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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
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
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
minus (k nabla119902)Ω119863
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
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
Advances in Numerical Analysis 5
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
Adding the interpolation error (64) to the projectionerror (65) we obtain the required result
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
(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
8 Advances in Numerical Analysis
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
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
Advances in Numerical Analysis 9
[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
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
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
minus (k nabla119902)Ω119863
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
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
Advances in Numerical Analysis 5
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
Adding the interpolation error (64) to the projectionerror (65) we obtain the required result
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
(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
8 Advances in Numerical Analysis
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
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
Advances in Numerical Analysis 9
[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
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
minus (k nabla119902)Ω119863
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
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
Advances in Numerical Analysis 5
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
Adding the interpolation error (64) to the projectionerror (65) we obtain the required result
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
(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
8 Advances in Numerical Analysis
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
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
Advances in Numerical Analysis 9
[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
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
Advances in Numerical Analysis 5
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
Adding the interpolation error (64) to the projectionerror (65) we obtain the required result
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
(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
8 Advances in Numerical Analysis
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
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
Advances in Numerical Analysis 9
[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
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
Adding the interpolation error (64) to the projectionerror (65) we obtain the required result
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
(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
8 Advances in Numerical Analysis
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
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
Advances in Numerical Analysis 9
[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
Adding the interpolation error (64) to the projectionerror (65) we obtain the required result
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
(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
8 Advances in Numerical Analysis
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
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
Advances in Numerical Analysis 9
[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
Adding the interpolation error (64) to the projectionerror (65) we obtain the required result
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
(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
8 Advances in Numerical Analysis
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
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
Advances in Numerical Analysis 9
[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
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
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
Advances in Numerical Analysis 9
[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
[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