research article analysis of subgrid stabilization method...

Research ArticleAnalysis of Subgrid Stabilization Method forStokes-Darcy Problems

Kamel Nafa

Department of Mathematics and Statistics Sultan Qaboos University College of Science PO Box 36Al-Khoudh 123 Muscat Oman

Correspondence should be addressed to Kamel Nafa nkamelsqueduom

Received 27 April 2016 Revised 31 July 2016 Accepted 16 August 2016

Academic Editor Weimin Han

Copyright copy 2016 Kamel Nafa This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A number of techniques used as remedy to the instability of the Galerkin finite element formulation for Stokes like problemsare found in the literature In this work we consider a coupled Stokes-Darcy problem where in one part of the domain the fluidmotion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and theconservation of mass Such systems can be discretized by heterogeneous mixed finite elements in the two parts A better methodfrom a computational point of view consists in using a unified approach on both subdomains Here the coupled Stokes-Darcyproblem is analyzed using equal-order velocity and pressure approximation combined with subgrid stabilizationWe prove that theobtained finite element solution is stable and converges to the classical solution with optimal rates for both velocity and pressure

1 Introduction

The transport of substances between surface water andgroundwater has attracted a lot of interest into the coupling ofviscous flows and porous media flows [1ndash5] In this work weconsider coupled problems in fluid dynamics where the fluidin one part of the domain is described by the Stokes equationsand in the other porous media part by the Darcy equationand mass conservation Velocity and pressure on these twoparts are mutually coupled by interface conditions derivedin [6] Such systems can be discretized by heterogeneousfinite elements as analyzed by Layton et al [1] In morerecent works unified approaches become more popular Forinstance discontinuous Galerkin methods were analyzed byGirault and Riviere [3] mixed methods by Karper et al [4]and local pressure gradient stabilized methods by Braack andNafa [7]

In this work we consider the 1198712-formulation of the

coupled Stokes-Darcy problem as in [4] but we discretizeby equal-order finite elements and use subgrid method andgrad-div term to stabilize the pressure and control the natural1198671(div) velocity norm on the Darcy subdomain

2 Formulations of the Stokes-DarcyCoupled Equations

21 Model Equations Let Ω sub 119877119889 119889 = 2 or 3 be a

bounded region with Lipschitz boundary 120597ΩΩ119878andΩ

119863are

respectively the fluid and porous media subdomains of Ω

such that Ω119878cap Ω119863

= 0 The subdomains have a commoninterface Γ = Ω

119878cap Ω119863 We denote by k = (k

119878 k119863) the

fluid velocity and by 119901 = (119901119878 119901119863) the fluid pressure where

k119894= k|Ω119894 119901119894= 119901|Ω119894 119894 = 119878 119863 The flow in the domain Ω

119878is

assumed to be of Stokes type and governed by the equations

minus2] div (119863 (k119878)) + nabla119901

119878= f in Ω

119878

div k119878= 0 in Ω

119878

(1)

with symmetric strain tensor 119863(k119878) = (12)(nablak

119878+ nablak119879119878)

external force f and constant viscosity ] gt 0 In the porousregion Ω

119863the filtration of an incompressible flow through

porous media is described by Darcy equations

119870minus1k119863+ nabla119901119863

= f in Ω119863

div k119863

= 119892 in Ω119863

(2)

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2016 Article ID 7389102 9 pageshttpdxdoiorg10115520167389102

2 Advances in Numerical Analysis

where the permeability 119870 = 119870(119909) is a positive definitesymmetric tensor and 119892 denotes an external Darcy force

22 Boundary Conditions On Γ119878

= 120597Ω119878 Γ we prescribe

homogeneous Dirichlet conditions for the velocity k119878

k119878= 0 on Γ

119878 (3)

The boundary of Ω119863is split into three parts 120597Ω

119863= Γ cup

Γ1198631

cupΓ1198632

We prescribe zero flux on Γ1198631

and a homogeneousDirichlet condition for the pressure on Γ

1198632

k119863sdot n119863

= 0 on Γ1198631

119901119863

= 0 on Γ1198632

(4)

where n119863denotes the outer normal vector on the boundary

pointing from Ω119863into Ω

119878 This boundary condition ensures

a zero mass flux

23 The Beavers-Joseph-Saffman Condition The flows in Ω119878

and Ω119863

are coupled across the interface Γ Conditionsdescribing the interaction of the flows are as follows [6 8]

(i) The continuity of the normal velocity

k119878sdot n119878= minusk119863sdot n119863 on Γ (5)

(ii) The balance of normal forces

minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863 on Γ (6)

(iii) The Beavers-Joseph-Saffman condition written interms of the strain tensor

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591 (7)

where = ]119870120591 sdot 120591 and 120572 is a dimensionless param-eter to be determined experimentally this conditionrelating the tangential slip velocity k

119878sdot 120591 to the normal

derivative of the tangential velocity component in theStokes region

3 Variational Formulation

As variational formulation we consider the so-called 1198712-

formulation used by Karper et al [4] and recently by [9 10]We denote

(kw)Ω

= intΩ

kw 119889119909 kw isin 1198712(Ω)119889

⟨V 119908⟩Γ = intΓ

V119908119889119904 V 119908 isin 1198712(Γ)

(8)

where 1198712(Ω) and119867

1(Ω) denote the usual Sobolev spaces

Next we define the spaces

H1Γ119878(Ω119878) = w isin (119867

1(Ω119878))119889

| w = 0 on Γ119878

H1 (div Ω119863) = w isin 119871

2(Ω119863)119889

| divw isin 1198712(Ω119863)

H1Γ1198631

(Ω119863) = w isin H1 (div Ω

119863) | w sdot n

119863= 0 on Γ

119863

(9)

Then multiplying the Stokes equations (1) by the test func-tionsw

119878isin H1Γ119878(Ω119878) 119902119878isin 1198712(Ω119878) respectively and integrating

by part on the domainΩ119878 we obtain

(2]119863(k119878) 119863 (w

119878))Ω119878

minus ⟨2]119863(k119878)n119878w119878⟩

minus (119901119878 divw

119878)Ω119878

+ ⟨119901119878w119878sdot n119878⟩Γ= (f w

119878)Ω119878

(div k119878 119902119878)Ω119878

= 0

(10)

Using the decompositionw119878= (w119878sdotn119878)n119878+(w119878sdot120591)120591 the fluid

normal stress condition (6) and the BJS interface condition(7) in (10) we obtain the weak formulation of the Stokesequations find k

119878isin H1Γ119878(Ω119878) 119901119878isin 1198712(Ω119878) such that

(2]119863(k119878) 119863 (w

119878))Ω119878

+]120572radic

⟨k119878sdot 120591w119878sdot 120591⟩Γ

minus (119901119878 divw

119878)Ω119878

+ ⟨119901119863w119878sdot n119878⟩Γ= (f w

119878)Ω119878

(div k119878 119902119878)Ω119878

= 0

(11)

forallw119878isin H1Γ119878(Ω119878) 119902119878isin 1198712(Ω119878)

Similarly taking 120575 gt 0 and testing theDarcy equations (2)by w119863

isin H1Γ1198631

(Ω119863) 119902119863

isin 1198712(Ω119863) respectively together with

the weighted grad-div term we obtain the weak formulationof Darcy equations find k

119863isin H1Γ119863(Ω119863) 119901119863

isin 1198671

1198632(Ω119863) such

that

(119870minus1k119863w119863)Ω119863

+ (nabla119901119863w119863)Ω119863

+ 120575 (div k119863 divw

119863)Ω119863

= 120575 (119892 divw119863)Ω119863

minus (k119863 nabla119902119863)Ω119863

+ ⟨k119863sdot n119863 119902119863⟩Γ= (119892 119902

119863)Ω119863

(12)

Summing up (11) and (12) the weak form of the coupledproblem is given by the following find k

119878isin H1Γ119878(Ω119878) 119901119878

isin

1198712(Ω119878) k119863

isin H1Γ119863(Ω119863) and 119901

119863isin 1198712(Ω119863) such that

(2]119863(k119878) 119863 (w

119878))Ω119878

minus (119901119878 divw

119878)Ω119878

+ (119870minus1k119863w119863)Ω119863

+ (nabla119901119863w119863)Ω119863

+ 120575 (div k119863 divw

119863)Ω119863

+]120572radic

⟨k119878sdot 120591w119878sdot 120591⟩Γ

+ ⟨119901119863w119878sdot n119878⟩Γ= (f w

119878)Ω119878

+ 120575 (119892 divw119863)Ω119863

(div k119878 119902119878)Ω119878

minus (k119863 nabla119902119863)Ω119863

minus ⟨k119878sdot n119878 119902119863⟩Γ= (119892 119902

119863)Ω119863

(13)

Advances in Numerical Analysis 3

To analyze the weak formulation of the coupled problemwe introduce the following spaces

V = k isin H (div Ω) | k119878isin (1198671(Ω119878))119889

k119878

= 0 on Γ119878 k sdot n

119863= 0 on Γ

1198631

119876 = 119902 isin 1198712(Ω) | 119901

119863isin 1198671(Ω119863) 119901 = 0 isin Γ

1198632

119883 = V times 119876

(14)

The velocity and pressure spaces V and 119876 are equipped withthe natural norms

kV = (nablak2Ω119878

+ k2Ω119863

+ div k2Ω119863

)12

10038171003817100381710038171199011003817100381710038171003817119876

= (1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863)12

(15)

Further due to the positive definiteness of119870with respectto the 119871

2(Ω119863) norm sdot

Ω119863 there exist positive real numbers

1198961and 1198962such that

1198961 k2

Ω119863le (119870minus1k k)

Ω119863

le 1198962 k2

Ω119863 forallk isin V (16)

Next we define the bilinear forms for k = (k119878 k119863) w =

(w119878w119863) in V and 119901 = (119901

119878 119901119863) 119902 = (119902

119878 119902119863) in 119876 on the two

parts of the domain by

A119878(k 119901w 119902) = (2]119863(k

119878) 119863 (w

119878))Ω119878

+]120572radic

⟨k119878sdot 120591w119878sdot 120591⟩Γ

minus (119901119878 divw

119878)Ω119878

+ (div k119878 119902119878)Ω119878

A119863(k 119901w 119902) = (119870

minus1k119863w119863)Ω119863

+ 120575 (div k119863 divw

119863)Ω119863

+ (nabla119901119863w119863)Ω119863

minus (k119863 nabla119902119863)Ω119863

(17)

Hence the bilinear form for the coupled problem is the sumof A119878(k 119901w 119902) A

119863(k 119901w 119902) and terms to enforce the

continuity of the normal part of the velocities across theinterface

A (k 119901w 119902) = A119878(k 119901w 119902) +A

119863(k 119901w 119902)

+ ⟨119901119863w119878sdot n119878⟩Γminus ⟨119902119863 k119878sdot n119878⟩Γ

(18)

Assuming for simplicity that f and 119892 are extended by zero tothe whole domain the variational formulation of the coupledStokes-Darcy system in compact form reads as follows find(k 119901) isin V times 119876 solution of

A (k 119901w 119902) = F (w 119902) forall (w 119902) isin V times 119876 (19)

with

F (w 119902) = (f w119878)Ω+ (119892 119902

119863)Ω+ 120575 (119892 divw

119863)Ω (20)

It can easily be shown that a sufficiently regular solution(k 119901) isin V times 119876 of (19) such that k

119878isin 1198672(Ω119878)119889 k119863

isin

1198671(Ω119863)119889 119901 isin 119867

1(Ω119878cup Ω119863) is also a classical solution of

(1) and (2) We note that there is an alternative variationalformulation to the one given here called119867(div)-formulationThe latter uses the term minus(119901 divw)

Ω119863+ (div k 119902)

Ω119863instead

of (w nabla119901)Ω119863

minus (k nabla119902)Ω119863

[4]The existence and uniqueness of the solution of problem

(19) follows from Brezzirsquos conditions for saddle point prob-lems [11] namely

119860 (k 119901 k 119901) ge lsaquoklsaquo2V

forallV isin V gt 0

(21)

inf119902isin1198712(Ω119878)

supkisin1198671(Ω119878)119889

(div k 119902)Ω119878

nablakΩ1198781003817100381710038171003817119902

1003817100381710038171003817Ω119878

ge 120573119878 (22)

inf119902isin1198671(Ω119863)

supkisin1198712(Ω119863)119889


kΩ1198631003817100381710038171003817nabla119902

1003817100381710038171003817Ω119863

ge 120573119863 (23)

with positive constants 120573119878and 120573

119863[7]

The following lemma is needed in the analysis below andis a consequence of the continuous inf-sup conditions (23)[10]

Lemma 1 For every (v 119901) isin 119883 there is w isin V such that w119878sdot

n119878= 0 on Γ satisfying

A (k 119901w 0) ge 1198882

10038171003817100381710038171199011003817100381710038171003817

2

119876minus 1198881 k2

V

wV le 1198883

10038171003817100381710038171199011003817100381710038171003817119876

(24)

with positive constants 1198881 1198882 and 119888

3

Proof Let (k 119901) isin 119883 Then due to Stokes inf-sup conditionthere exists w

119878isin 1198671(Ω119878)119889 with w

119878= 0 on Γ

119878and w

119878sdot n = 0

on Γ such that

minus (divw119878 119901)Ω119878

=1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

1003817100381710038171003817nablaw1198781003817100381710038171003817Ω119878

le 119888119878

10038171003817100381710038171199011003817100381710038171003817Ω119878

(25)

For the Darcy equation due to the condition 119901 = 0 on Γ1198632

there exists w

119863isin 1198671(Ω119863)119889 with w

119863sdot n = 0 on Γ

1198632and Γ

such that

minus (divw119863 119901)Ω119863

=1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

1003817100381710038171003817nablaw1198631003817100381710038171003817Ω119863

le 119888119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

(26)

Define

w =

w119878

in Ω119878

w119863

in Ω119863

(27)


and thenA (k 119901w 0) = (2]119863 (k) 119863 (w))

Ω119878minus (119901 divw)

Ω119878

+ (119870minus1kw)

Ω119863

+ (nabla119901w)Ω119863

+ 120575 (div k divw)Ω119863

ge minus2] 119863 (k)Ω119878

119863 (w)Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

minus 120575 div kΩ119863 divwΩ119863

ge minus2] nablakΩ119878 nablawΩ119878+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

minus 120575 div kΩ119863 nablawΩ119863

ge minus2]119888119878 nablakΩ119878

10038171003817100381710038171199011003817100381710038171003817Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198881199011198881198631198962 kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

minus 120575119888119863 div kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

(28)

where 119888119901denote the Poincare constant

Then using Youngrsquos inequality we obtain

A (k 119901w 0) ge minus]119888119878

1205761

nablak2Ω119878

+ (1 minus ]1198881198781205761)1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus

1198881199011198881198631198962

21205762

k2Ω119863

+ (1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863

minus120575119888119863

21205763

div k2Ω119863

(29)

Choosing 1205761 1205762 1205763positive constants such that

1205761lt

1

]119888119878

1205762lt

2

1198881199011198881198631198962

1205763lt

2 minus 11988811990111988811986311989621205762

120575119888119863

(30)

we obtain the required result

A (k 119901w 0) ge 1198882

10038171003817100381710038171199011003817100381710038171003817

2

119876minus 1198881 k2

V (31)

where

1198881= max

]119888119878

1205761

1198881199011198881198631198962

21205762

120575119888119863

21205763

1198882= min1 minus ]119888

1198781205761 1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2

(32)

In addition we also have

w2

V = nablaw2

Ω119878+ w

2

Ω119863+ divw

2

Ω119863

le 1198882

119878

10038171003817100381710038171199011003817100381710038171003817

2

Ω119878+ 1198882

119863(1198882

119901+ 1)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863le 1198883

10038171003817100381710038171199011003817100381710038171003817119876

(33)

where 1198882

3= max1198882

119878 1198882

119863(1198882

119901+ 1)

4 Finite Element Discretization

LetTℎbe a shape-regular partition of quadrilaterals for 119889 =

2 or hexahedra for 119889 = 3 [12 13] The diameter of element119879 isin T

ℎwill be denoted by ℎ

119879and the global mesh size is

defined by ℎ fl maxℎ119879

119879 in Tℎ Let fl (minus1 1)

119889 bethe reference element 119865

119879the mapping from to element 119879

and 119876119903() the space of all polynomials on with maximal

degree 119903 ge 0 in each coordinateWe assume that themeshTℎ

is obtained from a coarser mesh T2ℎ

by global refinementHenceT

2ℎconsists of patches of elements ofT

ℎ We define

the finite element space

119883119903

ℎfl V isin 119862 (Ω

119878) cup 119862 (Ω

119863) V|119879

∘ 119865119879in 119876119903() forall119879 isin T

ℎ

(34)

For the discrete spaces Vℎand 119876

ℎwe use the equal-order

finite element functions that are continuous in Ω119878and Ω

119863

and piecewise polynomials of degree 119903 ge 1

Vℎ= (119883119903

ℎ)119889

cap V

119876ℎ= 119883119903

ℎcap 119876 cap 119867

1(Ω)

(35)

We define the Scott-Zhang interpolation operator whichpreserves the boundary condition [13] as 119895ℎ

119903 1198671(Ω) rarr 119883

119903

ℎ

with stability and interpolation properties respectively as10038171003817100381710038171003817nabla119895ℎ

11990312060110038171003817100381710038171003817Ω

le 119888119904

100381610038161003816100381612060110038161003816100381610038161Ω

120601 isin 1198671(Ω) (36)

10038171003817100381710038171003817120601 minus 119895ℎ

11990312060110038171003817100381710038171003817119898Ω

le 119888119894ℎ119903+1minus119898 1003816100381610038161003816120601

1003816100381610038161003816119903+1Ω

120601 isin 119867119903+1

(Ω) 119898 = 0 or 1

(37)

where 119888119894 119888119904are positive constants

We will also use the inverse inequality

( sum

119879isinTℎ

ℎ2

119879

1003817100381710038171003817nabla1206011003817100381710038171003817119879

)

12

le 119888119868

10038171003817100381710038171206011003817100381710038171003817Ω

forall120601 isin 1198671(Ω) (38)

Similarly for vector functions we define the interpolationoperator

jℎ119903 1198671(Ω)119889997888rarr (119883

119903

ℎ)119889

(39)

with interpolation and stability properties as aboveIt is known that the standard Galerkin discretizations of

theDarcy system are not stable for equal-order elementsThisinstability stems from the violation of the discrete analogue


on to the inf-sup conditionOne possibility to circumvent thiscondition is to work with a modified bilinear formA

ℎ(sdot sdot) by

adding a stabilization term Sℎ(sdot sdot) that is

Aℎ(kℎ 119901ℎw 119902) = A (k

ℎ 119901ℎw 119902) +S

ℎ(119901ℎ 119902) (40)

such that the stabilized discrete problem reads

Aℎ(kℎ 119901ℎw 119902) = F (w 119902) forall (w 119902) isin V

ℎtimes 119876ℎ (41)

Unlike in [10] where a combination of a generalized minielement and local projection (LPS) is analyzed and in [14]where a method based on two local Gauss integrals for theStokes equations is used here we will analyze the problemusing a subgrid method [12 15 16]

For this method the filter with respect to the globalLagrange interpolant 119868

2ℎ onto a coarser mesh T

2ℎis used

Defining 1205812ℎ

= 119868 minus 1198682ℎthe subgrid stabilization term reads

Sℎ(119901ℎ 119902) = sum

119872isinT2ℎ

ℎ119872

(120574nabla1205812ℎ119901ℎ nabla1205812ℎ119902)119872

119903 ge 1 (42)

where 120574 is patchwise constantA more attractive method from the computational point

is obtained using only the fine mesh with smaller stencilDefining 120581

ℎ= 119868 minus 119868

ℎthe subgrid stabilization term reads

Sℎ(119901ℎ 119902) = sum

119870isinTℎ

ℎ119870(120574nabla120581ℎ119901ℎ nabla120581ℎ119902)119870 119903 ge 2 (43)

Next we prove the stability of the discrete coupled Stokes-Darcy problem with respect to the norm

lsaquo (k 119901) lsaquoℎ= (k2V +

10038171003817100381710038171199011003817100381710038171003817

2

119876+Sℎ(119901 119901))

12

(44)

5 Stability

Theorem 2 Let Tℎbe a quasi-regular partition [13] Then

the following discrete inf-sup condition holds for some positiveconstant independent of the mesh size ℎ

inf(kℎ 119901ℎ)isinVℎtimes119876ℎ00

sup(wℎ 119902ℎ)isinVℎtimes119876ℎ00

A (kℎ 119901ℎwℎ 119902ℎ)

lsaquo (kℎ 119901ℎ) lsaquoℎlsaquo (wℎ 119902ℎ) lsaquoℎ

ge

(45)

Proof First let (kℎ 119901ℎ) isin V

ℎtimes 119876ℎ and then the diagonal

testing combined with Kornrsquos inequality and the positivity of119870minus1 give

Aℎ(kℎ 119901ℎ kℎ 119901ℎ) = A (k

ℎ 119901ℎ kℎ 119901ℎ) +Sℎ(119901ℎ 119901ℎ)

ge k2V + Sℎ(119901ℎ 119901ℎ)

(46)

In addition let w be as in Lemma 1 corresponding to(kℎ 119901ℎ) isin Vℎtimes 119876ℎ and set z = jℎ

119903w minus w Then

A (kℎ 119901ℎ jℎ119903w 0) = A (k

ℎ 119901ℎw 0) +A (k

ℎ 119901ℎ z 0)

ge 1198882

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876minus 1198881

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

+A119878(kℎ 119901ℎ z 0)

+A119863(kℎ 119901ℎ z 0)

(47)

Next we estimate A119878(kℎ 119901ℎ z 0) and A

119863(kℎ 119901ℎ z 0) as

follows

A119878(kℎ 119901ℎ z 0) = (2]119863(k

ℎ) 119863 (z))

Ω119878+ (nabla119901

ℎ z)Ω119878

+]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

(48)

where the first two terms are bounded using Cauchy inequal-ity together with the interpolation stability and inverseinequalities

10038161003816100381610038161003816(]119863(k

ℎ) 119863 (z))

Ω119878

10038161003816100381610038161003816le ] 1003817100381710038171003817119863 (k

ℎ)1003817100381710038171003817Ω119878

119863 (z)Ω119878

le ] 1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878 le ]119888

119894

1003817100381710038171003817kℎ1003817100381710038171003817V nablawΩ119878

le ]1198883119888119894

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(nabla119901ℎ z)Ω119878

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879z2119879)

12

sdot ( sum

119879isinTℎ119879subΩ119878

ℎ2

119879

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817

2

119879)

12

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879ℎ2119903

119879nablaw2

119879)

12

119888119868

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 119888119888119894119888119868 nablawΩ119878

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 1198881198881198941198881198681198883

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

(49)

The boundary term is bounded using the trace theorem andthe119867

1- stability by

100381610038161003816100381610038161003816100381610038161003816

]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

100381610038161003816100381610038161003816100381610038161003816

le 1198882

Γ

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878

le 1198882

Γ1198881199041198883

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(50)

Hence by Young inequality with

1205981=

1198882

8]1198881198941198883

1205982=

1198882radic

8]1205721198882Γ1198881199041198883

(51)

we obtain

A119878(kℎ 119901ℎ z 0) le

1198882

81198884

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876+ 1198884

1003817100381710038171003817kℎ1003817100381710038171003817

2

V (52)

where 1198884= (4(]119888

3119888119894)2+ 025(119888

2

Γ1198881199041198883)2)1198882


For the Darcy bilinear form we have

A119863(kℎ 119901ℎ z 0) = (119870

minus1kℎ z)Ω119863

+ 120575 (div kℎ div z)

Ω119863

+ (nabla119901ℎ z)Ω119863

= (119870minus1kℎ z)Ω119863

+ 120575 (div kℎ div z)

Ω119863

+ (nabla (119901ℎminus 1205812ℎ119901ℎ) z)Ω119863

+ (nabla1205812ℎ119901ℎ z)Ω119863

le10038171003817100381710038171003817119870minus1kℎ

10038171003817100381710038171003817Ω119863zΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863zΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

zΩ119863

+1003817100381710038171003817nabla1205812ℎ119901ℎ

1003817100381710038171003817Ω119863zΩ119863

le 1198962

1003817100381710038171003817kℎ1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863(1 + 119888119904) wΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 119888119904

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817Ω119863

le 11989621198881198941198883

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1205751198883(1 + 119888119904)1003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1198881198941198883

1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

+ 119888119904

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(53)

Then by Young inequality and (52) we obtain

Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

51198882

8

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 119862 (1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(54)

Scaling jℎ119903w we obtain

Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 1198621(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(55)

Choosing (wℎ 119902ℎ) = (k

ℎ 119901ℎ) + (1(1 + 119862

1))(jℎ119903w 0) we obtain

Aℎ(kℎ 119901ℎwℎ 119902ℎ) ge

1003817100381710038171003817kℎ1003817100381710038171003817

2

V +1

1 + 1198621

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus1198621

1 + 1198621

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

=1

1 + 1198621

(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +1003817100381710038171003817119901ℎ

1003817100381710038171003817

2

119876)

=1

1 + 1198621

lsaquo (kℎ 119901ℎ) lsaquo2ℎ

lsaquowℎ 119902ℎlsaquoℎle lsaquo (k

ℎ 119901ℎ) lsaquoℎ

+1

1 + 1198621

lsaquo (jℎ119903w 0) lsaquo

ℎ

le lsaquo (kℎ 119901ℎ) lsaquoℎ+ 1198622

10038171003817100381710038171003817nablajℎ119903w10038171003817100381710038171003817Ω

le 1198623lsaquo (kℎ 119901ℎ) lsaquoℎ

(56)

which implies the required result


sup(wℎ119902ℎ)isinVℎtimes119876ℎ0

Aℎ(kℎ 119901ℎwℎ 119902ℎ)


ge (57)

with = 119862minus1

3(1 + 119862

1)

6 Error Analysis

Theorem 3 Assume that the solution (v 119901) of the Stokes-Darcy problem (19) is such that (v

119878 119901119878) isin V

119878cap 119867119903+1

(Ω119878)119889times

119876 cap 119867119897+1

(Ω119878) (v119863 119901119863) isin V119863cap 119867119903+1

(Ω119863)119889times 119876 cap 119867

119897+1(Ω119863)

and (vℎ 119901ℎ) is the solution of the stabilized problem (41)Then

the following error estimate holds with constants 1198881 1198882 119888

7

independent of ℎ

lsaquo (k minus kℎ 119901 minus 119901

ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(58)

Proof Using the stability estimate of Theorem 3 there exists(wℎ 119902ℎ) isin Vℎtimes 119876ℎ with lsaquo(w

ℎ 119902ℎ)lsaquoℎ

le satisfying

lsaquo (jℎ119903k minus kℎ 119895ℎ

119897119901 minus 119901ℎ) lsaquoℎ

le1

Aℎ(jℎ119903k minus kℎ 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

lsaquo (wℎ 119902ℎ) lsaquoℎ

le1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


+1

Aℎ(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)


(59)

Then by Galerkin orthogonality property the first term of(59) is bounded by

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


=Sℎ(119901 119902ℎ)


leSℎ(119901 119901)

12

Sℎ(119902ℎ 119902ℎ)12


le Sℎ(119901 119901)

12

(60)


Hence the approximation properties of 1205812ℎand 120581ℎimply

1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


le1

1003817100381710038171003817120574nabla1205812ℎ1199011003817100381710038171003817Ω

1003817100381710038171003817nabla1205812ℎ1199011003817100381710038171003817Ω

le 1198881minus1

12057412

ℎ119897+12 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω

(61)

To estimate the second term of (59) we consider separatelyeach individual term of the bilinear form (1)A

ℎ(jℎ119903k minus

k 119895ℎ119897119901 minus 119901w

ℎ 119902ℎ)

Next Cauchy schwarz and Poincare inequality for theboundary terms imply

1

A119878(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[]10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

+10038171003817100381710038171003817119895ℎ

119897119901 minus 119901

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817119902ℎ1003817100381710038171003817Ω119878

+]1205721198882Γ

radic

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

]

le minus1

119888119894 []ℎ119903 k119903+1Ω119878 + ℎ

119897+1 10038171003817100381710038171199011003817100381710038171003817119897Ω119878

+ ℎ119903k119903+1Ω119878 +

]1205721198882Γ

radic

ℎ119903k119903+1Ω119878]

1

A119863(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[1198962

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+ 12057510038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817divwℎ1003817100381710038171003817Ω119863

+10038171003817100381710038171003817nabla (119895ℎ

119897119901 minus 119901)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119902ℎ

1003817100381710038171003817Ω119863

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863]

le minus1

119888119894 [1198962ℎ119903+1

k119903+1Ω119863 + 120575ℎ119903k119903+1Ω119863

+ ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863+ ℎ119903+1

k119903+1Ω119863]

(62)

Thus


119897119901 minus 119901ℎ) lsaquoℎ

le (1] + 2) ℎ119903k119903+1Ω119878 + (

3ℎ + 4120575) ℎ119903k119903+1Ω119863

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119878

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863

(63)

Squaring the norm and applying Young inequality we obtain


119897119901 minus 119901ℎ) lsaquo2ℎ

le 4 (1] + 2)2

ℎ2119903

k2119903+1Ω119878

+ 4 (3ℎ + 4120575)2

ℎ2119903

k2119903+1Ω119863

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

(64)

Next we estimate the interpolation error by

lsaquo (k minus jℎ119903k 119901 minus 119895

ℎ

119897119901) lsaquo2ℎ

=10038171003817100381710038171003817nabla (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817(k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817div (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817119901 minus 119895ℎ

11989711990110038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817nabla (119901 minus 119895

ℎ

119897119901)

10038171003817100381710038171003817

2

Ω119863

+ Sℎ(1205812ℎ119901 1205812ℎ119901)

le 1198882

119894ℎ2119903

k2119903+1Ω119878

+ 1198882

119894ℎ2119903

(ℎ2+ 1) ℎ

2119903k2119903+1Ω119863

+ (2

119894ℎ2+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119878

+ (2

119894+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119863

(65)

Adding the interpolation error (64) to the projectionerror (65) we obtain the required result


ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(66)

Remark 4 We note that the analysis above holds true for thetriangular subgrid interpolation 119875

119903minus 119875119903minus 119875119903

Remark 5 Because of the presence of divergence of thevelocity and the gradient of the pressure in the discretenorm the velocity and pressure solutions are119874(ℎ

119903) and119874(ℎ

119897)

respectively So we expect the 1198712-asymptotic rates to be

119874(ℎ119903+1

) and 119874(ℎ119897+1

)

7 Numerical Results

As a test model problem we take Ω = (0 1) times (0 1) and splitit into Ω

119878= (0 12) times (0 1) and Ω

119863= (12 1) times (0 1) The

interface boundary is Γ = (05 119910) | 0 lt 119910 lt 1 We take


Table 1 Rates of convergence for velocity and pressure solution inthe Stokes subdomain

1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725

Table 2 Rates of convergence for velocity and pressure solution inthe Darcy subdomain

1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085

] = 1 120572 = 1 = 1 and 119870 = 119868 and the right hand sidesf 119892 such that the velocity and pressure solution in the twosubdomains are given by

u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)

Note that for this problem forcing terms are needed to balancethe equations notably additional terms are added to theinterface conditions in (6) and (7) as follows

minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)

The problem is solved using a 1198761minus 1198761velocity-pressure

approximation with a two-level subgrid stabilization on auniform mesh with 120575 = 04 Rates of convergence for thevelocity and pressure errors for ℎ = 18 116 132 164 and1128 are displayed in Tables 1 and 2

In Table 1 we see clearly that the velocity field in theStokes subdomain is of second-order accuracy with respectto the 119871

2-norm and first-order accuracy with respect to

1198671-seminorm and the pressure is of first-order accuracy

In addition In Table 2 we observe that the velocity fieldand its divergence are of first-order accuracy in the Darcysubdomain and the pressure is of first-order accuracy withrespect to the 119871

2-norm So clearly these results are in

agreement with the theoretical results of the previous sectionand are comparable to the ones found in [2 5]

Competing Interests

The author declares that they have no competing interests

Acknowledgments

The author acknowledges the financial support of the SultanQaboos University under Contract IGSCIDOMS1407

References

[1] W J Layton F Schieweck and I Yotov ldquoCoupling fluid flowwith porous media flowrdquo SIAM Journal on Numerical Analysisvol 40 no 6 pp 2195ndash2218 2003

[2] JMUrquiza D NrsquoDri A Garon andMCDelfour ldquoCouplingStokes and Darcy equationsrdquo Applied Numerical Mathematicsvol 58 no 5 pp 525ndash538 2008

[3] V Girault and B Riviere ldquoDG approximation of coupledNavier-Stokes and Darcy equations by Beaver-Joseph-Saffmaninterface conditionrdquo SIAM Journal on Numerical Analysis vol47 no 3 pp 2052ndash2089 2009

[4] T Karper K-AMardal and RWinther ldquoUnified finite elementdiscretizations of coupled Darcy-Stokes flowrdquo Numerical Meth-ods for Partial Differential Equations vol 25 no 2 pp 311ndash3262009

[5] G Pacquaut J Bruchon N Moulin and S Drapier ldquoCombin-ing a level-set method and amixed stabilized P1P1 formulationfor coupling Stokes-Darcy flowsrdquo International Journal forNumerical Methods in Fluids vol 69 no 2 pp 459ndash480 2012

[6] P G Saffman ldquoOn the boundary condition at the surface of aporous mediumrdquo Studies in Applied Mathematics vol 50 no 2pp 93ndash101 1971

[7] M Braack and K Nafa ldquoA uniform local projection finiteelement method for coupled Darcy-Stokes flowrdquo in Proceedingsof the 5th International Conference on Approximation Methodsand Numerical Modeling in Environment and Natural Resources(MAMERN rsquo13) Granada Spain April 2013

[8] A Mikelic and W Jager ldquoOn the interface boundary conditionof Beavers Joseph and Saffmanrdquo SIAM Journal on AppliedMathematics (SIAP) vol 60 no 4 pp 1111ndash1127 2000

[9] W Wang and C Xu ldquoSpectral methods based on new for-mulations for coupled Stokes and Darcy equationsrdquo Journal ofComputational Physics vol 257 pp 126ndash142 2014

[10] KNafa ldquoEqual order approximations enrichedwith bubbles forcoupled StokesndashDarcy problemrdquo Journal of Computational andApplied Mathematics vol 270 pp 275ndash282 2014

[11] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991

[12] A Ern and J-L Guermond Theory and Practice of FiniteElements vol 159 of Applied Mathematical Sciences SpringerNew York NY USA 2004


[13] S C Brenner and L R Scott The Mathematical Theory ofFinite Element Methods vol 15 of Texts in Applied MathematicsSpringer New York NY USA 3rd edition 2008

[14] R Li J Li Z Chen and Y Gao ldquoA stabilized finite elementmethod based on two local Gauss integrations for a coupledStokes-Darcy problemrdquo Journal of Computational and AppliedMathematics vol 292 pp 92ndash104 2016

[15] J L Guermond ldquoStabilization of Galerkin approximations ofmonotone operatorsrdquo IMA Journal of Numerical Analysis vol21 pp 165ndash197 2001

[16] S Badia and R Codina ldquoUnified stabilized finite elementformulations for the Stokes and the Darcy problemsrdquo SIAMJournal on Numerical Analysis vol 47 no 3 pp 1971ndash20002009

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Stochastic AnalysisInternational Journal of


where the permeability 119870 = 119870(119909) is a positive definitesymmetric tensor and 119892 denotes an external Darcy force

22 Boundary Conditions On Γ119878

= 120597Ω119878 Γ we prescribe

homogeneous Dirichlet conditions for the velocity k119878

k119878= 0 on Γ

119878 (3)

The boundary of Ω119863is split into three parts 120597Ω

119863= Γ cup

Γ1198631

cupΓ1198632

We prescribe zero flux on Γ1198631

and a homogeneousDirichlet condition for the pressure on Γ

1198632

k119863sdot n119863

= 0 on Γ1198631

119901119863

= 0 on Γ1198632

(4)

where n119863denotes the outer normal vector on the boundary

pointing from Ω119863into Ω

119878 This boundary condition ensures

a zero mass flux

23 The Beavers-Joseph-Saffman Condition The flows in Ω119878

and Ω119863

are coupled across the interface Γ Conditionsdescribing the interaction of the flows are as follows [6 8]

(i) The continuity of the normal velocity

k119878sdot n119878= minusk119863sdot n119863 on Γ (5)

(ii) The balance of normal forces

minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863 on Γ (6)

(iii) The Beavers-Joseph-Saffman condition written interms of the strain tensor

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591 (7)

where = ]119870120591 sdot 120591 and 120572 is a dimensionless param-eter to be determined experimentally this conditionrelating the tangential slip velocity k

119878sdot 120591 to the normal

derivative of the tangential velocity component in theStokes region

3 Variational Formulation

As variational formulation we consider the so-called 1198712-

formulation used by Karper et al [4] and recently by [9 10]We denote

(kw)Ω

= intΩ

kw 119889119909 kw isin 1198712(Ω)119889

⟨V 119908⟩Γ = intΓ

V119908119889119904 V 119908 isin 1198712(Γ)

(8)

where 1198712(Ω) and119867

1(Ω) denote the usual Sobolev spaces

Next we define the spaces

H1Γ119878(Ω119878) = w isin (119867

1(Ω119878))119889

| w = 0 on Γ119878

H1 (div Ω119863) = w isin 119871

2(Ω119863)119889

| divw isin 1198712(Ω119863)

H1Γ1198631

(Ω119863) = w isin H1 (div Ω

119863) | w sdot n

119863= 0 on Γ

119863

(9)

Then multiplying the Stokes equations (1) by the test func-tionsw

119878isin H1Γ119878(Ω119878) 119902119878isin 1198712(Ω119878) respectively and integrating

by part on the domainΩ119878 we obtain

(2]119863(k119878) 119863 (w

119878))Ω119878

minus ⟨2]119863(k119878)n119878w119878⟩

minus (119901119878 divw

119878)Ω119878

+ ⟨119901119878w119878sdot n119878⟩Γ= (f w

119878)Ω119878

(div k119878 119902119878)Ω119878

= 0

(10)

Using the decompositionw119878= (w119878sdotn119878)n119878+(w119878sdot120591)120591 the fluid

normal stress condition (6) and the BJS interface condition(7) in (10) we obtain the weak formulation of the Stokesequations find k

119878isin H1Γ119878(Ω119878) 119901119878isin 1198712(Ω119878) such that

(2]119863(k119878) 119863 (w

119878))Ω119878

+]120572radic

⟨k119878sdot 120591w119878sdot 120591⟩Γ

minus (119901119878 divw

119878)Ω119878

+ ⟨119901119863w119878sdot n119878⟩Γ= (f w

119878)Ω119878

(div k119878 119902119878)Ω119878

= 0

(11)

forallw119878isin H1Γ119878(Ω119878) 119902119878isin 1198712(Ω119878)

Similarly taking 120575 gt 0 and testing theDarcy equations (2)by w119863

isin H1Γ1198631

(Ω119863) 119902119863

isin 1198712(Ω119863) respectively together with

the weighted grad-div term we obtain the weak formulationof Darcy equations find k

119863isin H1Γ119863(Ω119863) 119901119863

isin 1198671

1198632(Ω119863) such

that

(119870minus1k119863w119863)Ω119863

+ (nabla119901119863w119863)Ω119863

+ 120575 (div k119863 divw

119863)Ω119863

= 120575 (119892 divw119863)Ω119863

minus (k119863 nabla119902119863)Ω119863

+ ⟨k119863sdot n119863 119902119863⟩Γ= (119892 119902

119863)Ω119863

(12)

Summing up (11) and (12) the weak form of the coupledproblem is given by the following find k

119878isin H1Γ119878(Ω119878) 119901119878

isin

1198712(Ω119878) k119863

isin H1Γ119863(Ω119863) and 119901

119863isin 1198712(Ω119863) such that

(2]119863(k119878) 119863 (w

119878))Ω119878

minus (119901119878 divw

119878)Ω119878

+ (119870minus1k119863w119863)Ω119863

+ (nabla119901119863w119863)Ω119863

+ 120575 (div k119863 divw

119863)Ω119863

+]120572radic

⟨k119878sdot 120591w119878sdot 120591⟩Γ

+ ⟨119901119863w119878sdot n119878⟩Γ= (f w

119878)Ω119878

+ 120575 (119892 divw119863)Ω119863

(div k119878 119902119878)Ω119878

minus (k119863 nabla119902119863)Ω119863

minus ⟨k119878sdot n119878 119902119863⟩Γ= (119892 119902

119863)Ω119863

(13)




k119878

= 0 on Γ119878 k sdot n

119863= 0 on Γ

1198631

119876 = 119902 isin 1198712(Ω) | 119901

119863isin 1198671(Ω119863) 119901 = 0 isin Γ

1198632

119883 = V times 119876

(14)



+ k2Ω119863

+ div k2Ω119863

)12

10038171003817100381710038171199011003817100381710038171003817119876

= (1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863)12

(15)





1198961 k2

Ω119863le (119870minus1k k)

Ω119863

le 1198962 k2



(w119878w119863) in V and 119901 = (119901

119878 119901119863) 119902 = (119902

119878 119902119863) in 119876 on the two


A119878(k 119901w 119902) = (2]119863(k

119878) 119863 (w

119878))Ω119878

+]120572radic

⟨k119878sdot 120591w119878sdot 120591⟩Γ

minus (119901119878 divw

119878)Ω119878

+ (div k119878 119902119878)Ω119878

A119863(k 119901w 119902) = (119870

minus1k119863w119863)Ω119863

+ 120575 (div k119863 divw

119863)Ω119863

+ (nabla119901119863w119863)Ω119863

minus (k119863 nabla119902119863)Ω119863

(17)




A (k 119901w 119902) = A119878(k 119901w 119902) +A

119863(k 119901w 119902)


(18)



with

F (w 119902) = (f w119878)Ω+ (119892 119902

119863)Ω+ 120575 (119892 divw

119863)Ω (20)


119878isin 1198672(Ω119878)119889 k119863

isin

1198671(Ω119863)119889 119901 isin 119867



Ω119863+ (div k 119902)

Ω119863instead

of (w nabla119901)Ω119863





forallV isin V gt 0

(21)

inf119902isin1198712(Ω119878)

supkisin1198671(Ω119878)119889

(div k 119902)Ω119878

nablakΩ1198781003817100381710038171003817119902

1003817100381710038171003817Ω119878

ge 120573119878 (22)

inf119902isin1198671(Ω119863)

supkisin1198712(Ω119863)119889


kΩ1198631003817100381710038171003817nabla119902

1003817100381710038171003817Ω119863

ge 120573119863 (23)


119863[7]




A (k 119901w 0) ge 1198882

10038171003817100381710038171199011003817100381710038171003817

2

119876minus 1198881 k2

V

wV le 1198883

10038171003817100381710038171199011003817100381710038171003817119876

(24)


3


119878isin 1198671(Ω119878)119889 with w

119878= 0 on Γ

119878and w

119878sdot n = 0

on Γ such that

minus (divw119878 119901)Ω119878

=1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

1003817100381710038171003817nablaw1198781003817100381710038171003817Ω119878

le 119888119878

10038171003817100381710038171199011003817100381710038171003817Ω119878

(25)


there exists w

119863isin 1198671(Ω119863)119889 with w

119863sdot n = 0 on Γ

1198632and Γ

such that

minus (divw119863 119901)Ω119863

=1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

1003817100381710038171003817nablaw1198631003817100381710038171003817Ω119863

le 119888119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

(26)

Define

w =

w119878

in Ω119878

w119863

in Ω119863

(27)


and thenA (k 119901w 0) = (2]119863 (k) 119863 (w))

Ω119878minus (119901 divw)

Ω119878

+ (119870minus1kw)

Ω119863

+ (nabla119901w)Ω119863

+ 120575 (div k divw)Ω119863

ge minus2] 119863 (k)Ω119878

119863 (w)Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863



1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863


ge minus2]119888119878 nablakΩ119878

10038171003817100381710038171199011003817100381710038171003817Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198881199011198881198631198962 kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

minus 120575119888119863 div kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

(28)



A (k 119901w 0) ge minus]119888119878

1205761

nablak2Ω119878

+ (1 minus ]1198881198781205761)1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus

1198881199011198881198631198962

21205762

k2Ω119863

+ (1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863

minus120575119888119863

21205763

div k2Ω119863

(29)


1205761lt

1

]119888119878

1205762lt

2

1198881199011198881198631198962

1205763lt

2 minus 11988811990111988811986311989621205762

120575119888119863

(30)


A (k 119901w 0) ge 1198882

10038171003817100381710038171199011003817100381710038171003817

2

119876minus 1198881 k2

V (31)

where

1198881= max

]119888119878

1205761

1198881199011198881198631198962

21205762

120575119888119863

21205763

1198882= min1 minus ]119888

1198781205761 1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2

(32)


w2

V = nablaw2

Ω119878+ w

2

Ω119863+ divw

2

Ω119863

le 1198882

119878

10038171003817100381710038171199011003817100381710038171003817

2

Ω119878+ 1198882

119863(1198882

119901+ 1)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863le 1198883

10038171003817100381710038171199011003817100381710038171003817119876

(33)

where 1198882

3= max1198882

119878 1198882

119863(1198882

119901+ 1)















ℎ We define


119883119903


119878) cup 119862 (Ω

119863) V|119879

∘ 119865119879in 119876119903() forall119879 isin T

ℎ

(34)




119863


Vℎ= (119883119903

ℎ)119889

cap V

119876ℎ= 119883119903

ℎcap 119876 cap 119867

1(Ω)

(35)


119903 1198671(Ω) rarr 119883

119903

ℎ


11990312060110038171003817100381710038171003817Ω

le 119888119904

100381610038161003816100381612060110038161003816100381610038161Ω

120601 isin 1198671(Ω) (36)

10038171003817100381710038171003817120601 minus 119895ℎ

11990312060110038171003817100381710038171003817119898Ω

le 119888119894ℎ119903+1minus119898 1003816100381610038161003816120601

1003816100381610038161003816119903+1Ω

120601 isin 119867119903+1

(Ω) 119898 = 0 or 1

(37)



( sum

119879isinTℎ

ℎ2

119879

1003817100381710038171003817nabla1206011003817100381710038171003817119879

)

12

le 119888119868

10038171003817100381710038171206011003817100381710038171003817Ω

forall120601 isin 1198671(Ω) (38)


jℎ119903 1198671(Ω)119889997888rarr (119883

119903

ℎ)119889

(39)





ℎ(sdot sdot) by


Aℎ(kℎ 119901ℎw 119902) = A (k

ℎ 119901ℎw 119902) +S

ℎ(119901ℎ 119902) (40)



ℎtimes 119876ℎ (41)




2ℎis used

Defining 1205812ℎ


Sℎ(119901ℎ 119902) = sum

119872isinT2ℎ

ℎ119872

(120574nabla1205812ℎ119901ℎ nabla1205812ℎ119902)119872

119903 ge 1 (42)



ℎ= 119868 minus 119868


Sℎ(119901ℎ 119902) = sum

119870isinTℎ




10038171003817100381710038171199011003817100381710038171003817

2

119876+Sℎ(119901 119901))

12

(44)

5 Stability





A (kℎ 119901ℎwℎ 119902ℎ)


ge

(45)




Aℎ(kℎ 119901ℎ kℎ 119901ℎ) = A (k

ℎ 119901ℎ kℎ 119901ℎ) +Sℎ(119901ℎ 119901ℎ)

ge k2V + Sℎ(119901ℎ 119901ℎ)

(46)



A (kℎ 119901ℎ jℎ119903w 0) = A (k

ℎ 119901ℎw 0) +A (k

ℎ 119901ℎ z 0)

ge 1198882

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876minus 1198881

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

+A119878(kℎ 119901ℎ z 0)

+A119863(kℎ 119901ℎ z 0)

(47)


119863(kℎ 119901ℎ z 0) as

follows

A119878(kℎ 119901ℎ z 0) = (2]119863(k

ℎ) 119863 (z))

Ω119878+ (nabla119901

ℎ z)Ω119878

+]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

(48)


10038161003816100381610038161003816(]119863(k

ℎ) 119863 (z))

Ω119878

10038161003816100381610038161003816le ] 1003817100381710038171003817119863 (k

ℎ)1003817100381710038171003817Ω119878

119863 (z)Ω119878

le ] 1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878 le ]119888

119894

1003817100381710038171003817kℎ1003817100381710038171003817V nablawΩ119878

le ]1198883119888119894

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(nabla119901ℎ z)Ω119878

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879z2119879)

12

sdot ( sum

119879isinTℎ119879subΩ119878

ℎ2

119879

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817

2

119879)

12

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879ℎ2119903

119879nablaw2

119879)

12

119888119868

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 119888119888119894119888119868 nablawΩ119878

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 1198881198881198941198881198681198883

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

(49)


1- stability by

100381610038161003816100381610038161003816100381610038161003816

]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

100381610038161003816100381610038161003816100381610038161003816

le 1198882

Γ

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878

le 1198882

Γ1198881199041198883

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(50)


1205981=

1198882

8]1198881198941198883

1205982=

1198882radic

8]1205721198882Γ1198881199041198883

(51)

we obtain

A119878(kℎ 119901ℎ z 0) le

1198882

81198884

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876+ 1198884

1003817100381710038171003817kℎ1003817100381710038171003817

2

V (52)

where 1198884= (4(]119888

3119888119894)2+ 025(119888

2

Γ1198881199041198883)2)1198882



A119863(kℎ 119901ℎ z 0) = (119870


+ 120575 (div kℎ div z)

Ω119863

+ (nabla119901ℎ z)Ω119863

= (119870minus1kℎ z)Ω119863

+ 120575 (div kℎ div z)

Ω119863


+ (nabla1205812ℎ119901ℎ z)Ω119863

le10038171003817100381710038171003817119870minus1kℎ

10038171003817100381710038171003817Ω119863zΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863zΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

zΩ119863

+1003817100381710038171003817nabla1205812ℎ119901ℎ

1003817100381710038171003817Ω119863zΩ119863

le 1198962

1003817100381710038171003817kℎ1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863(1 + 119888119904) wΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 119888119904

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817Ω119863

le 11989621198881198941198883

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1205751198883(1 + 119888119904)1003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1198881198941198883

1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

+ 119888119904

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(53)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

51198882

8

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 119862 (1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(54)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 1198621(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(55)


ℎ 119901ℎ) + (1(1 + 119862


Aℎ(kℎ 119901ℎwℎ 119902ℎ) ge

1003817100381710038171003817kℎ1003817100381710038171003817

2

V +1

1 + 1198621

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus1198621

1 + 1198621

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

=1

1 + 1198621

(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +1003817100381710038171003817119901ℎ

1003817100381710038171003817

2

119876)

=1

1 + 1198621




+1

1 + 1198621


ℎ


10038171003817100381710038171003817nablajℎ119903w10038171003817100381710038171003817Ω


(56)




Aℎ(kℎ 119901ℎwℎ 119902ℎ)


ge (57)

with = 119862minus1

3(1 + 119862

1)

6 Error Analysis


119878 119901119878) isin V

119878cap 119867119903+1

(Ω119878)119889times

119876 cap 119867119897+1

(Ω119878) (v119863 119901119863) isin V119863cap 119867119903+1

(Ω119863)119889times 119876 cap 119867

119897+1(Ω119863)



7

independent of ℎ


ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(58)



le satisfying


119897119901 minus 119901ℎ) lsaquoℎ

le1


119897119901 minus 119901w

ℎ 119902ℎ)


le1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


+1


119897119901 minus 119901w

ℎ 119902ℎ)


(59)


Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


=Sℎ(119901 119902ℎ)


leSℎ(119901 119901)

12

Sℎ(119902ℎ 119902ℎ)12


le Sℎ(119901 119901)

12

(60)



1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


le1

1003817100381710038171003817120574nabla1205812ℎ1199011003817100381710038171003817Ω

1003817100381710038171003817nabla1205812ℎ1199011003817100381710038171003817Ω

le 1198881minus1

12057412

ℎ119897+12 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω

(61)



k 119895ℎ119897119901 minus 119901w

ℎ 119902ℎ)


1

A119878(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[]10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

+10038171003817100381710038171003817119895ℎ

119897119901 minus 119901

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817119902ℎ1003817100381710038171003817Ω119878

+]1205721198882Γ

radic

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

]

le minus1

119888119894 []ℎ119903 k119903+1Ω119878 + ℎ

119897+1 10038171003817100381710038171199011003817100381710038171003817119897Ω119878

+ ℎ119903k119903+1Ω119878 +

]1205721198882Γ

radic

ℎ119903k119903+1Ω119878]

1

A119863(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[1198962

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+ 12057510038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817divwℎ1003817100381710038171003817Ω119863

+10038171003817100381710038171003817nabla (119895ℎ

119897119901 minus 119901)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119902ℎ

1003817100381710038171003817Ω119863

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863]

le minus1

119888119894 [1198962ℎ119903+1

k119903+1Ω119863 + 120575ℎ119903k119903+1Ω119863

+ ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863+ ℎ119903+1

k119903+1Ω119863]

(62)

Thus


119897119901 minus 119901ℎ) lsaquoℎ

le (1] + 2) ℎ119903k119903+1Ω119878 + (

3ℎ + 4120575) ℎ119903k119903+1Ω119863

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119878

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863

(63)



119897119901 minus 119901ℎ) lsaquo2ℎ

le 4 (1] + 2)2

ℎ2119903

k2119903+1Ω119878

+ 4 (3ℎ + 4120575)2

ℎ2119903

k2119903+1Ω119863

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

(64)



ℎ

119897119901) lsaquo2ℎ

=10038171003817100381710038171003817nabla (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817(k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817div (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817119901 minus 119895ℎ

11989711990110038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817nabla (119901 minus 119895

ℎ

119897119901)

10038171003817100381710038171003817

2

Ω119863

+ Sℎ(1205812ℎ119901 1205812ℎ119901)

le 1198882

119894ℎ2119903

k2119903+1Ω119878

+ 1198882

119894ℎ2119903

(ℎ2+ 1) ℎ

2119903k2119903+1Ω119863

+ (2

119894ℎ2+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119878

+ (2

119894+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119863

(65)



ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(66)


119903minus 119875119903minus 119875119903


119903) and119874(ℎ

119897)


119874(ℎ119903+1

) and 119874(ℎ119897+1

)

7 Numerical Results


119878= (0 12) times (0 1) and Ω

119863= (12 1) times (0 1) The




1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725


1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085


u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)


minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)









Competing Interests


Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












k119878

= 0 on Γ119878 k sdot n

119863= 0 on Γ

1198631

119876 = 119902 isin 1198712(Ω) | 119901

119863isin 1198671(Ω119863) 119901 = 0 isin Γ

1198632

119883 = V times 119876

(14)



+ k2Ω119863

+ div k2Ω119863

)12

10038171003817100381710038171199011003817100381710038171003817119876

= (1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863)12

(15)





1198961 k2

Ω119863le (119870minus1k k)

Ω119863

le 1198962 k2



(w119878w119863) in V and 119901 = (119901

119878 119901119863) 119902 = (119902

119878 119902119863) in 119876 on the two


A119878(k 119901w 119902) = (2]119863(k

119878) 119863 (w

119878))Ω119878

+]120572radic

⟨k119878sdot 120591w119878sdot 120591⟩Γ

minus (119901119878 divw

119878)Ω119878

+ (div k119878 119902119878)Ω119878

A119863(k 119901w 119902) = (119870

minus1k119863w119863)Ω119863

+ 120575 (div k119863 divw

119863)Ω119863

+ (nabla119901119863w119863)Ω119863

minus (k119863 nabla119902119863)Ω119863

(17)




A (k 119901w 119902) = A119878(k 119901w 119902) +A

119863(k 119901w 119902)


(18)



with

F (w 119902) = (f w119878)Ω+ (119892 119902

119863)Ω+ 120575 (119892 divw

119863)Ω (20)


119878isin 1198672(Ω119878)119889 k119863

isin

1198671(Ω119863)119889 119901 isin 119867



Ω119863+ (div k 119902)

Ω119863instead

of (w nabla119901)Ω119863





forallV isin V gt 0

(21)

inf119902isin1198712(Ω119878)

supkisin1198671(Ω119878)119889

(div k 119902)Ω119878

nablakΩ1198781003817100381710038171003817119902

1003817100381710038171003817Ω119878

ge 120573119878 (22)

inf119902isin1198671(Ω119863)

supkisin1198712(Ω119863)119889


kΩ1198631003817100381710038171003817nabla119902

1003817100381710038171003817Ω119863

ge 120573119863 (23)


119863[7]




A (k 119901w 0) ge 1198882

10038171003817100381710038171199011003817100381710038171003817

2

119876minus 1198881 k2

V

wV le 1198883

10038171003817100381710038171199011003817100381710038171003817119876

(24)


3


119878isin 1198671(Ω119878)119889 with w

119878= 0 on Γ

119878and w

119878sdot n = 0

on Γ such that

minus (divw119878 119901)Ω119878

=1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

1003817100381710038171003817nablaw1198781003817100381710038171003817Ω119878

le 119888119878

10038171003817100381710038171199011003817100381710038171003817Ω119878

(25)


there exists w

119863isin 1198671(Ω119863)119889 with w

119863sdot n = 0 on Γ

1198632and Γ

such that

minus (divw119863 119901)Ω119863

=1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

1003817100381710038171003817nablaw1198631003817100381710038171003817Ω119863

le 119888119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

(26)

Define

w =

w119878

in Ω119878

w119863

in Ω119863

(27)


and thenA (k 119901w 0) = (2]119863 (k) 119863 (w))

Ω119878minus (119901 divw)

Ω119878

+ (119870minus1kw)

Ω119863

+ (nabla119901w)Ω119863

+ 120575 (div k divw)Ω119863

ge minus2] 119863 (k)Ω119878

119863 (w)Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863



1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863


ge minus2]119888119878 nablakΩ119878

10038171003817100381710038171199011003817100381710038171003817Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198881199011198881198631198962 kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

minus 120575119888119863 div kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

(28)



A (k 119901w 0) ge minus]119888119878

1205761

nablak2Ω119878

+ (1 minus ]1198881198781205761)1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus

1198881199011198881198631198962

21205762

k2Ω119863

+ (1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863

minus120575119888119863

21205763

div k2Ω119863

(29)


1205761lt

1

]119888119878

1205762lt

2

1198881199011198881198631198962

1205763lt

2 minus 11988811990111988811986311989621205762

120575119888119863

(30)


A (k 119901w 0) ge 1198882

10038171003817100381710038171199011003817100381710038171003817

2

119876minus 1198881 k2

V (31)

where

1198881= max

]119888119878

1205761

1198881199011198881198631198962

21205762

120575119888119863

21205763

1198882= min1 minus ]119888

1198781205761 1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2

(32)


w2

V = nablaw2

Ω119878+ w

2

Ω119863+ divw

2

Ω119863

le 1198882

119878

10038171003817100381710038171199011003817100381710038171003817

2

Ω119878+ 1198882

119863(1198882

119901+ 1)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863le 1198883

10038171003817100381710038171199011003817100381710038171003817119876

(33)

where 1198882

3= max1198882

119878 1198882

119863(1198882

119901+ 1)















ℎ We define


119883119903


119878) cup 119862 (Ω

119863) V|119879

∘ 119865119879in 119876119903() forall119879 isin T

ℎ

(34)




119863


Vℎ= (119883119903

ℎ)119889

cap V

119876ℎ= 119883119903

ℎcap 119876 cap 119867

1(Ω)

(35)


119903 1198671(Ω) rarr 119883

119903

ℎ


11990312060110038171003817100381710038171003817Ω

le 119888119904

100381610038161003816100381612060110038161003816100381610038161Ω

120601 isin 1198671(Ω) (36)

10038171003817100381710038171003817120601 minus 119895ℎ

11990312060110038171003817100381710038171003817119898Ω

le 119888119894ℎ119903+1minus119898 1003816100381610038161003816120601

1003816100381610038161003816119903+1Ω

120601 isin 119867119903+1

(Ω) 119898 = 0 or 1

(37)



( sum

119879isinTℎ

ℎ2

119879

1003817100381710038171003817nabla1206011003817100381710038171003817119879

)

12

le 119888119868

10038171003817100381710038171206011003817100381710038171003817Ω

forall120601 isin 1198671(Ω) (38)


jℎ119903 1198671(Ω)119889997888rarr (119883

119903

ℎ)119889

(39)





ℎ(sdot sdot) by


Aℎ(kℎ 119901ℎw 119902) = A (k

ℎ 119901ℎw 119902) +S

ℎ(119901ℎ 119902) (40)



ℎtimes 119876ℎ (41)




2ℎis used

Defining 1205812ℎ


Sℎ(119901ℎ 119902) = sum

119872isinT2ℎ

ℎ119872

(120574nabla1205812ℎ119901ℎ nabla1205812ℎ119902)119872

119903 ge 1 (42)



ℎ= 119868 minus 119868


Sℎ(119901ℎ 119902) = sum

119870isinTℎ




10038171003817100381710038171199011003817100381710038171003817

2

119876+Sℎ(119901 119901))

12

(44)

5 Stability





A (kℎ 119901ℎwℎ 119902ℎ)


ge

(45)




Aℎ(kℎ 119901ℎ kℎ 119901ℎ) = A (k

ℎ 119901ℎ kℎ 119901ℎ) +Sℎ(119901ℎ 119901ℎ)

ge k2V + Sℎ(119901ℎ 119901ℎ)

(46)



A (kℎ 119901ℎ jℎ119903w 0) = A (k

ℎ 119901ℎw 0) +A (k

ℎ 119901ℎ z 0)

ge 1198882

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876minus 1198881

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

+A119878(kℎ 119901ℎ z 0)

+A119863(kℎ 119901ℎ z 0)

(47)


119863(kℎ 119901ℎ z 0) as

follows

A119878(kℎ 119901ℎ z 0) = (2]119863(k

ℎ) 119863 (z))

Ω119878+ (nabla119901

ℎ z)Ω119878

+]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

(48)


10038161003816100381610038161003816(]119863(k

ℎ) 119863 (z))

Ω119878

10038161003816100381610038161003816le ] 1003817100381710038171003817119863 (k

ℎ)1003817100381710038171003817Ω119878

119863 (z)Ω119878

le ] 1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878 le ]119888

119894

1003817100381710038171003817kℎ1003817100381710038171003817V nablawΩ119878

le ]1198883119888119894

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(nabla119901ℎ z)Ω119878

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879z2119879)

12

sdot ( sum

119879isinTℎ119879subΩ119878

ℎ2

119879

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817

2

119879)

12

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879ℎ2119903

119879nablaw2

119879)

12

119888119868

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 119888119888119894119888119868 nablawΩ119878

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 1198881198881198941198881198681198883

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

(49)


1- stability by

100381610038161003816100381610038161003816100381610038161003816

]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

100381610038161003816100381610038161003816100381610038161003816

le 1198882

Γ

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878

le 1198882

Γ1198881199041198883

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(50)


1205981=

1198882

8]1198881198941198883

1205982=

1198882radic

8]1205721198882Γ1198881199041198883

(51)

we obtain

A119878(kℎ 119901ℎ z 0) le

1198882

81198884

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876+ 1198884

1003817100381710038171003817kℎ1003817100381710038171003817

2

V (52)

where 1198884= (4(]119888

3119888119894)2+ 025(119888

2

Γ1198881199041198883)2)1198882



A119863(kℎ 119901ℎ z 0) = (119870


+ 120575 (div kℎ div z)

Ω119863

+ (nabla119901ℎ z)Ω119863

= (119870minus1kℎ z)Ω119863

+ 120575 (div kℎ div z)

Ω119863


+ (nabla1205812ℎ119901ℎ z)Ω119863

le10038171003817100381710038171003817119870minus1kℎ

10038171003817100381710038171003817Ω119863zΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863zΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

zΩ119863

+1003817100381710038171003817nabla1205812ℎ119901ℎ

1003817100381710038171003817Ω119863zΩ119863

le 1198962

1003817100381710038171003817kℎ1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863(1 + 119888119904) wΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 119888119904

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817Ω119863

le 11989621198881198941198883

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1205751198883(1 + 119888119904)1003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1198881198941198883

1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

+ 119888119904

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(53)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

51198882

8

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 119862 (1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(54)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 1198621(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(55)


ℎ 119901ℎ) + (1(1 + 119862


Aℎ(kℎ 119901ℎwℎ 119902ℎ) ge

1003817100381710038171003817kℎ1003817100381710038171003817

2

V +1

1 + 1198621

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus1198621

1 + 1198621

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

=1

1 + 1198621

(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +1003817100381710038171003817119901ℎ

1003817100381710038171003817

2

119876)

=1

1 + 1198621




+1

1 + 1198621


ℎ


10038171003817100381710038171003817nablajℎ119903w10038171003817100381710038171003817Ω


(56)




Aℎ(kℎ 119901ℎwℎ 119902ℎ)


ge (57)

with = 119862minus1

3(1 + 119862

1)

6 Error Analysis


119878 119901119878) isin V

119878cap 119867119903+1

(Ω119878)119889times

119876 cap 119867119897+1

(Ω119878) (v119863 119901119863) isin V119863cap 119867119903+1

(Ω119863)119889times 119876 cap 119867

119897+1(Ω119863)



7

independent of ℎ


ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(58)



le satisfying


119897119901 minus 119901ℎ) lsaquoℎ

le1


119897119901 minus 119901w

ℎ 119902ℎ)


le1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


+1


119897119901 minus 119901w

ℎ 119902ℎ)


(59)


Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


=Sℎ(119901 119902ℎ)


leSℎ(119901 119901)

12

Sℎ(119902ℎ 119902ℎ)12


le Sℎ(119901 119901)

12

(60)



1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


le1

1003817100381710038171003817120574nabla1205812ℎ1199011003817100381710038171003817Ω

1003817100381710038171003817nabla1205812ℎ1199011003817100381710038171003817Ω

le 1198881minus1

12057412

ℎ119897+12 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω

(61)



k 119895ℎ119897119901 minus 119901w

ℎ 119902ℎ)


1

A119878(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[]10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

+10038171003817100381710038171003817119895ℎ

119897119901 minus 119901

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817119902ℎ1003817100381710038171003817Ω119878

+]1205721198882Γ

radic

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

]

le minus1

119888119894 []ℎ119903 k119903+1Ω119878 + ℎ

119897+1 10038171003817100381710038171199011003817100381710038171003817119897Ω119878

+ ℎ119903k119903+1Ω119878 +

]1205721198882Γ

radic

ℎ119903k119903+1Ω119878]

1

A119863(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[1198962

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+ 12057510038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817divwℎ1003817100381710038171003817Ω119863

+10038171003817100381710038171003817nabla (119895ℎ

119897119901 minus 119901)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119902ℎ

1003817100381710038171003817Ω119863

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863]

le minus1

119888119894 [1198962ℎ119903+1

k119903+1Ω119863 + 120575ℎ119903k119903+1Ω119863

+ ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863+ ℎ119903+1

k119903+1Ω119863]

(62)

Thus


119897119901 minus 119901ℎ) lsaquoℎ

le (1] + 2) ℎ119903k119903+1Ω119878 + (

3ℎ + 4120575) ℎ119903k119903+1Ω119863

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119878

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863

(63)



119897119901 minus 119901ℎ) lsaquo2ℎ

le 4 (1] + 2)2

ℎ2119903

k2119903+1Ω119878

+ 4 (3ℎ + 4120575)2

ℎ2119903

k2119903+1Ω119863

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

(64)



ℎ

119897119901) lsaquo2ℎ

=10038171003817100381710038171003817nabla (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817(k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817div (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817119901 minus 119895ℎ

11989711990110038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817nabla (119901 minus 119895

ℎ

119897119901)

10038171003817100381710038171003817

2

Ω119863

+ Sℎ(1205812ℎ119901 1205812ℎ119901)

le 1198882

119894ℎ2119903

k2119903+1Ω119878

+ 1198882

119894ℎ2119903

(ℎ2+ 1) ℎ

2119903k2119903+1Ω119863

+ (2

119894ℎ2+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119878

+ (2

119894+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119863

(65)



ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(66)


119903minus 119875119903minus 119875119903


119903) and119874(ℎ

119897)


119874(ℎ119903+1

) and 119874(ℎ119897+1

)

7 Numerical Results


119878= (0 12) times (0 1) and Ω

119863= (12 1) times (0 1) The




1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725


1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085


u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)


minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)









Competing Interests


Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and thenA (k 119901w 0) = (2]119863 (k) 119863 (w))

Ω119878minus (119901 divw)

Ω119878

+ (119870minus1kw)

Ω119863

+ (nabla119901w)Ω119863

+ 120575 (div k divw)Ω119863

ge minus2] 119863 (k)Ω119878

119863 (w)Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863



1003817100381710038171003817

2

Ω119878

minus 1198962 kΩ119863

1003817100381710038171003817w1198631003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863


ge minus2]119888119878 nablakΩ119878

10038171003817100381710038171199011003817100381710038171003817Ω119878

+1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus 1198881199011198881198631198962 kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119901

1003817100381710038171003817

2

Ω119863

minus 120575119888119863 div kΩ119863

1003817100381710038171003817nabla1199011003817100381710038171003817Ω119863

(28)



A (k 119901w 0) ge minus]119888119878

1205761

nablak2Ω119878

+ (1 minus ]1198881198781205761)1003817100381710038171003817119901

1003817100381710038171003817

2

Ω119878

minus

1198881199011198881198631198962

21205762

k2Ω119863

+ (1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863

minus120575119888119863

21205763

div k2Ω119863

(29)


1205761lt

1

]119888119878

1205762lt

2

1198881199011198881198631198962

1205763lt

2 minus 11988811990111988811986311989621205762

120575119888119863

(30)


A (k 119901w 0) ge 1198882

10038171003817100381710038171199011003817100381710038171003817

2

119876minus 1198881 k2

V (31)

where

1198881= max

]119888119878

1205761

1198881199011198881198631198962

21205762

120575119888119863

21205763

1198882= min1 minus ]119888

1198781205761 1 minus

11988811990111988811986311989621205762

2minus

1205751198881198631205763

2

(32)


w2

V = nablaw2

Ω119878+ w

2

Ω119863+ divw

2

Ω119863

le 1198882

119878

10038171003817100381710038171199011003817100381710038171003817

2

Ω119878+ 1198882

119863(1198882

119901+ 1)

1003817100381710038171003817nabla1199011003817100381710038171003817

2

Ω119863le 1198883

10038171003817100381710038171199011003817100381710038171003817119876

(33)

where 1198882

3= max1198882

119878 1198882

119863(1198882

119901+ 1)















ℎ We define


119883119903


119878) cup 119862 (Ω

119863) V|119879

∘ 119865119879in 119876119903() forall119879 isin T

ℎ

(34)




119863


Vℎ= (119883119903

ℎ)119889

cap V

119876ℎ= 119883119903

ℎcap 119876 cap 119867

1(Ω)

(35)


119903 1198671(Ω) rarr 119883

119903

ℎ


11990312060110038171003817100381710038171003817Ω

le 119888119904

100381610038161003816100381612060110038161003816100381610038161Ω

120601 isin 1198671(Ω) (36)

10038171003817100381710038171003817120601 minus 119895ℎ

11990312060110038171003817100381710038171003817119898Ω

le 119888119894ℎ119903+1minus119898 1003816100381610038161003816120601

1003816100381610038161003816119903+1Ω

120601 isin 119867119903+1

(Ω) 119898 = 0 or 1

(37)



( sum

119879isinTℎ

ℎ2

119879

1003817100381710038171003817nabla1206011003817100381710038171003817119879

)

12

le 119888119868

10038171003817100381710038171206011003817100381710038171003817Ω

forall120601 isin 1198671(Ω) (38)


jℎ119903 1198671(Ω)119889997888rarr (119883

119903

ℎ)119889

(39)





ℎ(sdot sdot) by


Aℎ(kℎ 119901ℎw 119902) = A (k

ℎ 119901ℎw 119902) +S

ℎ(119901ℎ 119902) (40)



ℎtimes 119876ℎ (41)




2ℎis used

Defining 1205812ℎ


Sℎ(119901ℎ 119902) = sum

119872isinT2ℎ

ℎ119872

(120574nabla1205812ℎ119901ℎ nabla1205812ℎ119902)119872

119903 ge 1 (42)



ℎ= 119868 minus 119868


Sℎ(119901ℎ 119902) = sum

119870isinTℎ




10038171003817100381710038171199011003817100381710038171003817

2

119876+Sℎ(119901 119901))

12

(44)

5 Stability





A (kℎ 119901ℎwℎ 119902ℎ)


ge

(45)




Aℎ(kℎ 119901ℎ kℎ 119901ℎ) = A (k

ℎ 119901ℎ kℎ 119901ℎ) +Sℎ(119901ℎ 119901ℎ)

ge k2V + Sℎ(119901ℎ 119901ℎ)

(46)



A (kℎ 119901ℎ jℎ119903w 0) = A (k

ℎ 119901ℎw 0) +A (k

ℎ 119901ℎ z 0)

ge 1198882

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876minus 1198881

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

+A119878(kℎ 119901ℎ z 0)

+A119863(kℎ 119901ℎ z 0)

(47)


119863(kℎ 119901ℎ z 0) as

follows

A119878(kℎ 119901ℎ z 0) = (2]119863(k

ℎ) 119863 (z))

Ω119878+ (nabla119901

ℎ z)Ω119878

+]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

(48)


10038161003816100381610038161003816(]119863(k

ℎ) 119863 (z))

Ω119878

10038161003816100381610038161003816le ] 1003817100381710038171003817119863 (k

ℎ)1003817100381710038171003817Ω119878

119863 (z)Ω119878

le ] 1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878 le ]119888

119894

1003817100381710038171003817kℎ1003817100381710038171003817V nablawΩ119878

le ]1198883119888119894

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(nabla119901ℎ z)Ω119878

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879z2119879)

12

sdot ( sum

119879isinTℎ119879subΩ119878

ℎ2

119879

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817

2

119879)

12

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879ℎ2119903

119879nablaw2

119879)

12

119888119868

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 119888119888119894119888119868 nablawΩ119878

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 1198881198881198941198881198681198883

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

(49)


1- stability by

100381610038161003816100381610038161003816100381610038161003816

]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

100381610038161003816100381610038161003816100381610038161003816

le 1198882

Γ

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878

le 1198882

Γ1198881199041198883

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(50)


1205981=

1198882

8]1198881198941198883

1205982=

1198882radic

8]1205721198882Γ1198881199041198883

(51)

we obtain

A119878(kℎ 119901ℎ z 0) le

1198882

81198884

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876+ 1198884

1003817100381710038171003817kℎ1003817100381710038171003817

2

V (52)

where 1198884= (4(]119888

3119888119894)2+ 025(119888

2

Γ1198881199041198883)2)1198882



A119863(kℎ 119901ℎ z 0) = (119870


+ 120575 (div kℎ div z)

Ω119863

+ (nabla119901ℎ z)Ω119863

= (119870minus1kℎ z)Ω119863

+ 120575 (div kℎ div z)

Ω119863


+ (nabla1205812ℎ119901ℎ z)Ω119863

le10038171003817100381710038171003817119870minus1kℎ

10038171003817100381710038171003817Ω119863zΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863zΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

zΩ119863

+1003817100381710038171003817nabla1205812ℎ119901ℎ

1003817100381710038171003817Ω119863zΩ119863

le 1198962

1003817100381710038171003817kℎ1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863(1 + 119888119904) wΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 119888119904

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817Ω119863

le 11989621198881198941198883

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1205751198883(1 + 119888119904)1003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1198881198941198883

1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

+ 119888119904

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(53)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

51198882

8

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 119862 (1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(54)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 1198621(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(55)


ℎ 119901ℎ) + (1(1 + 119862


Aℎ(kℎ 119901ℎwℎ 119902ℎ) ge

1003817100381710038171003817kℎ1003817100381710038171003817

2

V +1

1 + 1198621

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus1198621

1 + 1198621

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

=1

1 + 1198621

(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +1003817100381710038171003817119901ℎ

1003817100381710038171003817

2

119876)

=1

1 + 1198621




+1

1 + 1198621


ℎ


10038171003817100381710038171003817nablajℎ119903w10038171003817100381710038171003817Ω


(56)




Aℎ(kℎ 119901ℎwℎ 119902ℎ)


ge (57)

with = 119862minus1

3(1 + 119862

1)

6 Error Analysis


119878 119901119878) isin V

119878cap 119867119903+1

(Ω119878)119889times

119876 cap 119867119897+1

(Ω119878) (v119863 119901119863) isin V119863cap 119867119903+1

(Ω119863)119889times 119876 cap 119867

119897+1(Ω119863)



7

independent of ℎ


ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(58)



le satisfying


119897119901 minus 119901ℎ) lsaquoℎ

le1


119897119901 minus 119901w

ℎ 119902ℎ)


le1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


+1


119897119901 minus 119901w

ℎ 119902ℎ)


(59)


Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


=Sℎ(119901 119902ℎ)


leSℎ(119901 119901)

12

Sℎ(119902ℎ 119902ℎ)12


le Sℎ(119901 119901)

12

(60)



1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


le1

1003817100381710038171003817120574nabla1205812ℎ1199011003817100381710038171003817Ω

1003817100381710038171003817nabla1205812ℎ1199011003817100381710038171003817Ω

le 1198881minus1

12057412

ℎ119897+12 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω

(61)



k 119895ℎ119897119901 minus 119901w

ℎ 119902ℎ)


1

A119878(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[]10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

+10038171003817100381710038171003817119895ℎ

119897119901 minus 119901

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817119902ℎ1003817100381710038171003817Ω119878

+]1205721198882Γ

radic

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

]

le minus1

119888119894 []ℎ119903 k119903+1Ω119878 + ℎ

119897+1 10038171003817100381710038171199011003817100381710038171003817119897Ω119878

+ ℎ119903k119903+1Ω119878 +

]1205721198882Γ

radic

ℎ119903k119903+1Ω119878]

1

A119863(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[1198962

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+ 12057510038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817divwℎ1003817100381710038171003817Ω119863

+10038171003817100381710038171003817nabla (119895ℎ

119897119901 minus 119901)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119902ℎ

1003817100381710038171003817Ω119863

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863]

le minus1

119888119894 [1198962ℎ119903+1

k119903+1Ω119863 + 120575ℎ119903k119903+1Ω119863

+ ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863+ ℎ119903+1

k119903+1Ω119863]

(62)

Thus


119897119901 minus 119901ℎ) lsaquoℎ

le (1] + 2) ℎ119903k119903+1Ω119878 + (

3ℎ + 4120575) ℎ119903k119903+1Ω119863

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119878

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863

(63)



119897119901 minus 119901ℎ) lsaquo2ℎ

le 4 (1] + 2)2

ℎ2119903

k2119903+1Ω119878

+ 4 (3ℎ + 4120575)2

ℎ2119903

k2119903+1Ω119863

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

(64)



ℎ

119897119901) lsaquo2ℎ

=10038171003817100381710038171003817nabla (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817(k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817div (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817119901 minus 119895ℎ

11989711990110038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817nabla (119901 minus 119895

ℎ

119897119901)

10038171003817100381710038171003817

2

Ω119863

+ Sℎ(1205812ℎ119901 1205812ℎ119901)

le 1198882

119894ℎ2119903

k2119903+1Ω119878

+ 1198882

119894ℎ2119903

(ℎ2+ 1) ℎ

2119903k2119903+1Ω119863

+ (2

119894ℎ2+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119878

+ (2

119894+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119863

(65)



ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(66)


119903minus 119875119903minus 119875119903


119903) and119874(ℎ

119897)


119874(ℎ119903+1

) and 119874(ℎ119897+1

)

7 Numerical Results


119878= (0 12) times (0 1) and Ω

119863= (12 1) times (0 1) The




1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725


1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085


u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)


minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)









Competing Interests


Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ(sdot sdot) by


Aℎ(kℎ 119901ℎw 119902) = A (k

ℎ 119901ℎw 119902) +S

ℎ(119901ℎ 119902) (40)



ℎtimes 119876ℎ (41)




2ℎis used

Defining 1205812ℎ


Sℎ(119901ℎ 119902) = sum

119872isinT2ℎ

ℎ119872

(120574nabla1205812ℎ119901ℎ nabla1205812ℎ119902)119872

119903 ge 1 (42)



ℎ= 119868 minus 119868


Sℎ(119901ℎ 119902) = sum

119870isinTℎ




10038171003817100381710038171199011003817100381710038171003817

2

119876+Sℎ(119901 119901))

12

(44)

5 Stability





A (kℎ 119901ℎwℎ 119902ℎ)


ge

(45)




Aℎ(kℎ 119901ℎ kℎ 119901ℎ) = A (k

ℎ 119901ℎ kℎ 119901ℎ) +Sℎ(119901ℎ 119901ℎ)

ge k2V + Sℎ(119901ℎ 119901ℎ)

(46)



A (kℎ 119901ℎ jℎ119903w 0) = A (k

ℎ 119901ℎw 0) +A (k

ℎ 119901ℎ z 0)

ge 1198882

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876minus 1198881

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

+A119878(kℎ 119901ℎ z 0)

+A119863(kℎ 119901ℎ z 0)

(47)


119863(kℎ 119901ℎ z 0) as

follows

A119878(kℎ 119901ℎ z 0) = (2]119863(k

ℎ) 119863 (z))

Ω119878+ (nabla119901

ℎ z)Ω119878

+]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

(48)


10038161003816100381610038161003816(]119863(k

ℎ) 119863 (z))

Ω119878

10038161003816100381610038161003816le ] 1003817100381710038171003817119863 (k

ℎ)1003817100381710038171003817Ω119878

119863 (z)Ω119878

le ] 1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878 le ]119888

119894

1003817100381710038171003817kℎ1003817100381710038171003817V nablawΩ119878

le ]1198883119888119894

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(nabla119901ℎ z)Ω119878

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879z2119879)

12

sdot ( sum

119879isinTℎ119879subΩ119878

ℎ2

119879

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817

2

119879)

12

le ( sum

119879isinTℎ119879subΩ119878

ℎminus2

119879ℎ2119903

119879nablaw2

119879)

12

119888119868

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 119888119888119894119888119868 nablawΩ119878

1003817100381710038171003817119901ℎ1003817100381710038171003817Ω119878

le 1198881198881198941198881198681198883

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

(49)


1- stability by

100381610038161003816100381610038161003816100381610038161003816

]120572radic

⟨kℎ119878

sdot 120591 z119878sdot 120591⟩Γ

100381610038161003816100381610038161003816100381610038161003816

le 1198882

Γ

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V nablazΩ119878

le 1198882

Γ1198881199041198883

]120572radic

1003817100381710038171003817kℎ1003817100381710038171003817V

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(50)


1205981=

1198882

8]1198881198941198883

1205982=

1198882radic

8]1205721198882Γ1198881199041198883

(51)

we obtain

A119878(kℎ 119901ℎ z 0) le

1198882

81198884

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876+ 1198884

1003817100381710038171003817kℎ1003817100381710038171003817

2

V (52)

where 1198884= (4(]119888

3119888119894)2+ 025(119888

2

Γ1198881199041198883)2)1198882



A119863(kℎ 119901ℎ z 0) = (119870


+ 120575 (div kℎ div z)

Ω119863

+ (nabla119901ℎ z)Ω119863

= (119870minus1kℎ z)Ω119863

+ 120575 (div kℎ div z)

Ω119863


+ (nabla1205812ℎ119901ℎ z)Ω119863

le10038171003817100381710038171003817119870minus1kℎ

10038171003817100381710038171003817Ω119863zΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863zΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

zΩ119863

+1003817100381710038171003817nabla1205812ℎ119901ℎ

1003817100381710038171003817Ω119863zΩ119863

le 1198962

1003817100381710038171003817kℎ1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863(1 + 119888119904) wΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 119888119904

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817Ω119863

le 11989621198881198941198883

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1205751198883(1 + 119888119904)1003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1198881198941198883

1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

+ 119888119904

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(53)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

51198882

8

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 119862 (1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(54)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 1198621(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(55)


ℎ 119901ℎ) + (1(1 + 119862


Aℎ(kℎ 119901ℎwℎ 119902ℎ) ge

1003817100381710038171003817kℎ1003817100381710038171003817

2

V +1

1 + 1198621

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus1198621

1 + 1198621

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

=1

1 + 1198621

(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +1003817100381710038171003817119901ℎ

1003817100381710038171003817

2

119876)

=1

1 + 1198621




+1

1 + 1198621


ℎ


10038171003817100381710038171003817nablajℎ119903w10038171003817100381710038171003817Ω


(56)




Aℎ(kℎ 119901ℎwℎ 119902ℎ)


ge (57)

with = 119862minus1

3(1 + 119862

1)

6 Error Analysis


119878 119901119878) isin V

119878cap 119867119903+1

(Ω119878)119889times

119876 cap 119867119897+1

(Ω119878) (v119863 119901119863) isin V119863cap 119867119903+1

(Ω119863)119889times 119876 cap 119867

119897+1(Ω119863)



7

independent of ℎ


ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(58)



le satisfying


119897119901 minus 119901ℎ) lsaquoℎ

le1


119897119901 minus 119901w

ℎ 119902ℎ)


le1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


+1


119897119901 minus 119901w

ℎ 119902ℎ)


(59)


Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


=Sℎ(119901 119902ℎ)


leSℎ(119901 119901)

12

Sℎ(119902ℎ 119902ℎ)12


le Sℎ(119901 119901)

12

(60)



1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


le1

1003817100381710038171003817120574nabla1205812ℎ1199011003817100381710038171003817Ω

1003817100381710038171003817nabla1205812ℎ1199011003817100381710038171003817Ω

le 1198881minus1

12057412

ℎ119897+12 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω

(61)



k 119895ℎ119897119901 minus 119901w

ℎ 119902ℎ)


1

A119878(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[]10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

+10038171003817100381710038171003817119895ℎ

119897119901 minus 119901

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817119902ℎ1003817100381710038171003817Ω119878

+]1205721198882Γ

radic

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

]

le minus1

119888119894 []ℎ119903 k119903+1Ω119878 + ℎ

119897+1 10038171003817100381710038171199011003817100381710038171003817119897Ω119878

+ ℎ119903k119903+1Ω119878 +

]1205721198882Γ

radic

ℎ119903k119903+1Ω119878]

1

A119863(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[1198962

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+ 12057510038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817divwℎ1003817100381710038171003817Ω119863

+10038171003817100381710038171003817nabla (119895ℎ

119897119901 minus 119901)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119902ℎ

1003817100381710038171003817Ω119863

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863]

le minus1

119888119894 [1198962ℎ119903+1

k119903+1Ω119863 + 120575ℎ119903k119903+1Ω119863

+ ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863+ ℎ119903+1

k119903+1Ω119863]

(62)

Thus


119897119901 minus 119901ℎ) lsaquoℎ

le (1] + 2) ℎ119903k119903+1Ω119878 + (

3ℎ + 4120575) ℎ119903k119903+1Ω119863

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119878

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863

(63)



119897119901 minus 119901ℎ) lsaquo2ℎ

le 4 (1] + 2)2

ℎ2119903

k2119903+1Ω119878

+ 4 (3ℎ + 4120575)2

ℎ2119903

k2119903+1Ω119863

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

(64)



ℎ

119897119901) lsaquo2ℎ

=10038171003817100381710038171003817nabla (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817(k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817div (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817119901 minus 119895ℎ

11989711990110038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817nabla (119901 minus 119895

ℎ

119897119901)

10038171003817100381710038171003817

2

Ω119863

+ Sℎ(1205812ℎ119901 1205812ℎ119901)

le 1198882

119894ℎ2119903

k2119903+1Ω119878

+ 1198882

119894ℎ2119903

(ℎ2+ 1) ℎ

2119903k2119903+1Ω119863

+ (2

119894ℎ2+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119878

+ (2

119894+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119863

(65)



ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(66)


119903minus 119875119903minus 119875119903


119903) and119874(ℎ

119897)


119874(ℎ119903+1

) and 119874(ℎ119897+1

)

7 Numerical Results


119878= (0 12) times (0 1) and Ω

119863= (12 1) times (0 1) The




1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725


1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085


u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)


minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)









Competing Interests


Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











A119863(kℎ 119901ℎ z 0) = (119870


+ 120575 (div kℎ div z)

Ω119863

+ (nabla119901ℎ z)Ω119863

= (119870minus1kℎ z)Ω119863

+ 120575 (div kℎ div z)

Ω119863


+ (nabla1205812ℎ119901ℎ z)Ω119863

le10038171003817100381710038171003817119870minus1kℎ

10038171003817100381710038171003817Ω119863zΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863zΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

zΩ119863

+1003817100381710038171003817nabla1205812ℎ119901ℎ

1003817100381710038171003817Ω119863zΩ119863

le 1198962

1003817100381710038171003817kℎ1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 1205751003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863(1 + 119888119904) wΩ119863

+1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

119888119894 wΩ119863

+ 119888119904

1003817100381710038171003817nabla119901ℎ

1003817100381710038171003817Ω119863

le 11989621198881198941198883

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1205751198883(1 + 119888119904)1003817100381710038171003817div kℎ

1003817100381710038171003817Ω119863

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

+ 1198881198941198883

1003817100381710038171003817nabla (119901ℎminus 1205812ℎ119901ℎ)1003817100381710038171003817Ω119863

+ 119888119904

1003817100381710038171003817119901ℎ1003817100381710038171003817119876

(53)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

51198882

8

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 119862 (1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(54)


Aℎ(kℎ 119901ℎ jℎ119903w 0) ge

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus 1198621(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +Sℎ(119901ℎ 119901ℎ))

(55)


ℎ 119901ℎ) + (1(1 + 119862


Aℎ(kℎ 119901ℎwℎ 119902ℎ) ge

1003817100381710038171003817kℎ1003817100381710038171003817

2

V +1

1 + 1198621

1003817100381710038171003817119901ℎ1003817100381710038171003817

2

119876

minus1198621

1 + 1198621

1003817100381710038171003817kℎ1003817100381710038171003817

2

V

=1

1 + 1198621

(1003817100381710038171003817kℎ

1003817100381710038171003817

2

V +1003817100381710038171003817119901ℎ

1003817100381710038171003817

2

119876)

=1

1 + 1198621




+1

1 + 1198621


ℎ


10038171003817100381710038171003817nablajℎ119903w10038171003817100381710038171003817Ω


(56)




Aℎ(kℎ 119901ℎwℎ 119902ℎ)


ge (57)

with = 119862minus1

3(1 + 119862

1)

6 Error Analysis


119878 119901119878) isin V

119878cap 119867119903+1

(Ω119878)119889times

119876 cap 119867119897+1

(Ω119878) (v119863 119901119863) isin V119863cap 119867119903+1

(Ω119863)119889times 119876 cap 119867

119897+1(Ω119863)



7

independent of ℎ


ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(58)



le satisfying


119897119901 minus 119901ℎ) lsaquoℎ

le1


119897119901 minus 119901w

ℎ 119902ℎ)


le1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


+1


119897119901 minus 119901w

ℎ 119902ℎ)


(59)


Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


=Sℎ(119901 119902ℎ)


leSℎ(119901 119901)

12

Sℎ(119902ℎ 119902ℎ)12


le Sℎ(119901 119901)

12

(60)



1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


le1

1003817100381710038171003817120574nabla1205812ℎ1199011003817100381710038171003817Ω

1003817100381710038171003817nabla1205812ℎ1199011003817100381710038171003817Ω

le 1198881minus1

12057412

ℎ119897+12 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω

(61)



k 119895ℎ119897119901 minus 119901w

ℎ 119902ℎ)


1

A119878(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[]10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

+10038171003817100381710038171003817119895ℎ

119897119901 minus 119901

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817119902ℎ1003817100381710038171003817Ω119878

+]1205721198882Γ

radic

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

]

le minus1

119888119894 []ℎ119903 k119903+1Ω119878 + ℎ

119897+1 10038171003817100381710038171199011003817100381710038171003817119897Ω119878

+ ℎ119903k119903+1Ω119878 +

]1205721198882Γ

radic

ℎ119903k119903+1Ω119878]

1

A119863(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[1198962

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+ 12057510038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817divwℎ1003817100381710038171003817Ω119863

+10038171003817100381710038171003817nabla (119895ℎ

119897119901 minus 119901)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119902ℎ

1003817100381710038171003817Ω119863

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863]

le minus1

119888119894 [1198962ℎ119903+1

k119903+1Ω119863 + 120575ℎ119903k119903+1Ω119863

+ ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863+ ℎ119903+1

k119903+1Ω119863]

(62)

Thus


119897119901 minus 119901ℎ) lsaquoℎ

le (1] + 2) ℎ119903k119903+1Ω119878 + (

3ℎ + 4120575) ℎ119903k119903+1Ω119863

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119878

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863

(63)



119897119901 minus 119901ℎ) lsaquo2ℎ

le 4 (1] + 2)2

ℎ2119903

k2119903+1Ω119878

+ 4 (3ℎ + 4120575)2

ℎ2119903

k2119903+1Ω119863

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

(64)



ℎ

119897119901) lsaquo2ℎ

=10038171003817100381710038171003817nabla (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817(k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817div (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817119901 minus 119895ℎ

11989711990110038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817nabla (119901 minus 119895

ℎ

119897119901)

10038171003817100381710038171003817

2

Ω119863

+ Sℎ(1205812ℎ119901 1205812ℎ119901)

le 1198882

119894ℎ2119903

k2119903+1Ω119878

+ 1198882

119894ℎ2119903

(ℎ2+ 1) ℎ

2119903k2119903+1Ω119863

+ (2

119894ℎ2+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119878

+ (2

119894+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119863

(65)



ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(66)


119903minus 119875119903minus 119875119903


119903) and119874(ℎ

119897)


119874(ℎ119903+1

) and 119874(ℎ119897+1

)

7 Numerical Results


119878= (0 12) times (0 1) and Ω

119863= (12 1) times (0 1) The




1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725


1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085


u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)


minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)









Competing Interests


Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

Aℎ(k minus k

ℎ 119901 minus 119901

ℎwℎ 119902ℎ)


le1

1003817100381710038171003817120574nabla1205812ℎ1199011003817100381710038171003817Ω

1003817100381710038171003817nabla1205812ℎ1199011003817100381710038171003817Ω

le 1198881minus1

12057412

ℎ119897+12 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω

(61)



k 119895ℎ119897119901 minus 119901w

ℎ 119902ℎ)


1

A119878(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[]10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

+10038171003817100381710038171003817119895ℎ

119897119901 minus 119901

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817119902ℎ1003817100381710038171003817Ω119878

+]1205721198882Γ

radic

10038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119878

1003817100381710038171003817nablawℎ1003817100381710038171003817Ω119878

]

le minus1

119888119894 []ℎ119903 k119903+1Ω119878 + ℎ

119897+1 10038171003817100381710038171199011003817100381710038171003817119897Ω119878

+ ℎ119903k119903+1Ω119878 +

]1205721198882Γ

radic

ℎ119903k119903+1Ω119878]

1

A119863(jℎ119903k minus k 119895ℎ

119897119901 minus 119901w

ℎ 119902ℎ)

le minus1

[1198962

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+ 12057510038171003817100381710038171003817nabla (jℎ119903k minus k)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817divwℎ1003817100381710038171003817Ω119863

+10038171003817100381710038171003817nabla (119895ℎ

119897119901 minus 119901)

10038171003817100381710038171003817Ω119863

1003817100381710038171003817wℎ1003817100381710038171003817Ω119863

+1003817100381710038171003817nabla119902ℎ

1003817100381710038171003817Ω119863

10038171003817100381710038171003817jℎ119903k minus k

10038171003817100381710038171003817Ω119863]

le minus1

119888119894 [1198962ℎ119903+1

k119903+1Ω119863 + 120575ℎ119903k119903+1Ω119863

+ ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863+ ℎ119903+1

k119903+1Ω119863]

(62)

Thus


119897119901 minus 119901ℎ) lsaquoℎ

le (1] + 2) ℎ119903k119903+1Ω119878 + (

3ℎ + 4120575) ℎ119903k119903+1Ω119863

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119878

+ (5+ 612057412

ℎ12

+ 7ℎ) ℎ119897 1003817100381710038171003817119901

1003817100381710038171003817119897+1Ω119863

(63)



119897119901 minus 119901ℎ) lsaquo2ℎ

le 4 (1] + 2)2

ℎ2119903

k2119903+1Ω119878

+ 4 (3ℎ + 4120575)2

ℎ2119903

k2119903+1Ω119863

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ 4 (5+ 612057412

ℎ12

+ 7ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

(64)



ℎ

119897119901) lsaquo2ℎ

=10038171003817100381710038171003817nabla (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817(k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817div (k minus jℎ

119903k)

10038171003817100381710038171003817

2

Ω119863

+10038171003817100381710038171003817119901 minus 119895ℎ

11989711990110038171003817100381710038171003817

2

Ω119878

+10038171003817100381710038171003817nabla (119901 minus 119895

ℎ

119897119901)

10038171003817100381710038171003817

2

Ω119863

+ Sℎ(1205812ℎ119901 1205812ℎ119901)

le 1198882

119894ℎ2119903

k2119903+1Ω119878

+ 1198882

119894ℎ2119903

(ℎ2+ 1) ℎ

2119903k2119903+1Ω119863

+ (2

119894ℎ2+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119878

+ (2

119894+ 120574ℎ) ℎ

2119897 10038171003817100381710038171199011003817100381710038171003817

2

119897+1Ω119863

(65)



ℎ) lsaquoℎle (1198881] + 1198882)2

ℎ2119903

k2119903+1Ω119878

+ (1198883ℎ + 1198884120575)2

ℎ2119903

k2119903+1Ω119863

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119878

+ (1198885+ 119888612057412

ℎ12

+ 1198887ℎ)2

ℎ2119897 1003817100381710038171003817119901

1003817100381710038171003817

2

119897+1Ω119863

12

(66)


119903minus 119875119903minus 119875119903


119903) and119874(ℎ

119897)


119874(ℎ119903+1

) and 119874(ℎ119897+1

)

7 Numerical Results


119878= (0 12) times (0 1) and Ω

119863= (12 1) times (0 1) The




1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725


1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085


u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)


minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)









Competing Interests


Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119878

1003817100381710038171003817nabla (u minus uℎ)10038171003817100381710038170Ω119878

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119878

ℎ = 1

8mdash mdash mdash

ℎ = 1

1619303 10284 08480

ℎ = 1

3219735 10208 09149

ℎ = 1

6419890 10119 09511

ℎ = 1

12819951 10055 09725


1003817100381710038171003817u minus uℎ

10038171003817100381710038170Ω119863

1003817100381710038171003817div (u minus uℎ)10038171003817100381710038170Ω119863

1003817100381710038171003817119901 minus 119901ℎ

10038171003817100381710038170Ω119863

ℎ = 1

8mdash mdash mdash

ℎ = 1

1608813 08412 10416

ℎ = 1

3209534 09235 10318

ℎ = 1

6409642 09514 10167

ℎ = 1

12809857 09657 10085


u119878= (1199104119890119909 119890119910 cos (2119909)) (119909 119910) isin Ω

119878

u119863

= (1199104119890119909 41199103119890119909) (119909 119910) isin Ω

119863

119901 = 1199104119890119909 (119909 119910) isin Ω

(67)


minus (minus119901119878119868 + 2]119863(k

119878))n119878sdot n119878= 119901119863+ 1198921 on Γ

k119878sdot 120591 = minus

2radic

120572(119863 (k119878) sdot n119878) sdot 120591

on Γ

(68)

where1198921= minus2119910

4119890119909 and119892

2= 119890119910 cos(2119909)+4119910

3119890119909minus2119890119910 sin(2119909)









Competing Interests


Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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