381710038170Ω+1003817100381710038171003817120601 minus 120601ℎ

10038171003817100381710038170Ω

le 119862( infVℎisin119881ℎ

1003817100381710038171003817119906 minus 119906ℎ10038171003817100381710038171Ω

+ inf120591ℎisin[119871ℎ]119889

1003817100381710038171003817120590 minus 120591ℎ10038171003817100381710038170Ω

+ inf120595ℎisin[119872ℎ]119889

1003817100381710038171003817120601 minus 120595ℎ10038171003817100381710038170Ω

)

(59)

The optimal convergence of the discrete solution is thenguaranteed by the standard approximation property of119881ℎ and119871ℎ and Assumption 1(iii)

In the following we give these basis functions for lin-ear simplicial finite elements in two and three dimensions[18] The parallelotope case is handled by using the tensorproduct construction of the one-dimensional basis functionspresented in [20 26] These basis functions are very useful inthe mortar finite element method and extensively studied inone- and two-dimensional cases [16 23 26]

Note that we have imposed the condition dim119872ℎ =

dim 119871ℎ to get that our Grammatrix is a square matrixThere-fore the local basis functions for linearbilineartrilinearfinite spaces are associated with the vertices of trianglestetrahedra quadrilaterals and hexahedra For example forthe reference triangle = (119909 119910) 0 lt 119909 0 lt 119910 119909 + 119910 lt 1we have

1205831 = 3 minus 4119909 minus 4119910 1205832 = 4119909 minus 1

1205833 = 4119910 minus 1

(60)

where the basis functions 1205831 1205832 and 1205833 are associated withthree vertices (0 0) (1 0) and (0 1) of the reference triangle

For the reference tetrahedron = (119909 119910 119911) 0 lt 119909 0 lt

119910 0 lt 119911 119909 + 119910 + 119911 lt 1 we have

1205831 = 4 minus 5119909 minus 5119910 minus 5119911 1205832 = 5119909 minus 1

1205833 = 5119910 minus 1 1205834 = 5119911 minus 1

(61)

where the basis functions 1205831 1205832 1205833 and 1205834 associated withfour vertices (0 0 0) (1 0 0) (0 1 0) and (0 0 1) of thereference tetrahedron (see also [17 18]) Note that the basisfunctions are constructed in such a way that the biorthogo-nality condition is satisfied on the reference element

The global basis functions for the test space are con-structed by gluing the local basis functions together followingexactly the same procedure of constructing global finiteelement basis functions from the local ones These globalbasis functions then satisfy the condition of biorthogonalitywith global finite element basis functions of 119878ℎ These basisfunctions also satisfy Assumption 1(i)ndash(iv) (see [17 18])

33 Static Condensation Here we want to eliminate 120590ℎ andLagrange multiplier 120601

ℎfrom the saddle-point problem (57)

We make use of the operator 119876ℎ defined previously Usingthe biorthogonality relation between the basis functions of 119871ℎand 119872ℎ the action of operator 119876ℎ on a function V isin 119871

2(Ω)

can be written as

119876ℎV =119899

sum

119894=1

intΩ120583119894V 119889119909

119888119894

120593119894 (62)

which tells that the operator119876ℎ is local in the sense to be givenbelow (see also [27])

Using the property of operator 119876ℎ we can eliminate thedegrees of freedom corresponding to 120590ℎ and 120601ℎ so that ourproblem is to find 119906ℎ isin 119878ℎ such that

119869 (119906ℎ) = minVℎisin119881ℎ

119869 (Vℎ) (63)

where

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ) (64)

for the nonstabilized formulation (9) and

119869 (Vℎ) =1003817100381710038171003817nabla (119876ℎ (nablaVℎ))

1003817100381710038171003817

2

0Ω

+1003817100381710038171003817119876ℎ (nablaVℎ) minus nablaVℎ

1003817100381710038171003817

2

0Ωminus 2ℓ (Vℎ)

(65)

for the stabilized formulation (18)Let the bilinear form 119860(sdot sdot) be defined as

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909 (66)

for the non-stabilized formulation (9) and

119860 (119906ℎ Vℎ) = intΩ

120590ℎ 120591ℎ119889119909

+ intΩ

(120590ℎ minus nabla119906ℎ) sdot (120591ℎ minus nablaVℎ) 119889119909

(67)

8 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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 Advances in Numerical Analysis

for the stabilized formulation (18) with 120590ℎ = 119876ℎ(nabla119906ℎ)

and 120591ℎ = 119876ℎ(nablaVℎ) Due to the biorthogonality or quasi-biorthogonality condition the action of 119876ℎ is computed byinverting a diagonal matrix

Since the bilinear form119860(sdot sdot) is symmetric theminimiza-tion problem (63) is equivalent to the variational problem offinding 119906ℎ isin 119878ℎ such that [3 15]

119860 (119906ℎ Vℎ) = ℓ (Vℎ) Vℎ isin 119881ℎ (68)

The coercivity of119860(sdot sdot) and the stability result of Lemma 9immediately yield the coercivity of the bilinear form119860(sdot sdot) onthe space 119881ℎ with the norm radic 119906ℎ

2

0Ω+ nabla119876ℎ(nabla119906ℎ)

2

0Ωfor

119906ℎ isin 119881ℎ That is

119860 (119906ℎ 119906ℎ) ge 119862 (1003817100381710038171003817119906ℎ

1003817100381710038171003817

2

0Ω+1003817100381710038171003817nabla119876ℎ (nabla119906ℎ)

1003817100381710038171003817

2

0Ω) 119906ℎ isin 119881ℎ

(69)

4 Conclusion

We have considered two three-field formulations of Pois-son problem and shown the well-posedness of them Wehave proposed efficient finite element schemes for both ofthese formulations one for the stabilized three-field for-mulation based on biorthogonal systems and the other forthe non-stabilized three-field formulation based on quasi-biorthogonal systems Optimal a priori error estimates areproved for both approaches and a reduced discrete problemis presented

Appendix

If we want to enrich the finite element space with thebubble function to attain stability it is not possible to use abiorthogonal system We consider the local basis functionsof 119871ℎ for the reference triangle = (119909 119910) 0 le 119909 0 le

119910 119909+119910 le 1 Let 1205931 1205934 be defined as in (39)Then usingthe biorthogonal relation for the four local basis functions1205831 1205834 of 119872ℎ with the four basis functions 1205931 1205934of 119871ℎ for the triangle andsum

3

119894=1120593119894 = 1 on 1205834 should satisfy

int

1205834(

3

sum

119894=1

120593119894)119889119909 = int

1205834 119889119909 = 0 (A1)

since it is associated with the barycenter of the triangle Thisleads to instability in the formulation as we suggest that thefunction 1205834 should have nonzero integral over (see (46))This explains why we cannot use a biorthogonal system Wehave a similar issue in the three-dimensional case assum4

119894=1120593119894 =

1 resulting in

int

1205835 119889119909 = 0 (A2)

if we impose biorthogonality

Acknowledgments

Support from the new staff Grant of the University ofNewcastle is gratefully acknowledged The author is grateful

to the anonymous referees for their valuable suggestions toimprove the quality of the earlier version of this work

References

[1] Y Achdou C Bernardi and F Coquel ldquoA priori and a posteriorianalysis of finite volume discretizations of Darcyrsquos equationsrdquoNumerische Mathematik vol 96 no 1 pp 17ndash42 2003

[2] P B Bochev and C R Dohrmann ldquoA computational study ofstabilized low-order1198620 finite element approximations of Darcyequationsrdquo Computational Mechanics vol 38 no 4-5 pp 310ndash333 2006

[3] D Braess Finite Elements Theory Fast Solver and Applicationsin Solid Mechanics Cambridge University Press CambridgeUK 2nd edition 2001

[4] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[5] F Brezzi T J R Hughes L D Marini and A Masud ldquoMixeddiscontinuous Galerkin methods for Darcy flowrdquo Journal ofScientific Computing vol 22-23 pp 119ndash145 2005

[6] K AMardal X-C Tai and RWinther ldquoA robust finite elementmethod for Darcy-Stokes flowrdquo SIAM Journal on NumericalAnalysis vol 40 no 5 pp 1605ndash1631 2002

[7] A Masud and T J R Hughes ldquoA stabilized mixed finiteelement method for Darcy flowrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 39-40 pp 4341ndash43702002

[8] J M Urquiza D NrsquoDri A Garon and M C Delfour ldquoAnumerical study of primal mixed finite element approximationsof Darcy equationsrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 22 no 8 pp 901ndash915 2006

[9] B P Lamichhane and E P Stephan ldquoA symmetric mixed finiteelement method for nearly incompressible elasticity based onbiorthogonal systemsrdquo Numerical Methods for Partial Differen-tial Equations vol 28 no 4 pp 1336ndash1353 2012

[10] S C Brenner and L R ScottTheMathematicalTheory of FiniteElement Methods Springer New York NY USA 1994

[11] B P Lamichhane B D Reddy and B I Wohlmuth ldquoCon-vergence in the incompressible limit of finite element approx-imations based on the Hu-Washizu formulationrdquo NumerischeMathematik vol 104 no 2 pp 151ndash175 2006

[12] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[13] K Washizu Variational Methods in Elasticity and PlasticityPergamon Press 3rd edition 1982

[14] D N Arnold and F Brezzi ldquoSome new elements for theReissner-Mindlin plate modelrdquo in Boundary Value Problems forPartial Differerntial Equations and Applications pp 287ndash292Masson Paris France 1993

[15] P G Ciarlet The Finite Element Method for Elliptic ProblemsNorth-Holland Amsterdam The Netherlands 1978

[16] B P Lamichhane Higher order mortar finite elements withdual lagrange multiplier spaces and applications [PhD thesis]University of Stuttgart 2006

[17] B P Lamichhane ldquoA stabilized mixed finite element methodfor the biharmonic equation based on biorthogonal systemsrdquoJournal of Computational and AppliedMathematics vol 235 no17 pp 5188ndash5197 2011

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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

Advances in Numerical Analysis 9

[18] B P Lamichhane ldquoA mixed finite element method for thebiharmonic problem using biorthogonal or quasi-biorthogonalsystemsrdquo Journal of Scientific Computing vol 46 no 3 pp 379ndash396 2011

[19] B P Lamichhane S Roberts and L Stals ldquoA mixed finiteelement discretisation of thin-plate splinesrdquo ANZIAM Journalvol 52 pp C518ndashC534 2011 Proceedings of the 15th BiennialComputational Techniques and Applications Conference

[20] B P Lamichhane and B IWohlmuth ldquoBiorthogonal bases withlocal support and approximation propertiesrdquo Mathematics ofComputation vol 76 no 257 pp 233ndash249 2007

[21] A Galantai Projectors and Projection Methods vol 6 KluwerAcademic Publishers Dordrecht The Netherlands 2004

[22] D B Szyld ldquoThe many proofs of an identity on the norm ofoblique projectionsrdquo Numerical Algorithms vol 42 no 3-4 pp309ndash323 2006

[23] C Kim R D Lazarov J E Pasciak and P S VassilevskildquoMultiplier spaces for themortar finite elementmethod in threedimensionsrdquo SIAM Journal on Numerical Analysis vol 39 no 2pp 519ndash538 2001

[24] D N Arnold F Brezzi and M Fortin ldquoA stable finite elementfor the Stokes equationsrdquo Calcolo vol 21 no 4 pp 337ndash3441984

[25] B P Lamichhane ldquoA mixed finite element method for non-linear and nearly incompressible elasticity based on biorthog-onal systemsrdquo International Journal for Numerical Methods inEngineering vol 79 no 7 pp 870ndash886 2009

[26] B I Wohlmuth Discretization Methods and Iterative SolversBased on Domain Decomposition vol 17 of Lecture Notes inComputational Science and Engineering Springer HeidelbergGermany 2001

[27] M Ainsworth and J T Oden A Posteriori Error Estimationin Finite Element Analysis Wiley-Interscience New York NYUSA 2000

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