A number of techniques, used as remedy to the instability of the Galerkin finite element formulation for Stokes like problems, are found in the literature. In this work we consider a coupled Stokes-Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous mixed finite elements in the two parts. A better method, from a computational point of view, consists in using a unified approach on both subdomains. Here, the coupled Stokes-Darcy problem is analyzed using equal-order velocity and pressure approximation combined with subgrid stabilization. We prove that the obtained finite element solution is stable and converges to the classical solution with optimal rates for both velocity and pressure.
Sultan Qaboos UniversityIG/SCI/DOMS/14/071. Introduction
The transport of substances between surface water and groundwater has attracted a lot of interest into the coupling of viscous flows and porous media flows [1–5]. In this work we consider coupled problems in fluid dynamics where the fluid in one part of the domain is described by the Stokes equations and in the other, porous media part, by the Darcy equation and mass conservation. Velocity and pressure on these two parts are mutually coupled by interface conditions derived in [6]. Such systems can be discretized by heterogeneous finite elements as analyzed by Layton et al. [1]. In more recent works, unified approaches become more popular. For instance, discontinuous Galerkin methods were analyzed by Girault and Rivière [3], mixed methods by Karper et al. [4], and local pressure gradient stabilized methods by Braack and Nafa [7].
In this work, we consider the L2-formulation of the coupled Stokes-Darcy problem as in [4], but we discretize by equal-order finite elements and use subgrid method and grad-div term to stabilize the pressure and control the natural H1(div) velocity norm on the Darcy subdomain.
2. Formulations of the Stokes-Darcy Coupled Equations2.1. Model Equations
Let Ω⊂Rd, d=2 or 3, be a bounded region with Lipschitz boundary ∂Ω. ΩS and ΩD are, respectively, the fluid and porous media subdomains of Ω such that ΩS∩ΩD=∅. The subdomains have a common interface Γ=Ω¯S∩Ω¯D. We denote by v=(vS,vD) the fluid velocity and by p=(pS,pD) the fluid pressure, where vi=v|Ωi, pi=p|Ωi, i=S,D. The flow in the domain ΩS is assumed to be of Stokes type and governed by the equations (1)-2νdivDvS+∇pS=f,inΩSdivvS=0,inΩSwith symmetric strain tensor D(vS)=1/2(∇vS+∇vST), external force f, and constant viscosity ν>0. In the porous region ΩD the filtration of an incompressible flow through porous media is described by Darcy equations (2)K-1vD+∇pD=f,inΩDdivvD=g,inΩD,where the permeability K=K(x) is a positive definite symmetric tensor and g denotes an external Darcy force.
2.2. Boundary Conditions
On ΓS=∂ΩS∖Γ, we prescribe homogeneous Dirichlet conditions for the velocity vS. (3)vS=0,onΓS.The boundary of ΩD is split into three parts ∂ΩD=Γ∪ΓD,1∪ΓD,2. We prescribe zero flux on ΓD,1 and a homogeneous Dirichlet condition for the pressure on ΓD,2.(4)vD·nD=0,onΓD,1pD=0,onΓD,2,where nD denotes the outer normal vector on the boundary pointing from ΩD into ΩS. This boundary condition ensures a zero mass flux.
2.3. The Beavers-Joseph-Saffman Condition
The flows in ΩS and ΩD are coupled across the interface Γ. Conditions describing the interaction of the flows are as follows [6, 8]:
The continuity of the normal velocity:(5)vS·nS=-vD·nD,onΓ
The balance of normal forces:(6)--pSI+2νDvSnS·nS=pD,onΓ
The Beavers-Joseph-Saffman condition written in terms of the strain tensor: (7)vS·τ=-2k~αDvS·nS·τ,
where k~=νKτ·τ and α is a dimensionless parameter to be determined experimentally, this condition relating the tangential slip velocity vS·τ to the normal derivative of the tangential velocity component in the Stokes region
3. Variational Formulation
As variational formulation we consider the so-called L2- formulation used by Karper et al. [4] and recently by [9, 10]. We denote (8)v,wΩ=∫Ωvwdx,v,w∈L2Ωd,v,wΓ=∫Γvwds,v,w∈L2Γ,where L2(Ω) and H1(Ω) denote the usual Sobolev spaces.
Next, we define the spaces (9)HΓS1ΩS=w∈H1ΩSd∣w=0onΓSH1div,ΩD=w∈L2ΩDd∣divw∈L2ΩD,HΓD,11ΩD=w∈H1div,ΩD∣w·nD=0onΓD.Then, multiplying the Stokes equations (1) by the test functions wS∈HΓS1(ΩS), qS∈L2(ΩS), respectively, and integrating by part on the domain ΩS, we obtain (10)2νDvS,DwSΩS-2νDvSnS,wS-pS,divwSΩS+pS,wS·nSΓ=f,wSΩS,divvS,qSΩS=0.Using the decomposition wS=wS·nSnS+wS·ττ, the fluid normal stress condition (6), and the BJS interface condition (7) in (10), we obtain the weak formulation of the Stokes equations: find vS∈HΓS1(ΩS), pS∈L2(ΩS) such that (11)2νDvS,DwSΩS+ναk~vS·τ,wS·τΓ-pS,divwSΩS+pD,wS·nSΓ=f,wSΩS,divvS,qSΩS=0,∀wS∈HΓS1(ΩS), qS∈L2(ΩS).
Similarly, taking δ>0 and testing the Darcy equations (2) by wD∈HΓD,11(ΩD), qD∈L2(ΩD), respectively, together with the weighted grad-div term we obtain the weak formulation of Darcy equations: find vD∈HΓD1(ΩD), pD∈HD,21(ΩD) such that (12)K-1vD,wDΩD+∇pD,wDΩD+δdivvD,divwDΩD=δg,divwDΩD,-vD,∇qDΩD+vD·nD,qDΓ=g,qDΩD.Summing up (11) and (12) the weak form of the coupled problem is given by the following: find vS∈HΓS1(ΩS), pS∈L2(ΩS), vD∈HΓD1(ΩD), and pD∈L2(ΩD) such that(13)2νDvS,DwSΩS-pS,divwSΩS+K-1vD,wDΩD+∇pD,wDΩD+δdivvD,divwDΩD+ναk~vS·τ,wS·τΓ+pD,wS·nSΓ=f,wSΩS+δg,divwDΩD,divvS,qSΩS-vD,∇qDΩD-vS·nS,qDΓ=g,qDΩD.
To analyze the weak formulation of the coupled problem we introduce the following spaces(14)V=v∈Hdiv,Ω∣vS∈H1ΩSd,vS=0onΓS,v·nD=0onΓD,1,Q=q∈L2Ω∣pD∈H1ΩD,p=0∈ΓD,2,X=V×Q.The velocity and pressure spaces V and Q are equipped with the natural norms(15)vV=∇vΩS2+vΩD2+divvΩD21/2,pQ=pΩS2+∇pΩD21/2.
Further, due to the positive definiteness of K with respect to the L2(ΩD) norm ·ΩD, there exist positive real numbers k1 and k2 such that (16)k1vΩD2≤K-1v,vΩD≤k2vΩD2,∀v∈V.
Next, we define the bilinear forms for v=(vS,vD), w=(wS,wD) in V and p=(pS,pD), q=(qS,qD) in Q on the two parts of the domain by (17)ASv,p;w,q=2νDvS,DwSΩS+ναk~vS·τ,wS·τΓ-pS,divwSΩS+divvS,qSΩS,ADv,p;w,q=K-1vD,wDΩD+δdivvD,divwDΩD+∇pD,wDΩD-vD,∇qDΩD.Hence, the bilinear form for the coupled problem is the sum of AS(v,p;w,q), AD(v,p;w,q), and terms to enforce the continuity of the normal part of the velocities across the interface. (18)Av,p;w,q=ASv,p;w,q+ADv,p;w,q+pD,wS·nSΓ-qD,vS·nSΓ.Assuming, for simplicity, that f and g are extended by zero to the whole domain, the variational formulation of the coupled Stokes-Darcy system in compact form reads as follows: find (v,p)∈V×Q solution of(19)Av,p;w,q=Fw,q,∀w,q∈V×Q,with(20)Fw,q=f,wSΩ+g,qDΩ+δg,divwDΩ.It can easily be shown that a sufficiently regular solution (v,p)∈V×Q of (19) such that vS∈H2(ΩS)d, vD∈H1(ΩD)d, p∈H1(ΩS∪ΩD) is also a classical solution of (1) and (2). We note that there is an alternative variational formulation to the one given here called H(div)-formulation. The latter uses the term -(p,divw)ΩD+(divv,q)ΩD instead of (w,∇p)ΩD-(v,∇q)ΩD [4].
The existence and uniqueness of the solution of problem (19) follows from Brezzi’s conditions for saddle point problems [11]; namely, (21)Av,p;v,p≥γ~⦀v⦀V2,∀v∈V,γ~>0,(22)infq∈L2ΩSsupv∈H1ΩSddivv,qΩS∇vΩSqΩS≥βS,(23)infq∈H1ΩDsupv∈L2ΩDd-v,∇qΩDvΩD∇qΩD≥βD.with positive constants βS and βD [7].
The following lemma is needed in the analysis below and is a consequence of the continuous inf-sup conditions (23) [10].
Lemma 1.
For every (v,p)∈X there is w∈V such that wS·nS=0 on Γ, satisfying (24)Av,p;w,0≥c2pQ2-c1vV2,wV≤c3pQ,with positive constants c1, c2, and c3.
Proof.
Let (v,p)∈X. Then, due to Stokes inf-sup condition there exists wS∈H1(ΩS)d with wS=0 on ΓS and wS·n=0 on Γ such that (25)-divwS,pΩS=pΩS2,∇wSΩS≤cSpΩS.For the Darcy equation, due to the condition p=0 on ΓD,2, there exists wD∈H1(ΩD)d with wD·n=0 on ΓD,2 and Γ, such that (26)-divwD,pΩD=∇pΩD2,∇wDΩD≤cD∇pΩD.Define (27)w=wSinΩSwDinΩD,and then (28)Av,p;w,0=2νDv,DwΩS-p,divwΩS+K-1v,wΩD+∇p,wΩD+δdivv,divwΩD≥-2νDvΩSDwΩS+pΩS2-k2vΩDwDΩD+∇pΩD2-δdivvΩDdivwΩD≥-2ν∇vΩS∇wΩS+pΩS2-k2vΩDwDΩD+∇pΩD2-δdivvΩD∇wΩD≥-2νcS∇vΩSpΩS+pΩS2-cpcDk2vΩD∇pΩD+∇pΩD2-δcDdivvΩD∇pΩD,where cp denote the Poincaré constant.
Then, using Young’s inequality we obtain(29)Av,p;w,0≥-νcSε1∇vΩS2+1-νcSε1pΩS2-cpcDk22ε2vΩD2+1-cpcDk2ε22-δcDε32∇pΩD2-δcD2ε3divvΩD2.
Choosing ε1, ε2, ε3 positive constants such that (30)ε1<1νcS,ε2<2cpcDk2,ε3<2-cpcDk2ε2δcD,we obtain the required result (31)Av,p;w,0≥c2pQ2-c1vV2,where (32)c1=maxνcSε1,cpcDk22ε2,δcD2ε3,c2=min1-νcSε1,1-cpcDk2ε22-δcDε32.In addition, we also have(33)wV2=∇wΩS2+wΩD2+divwΩD2≤cS2pΩS2+cD2cp2+1∇pΩD2≤c3pQ,where c32=maxcS2,cD2(cp2+1).
4. Finite Element Discretization
Let Th be a shape-regular partition of quadrilaterals for d=2 or hexahedra for d=3 [12, 13]. The diameter of element T∈Th will be denoted by hT and the global mesh size is defined by h≔max{hT:TinTh}. Let T^≔(-1;1)d be the reference element, FT the mapping from T^ to element T, and Qr(T^) the space of all polynomials on T^ with maximal degree r≥0 in each coordinate. We assume that the mesh Th is obtained from a coarser mesh T2h by global refinement. Hence, T2h consists of patches of elements of Th. We define the finite element space (34)Xhr≔v∈CΩS∪CΩD:vT∘FTinQrT^,∀T∈Th.For the discrete spaces Vh and Qh we use the equal-order finite element functions that are continuous in ΩS and ΩD and piecewise polynomials of degree r≥1. (35)Vh=Xhrd∩V,Qh=Xhr∩Q∩H1Ω.We define the Scott-Zhang interpolation operator which preserves the boundary condition [13], as jrh:H1(Ω)→Xhr with stability and interpolation properties, respectively, as(36)∇jrhϕΩ≤csϕ1,Ω,ϕ∈H1Ω.(37)ϕ-jrhϕm,Ω≤cihr+1-mϕr+1,Ω,ϕ∈Hr+1Ω,m=0 or 1,where ci, cs are positive constants.
We will also use the inverse inequality (38)∑T∈ThhT2∇ϕT1/2≤cIϕΩ,∀ϕ∈H1Ω.
Similarly, for vector functions we define the interpolation operator(39)jrh:H1Ωd⟶Xhrd,with interpolation and stability properties as above.
It is known that the standard Galerkin discretizations of the Darcy system are not stable for equal-order elements. This instability stems from the violation of the discrete analogue on to the inf-sup condition. One possibility to circumvent this condition is to work with a modified bilinear form Ah(·;·) by adding a stabilization term Sh(·;·); that is,(40)Ahvh,ph;w,q=Avh,ph;w,q+Shph;q,such that the stabilized discrete problem reads (41)Ahvh,ph;w,q=Fw,q∀w,q∈Vh×Qh.Unlike in [10] where a combination of a generalized mini element and local projection (LPS) is analyzed and in [14] where a method based on two local Gauss integrals for the Stokes equations is used, here we will analyze the problem using a subgrid method [12, 15, 16].
For this method the filter, with respect to the global Lagrange interpolant I2h, onto a coarser mesh T2h is used. Defining κ2h=I-I2h the subgrid stabilization term reads(42)Shph;q=∑M∈T2hhMγ∇κ2hph,∇κ2hqM,r≥1,where γ is patchwise constant.
A more attractive method from the computational point is obtained using only the fine mesh with smaller stencil. Defining κh=I-Ih the subgrid stabilization term reads (43)Shph;q=∑K∈ThhKγ∇κhph,∇κhqK,r≥2.
Next, we prove the stability of the discrete coupled Stokes-Darcy problem with respect to the norm (44)⦀v,p⦀h=vV2+pQ2+Shp;p1/2.
5. StabilityTheorem 2.
Let Th be a quasi-regular partition [13]. Then, the following discrete inf-sup condition holds for some positive constant β~ independent of the mesh size h. (45)infvh,ph∈Vh×Qh∖0,0supwh,qh∈Vh×Qh∖0,0Avh,ph;wh,qh⦀vh,ph⦀h⦀wh,qh⦀h≥β~.
Proof.
First, let (vh,ph)∈Vh×Qh, and then the diagonal testing combined with Korn’s inequality and the positivity of K-1 give (46)Ahvh,ph;vh,ph=Avh,ph;vh,ph+Shph;ph≥α~vV2+Shph;ph.In addition, let w be as in Lemma 1, corresponding to (vh,ph)∈Vh×Qh, and set z=jrhw-w. Then,(47)Avh,ph;jrhw,0=Avh,ph;w,0+Avh,ph;z,0≥c2phQ2-c1vhV2+ASvh,ph;z,0+ADvh,ph;z,0.Next, we estimate AS(vh,ph;z,0) and AD(vh,ph;z,0) as follows:(48)ASvh,ph;z,0=2νDvh,DzΩS+∇ph,zΩS+ναk~vhS·τ,zS·τΓ,where the first two terms are bounded using Cauchy inequality together with the interpolation, stability, and inverse inequalities(49)νDvh,DzΩS≤νDvhΩSDzΩS≤νvhV∇zΩS≤νcivhV∇wΩS≤νc3civhVphQ,∇ph,zΩS≤∑T∈Th,T⊂ΩShT-2zT21/2∑T∈Th,T⊂ΩShT2∇phT21/2≤∑T∈Th,T⊂ΩShT-2hT2r∇wT21/2cIphΩS≤ccicI∇wΩSphΩS≤ccicIc3phQ2.The boundary term is bounded using the trace theorem and the H1- stability by(50)ναk~vhS·τ,zS·τΓ≤cΓ2ναk~vhV∇zΩS≤cΓ2csc3ναk~vhVphQ.Hence, by Young inequality with (51)ϵ1=c28νcic3,ϵ2=c2k~8ναcΓ2csc3we obtain (52)ASvh,ph;z,0≤c28c4phQ2+c4vhV2,where c4=(4(νc3ci)2+0.25(cΓ2csc3)2)/c2.
For the Darcy bilinear form we have(53)ADvh,ph;z,0=K-1vh,zΩD+δdivvh,divzΩD+∇ph,zΩD=K-1vh,zΩD+δdivvh,divzΩD+∇ph-κ2hph,zΩD+∇κ2hph,zΩD≤K-1vhΩDzΩD+δdivvhΩDzΩD+∇ph-κ2hphΩDzΩD+∇κ2hphΩDzΩD≤k2vhΩDciwΩD+δdivvhΩD1+cswΩD+∇ph-κ2hphΩDciwΩD+cs∇phΩD≤k2cic3phQ+δc31+csdivvhΩDphQ+cic3∇ph-κ2hphΩD+csphQ.Then, by Young inequality and (52) we obtain (54)Ahvh,ph;jrhw,0≥5c28phQ2-CvhV2+Shph;ph.Scaling jrhw we obtain (55)Ahvh,ph;jrhw,0≥phQ2-C1vhV2+Shph;ph.Choosing (wh,qh)=(vh,ph)+1/1+C1(jrhw,0) we obtain (56)Ahvh,ph;wh,qh≥vhV2+11+C1phQ2-C11+C1vhV2=11+C1vhV2+phQ2=11+C1⦀vh,ph⦀h2,⦀wh,qh⦀h≤⦀vh,ph⦀h+11+C1⦀jrhw,0⦀h≤⦀vh,ph⦀h+C2∇jrhwΩ≤C3⦀vh,ph⦀hwhich implies the required result (57)infvh,ph∈Vh×Qh∖0supwh,qh∈Vh×Qh∖0Ahvh,ph;wh,qh⦀vh,ph⦀h⦀wh,qh⦀h≥β~,
with β~=C3-1/(1+C1).
6. Error AnalysisTheorem 3.
Assume that the solution (v,p) of the Stokes-Darcy problem (19) is such that (vS,pS)∈VS∩Hr+1(ΩS)d×Q∩Hl+1(ΩS), (vD,pD)∈VD∩Hr+1(ΩD)d×Q∩Hl+1(ΩD), and (vh,ph) is the solution of the stabilized problem (41). Then, the following error estimate holds with constants c1,c2,…,c7 independent of h: (58)⦀v-vh,p-ph⦀h≤c1ν+c22h2rvr+1,ΩS2+c3h+c4δ2h2rvr+1,ΩD2+c5+c6γ1/2h1/2+c7h2h2lpl+1,ΩS2+c5+c6γ1/2h1/2+c7h2h2lpl+1,ΩD21/2.
Proof.
Using the stability estimate of Theorem 3, there exists (wh,qh)∈Vh×Qh, with ⦀(wh,qh)⦀h≤C~ satisfying (59)⦀jrhv-vh,jlhp-ph⦀h≤1β~Ahjrhv-vh,jlhp-p;wh,qh⦀wh,qh⦀h≤1β~Ahv-vh,p-ph;wh,qh⦀wh,qh⦀h+1β~Ahjrhv-v,jlhp-p;wh,qh⦀wh,qh⦀h.Then, by Galerkin orthogonality property, the first term of (59) is bounded by (60)Ahv-vh,p-ph;wh,qh⦀wh,qh⦀h=Shp;qh⦀wh,qh⦀h≤Shp;p1/2Shqh;qh1/2⦀wh,qh⦀h≤Shp;p1/2.Hence, the approximation properties of κ2h and κh imply (61)1β~Ahv-vh,p-ph;wh,qh⦀wh,qh⦀h≤1β~γ∇κ2hpΩ∇κ2hpΩ≤c1β~-1γ1/2hl+1/2pl+1,Ω.To estimate the second term of (59) we consider separately each individual term of the bilinear form 1/β~Ah(jrhv-v,jlhp-p;wh,qh).
Next, Cauchy schwarz and Poincaré inequality for the boundary terms imply (62)1β~ASjrhv-v,jlhp-p;wh,qh≤β~-1ν∇jrhv-vΩS∇whΩS+jlhp-pΩS∇whΩS∇jrhv-vΩSqhΩS+ναcΓ2k~∇jrhv-vΩS∇whΩS≤β~-1ciC~νhrvr+1,ΩS+hl+1pl,ΩS+hrvr+1,ΩS+ναcΓ2k~hrvr+1,ΩS,1β~ADjrhv-v,jlhp-p;wh,qh≤β~-1k2jrhv-vΩDwhΩD+δ∇jrhv-vΩDdivwhΩD+∇jlhp-pΩDwhΩD+∇qhΩDjrhv-vΩD≤β~-1ciC~k2hr+1vr+1,ΩD+δhrvr+1,ΩD+hlpl+1,ΩD+hr+1vr+1,ΩD.Thus, (63)⦀jrhv-vh,jlhp-ph⦀h≤c~1ν+c~2hrvr+1,ΩS+c~3h+c~4δhrvr+1,ΩD+c~5+c~6γ1/2h1/2+c~7hhlpl+1,ΩS+c~5+c~6γ1/2h1/2+c~7hhlpl+1,ΩD.Squaring the norm and applying Young inequality we obtain (64)⦀jrhv-vh,jlhp-ph⦀h2≤4c~1ν+c~22h2rvr+1,ΩS2+4c~3h+c~4δ2h2rvr+1,ΩD2+4c~5+c~6γ1/2h1/2+c~7h2h2lpl+1,ΩS2+4c~5+c~6γ1/2h1/2+c~7h2h2lpl+1,ΩD2.Next, we estimate the interpolation error by (65)⦀v-jrhv,p-jlhp⦀h2=∇v-jrhvΩS2+v-jrhvΩD2+divv-jrhvΩD2+p-jlhpΩS2+∇p-jlhpΩD2+Shκ2hp,κ2hp≤ci2h2rvr+1,ΩS2+ci2h2rh2+1h2rvr+1,ΩD2+c~i2h2+γhh2lpl+1,ΩS2+c~i2+γhh2lpl+1,ΩD2.
Adding the interpolation error (64) to the projection error (65) we obtain the required result (66)⦀v-vh,p-ph⦀h≤c1ν+c22h2rvr+1,ΩS2+c3h+c4δ2h2rvr+1,ΩD2+c5+c6γ1/2h1/2+c7h2h2lpl+1,ΩS2+c5+c6γ1/2h1/2+c7h2h2lpl+1,ΩD21/2.
Remark 4.
We note that the analysis above holds true for the triangular subgrid interpolation Pr-Pr-Pr.
Remark 5.
Because of the presence of divergence of the velocity and the gradient of the pressure in the discrete norm, the velocity and pressure solutions are O(hr) and O(hl), respectively. So, we expect the L2-asymptotic rates to be O(hr+1) and O(hl+1).
7. Numerical Results
As a test model problem we take Ω=(0,1)×(0,1) and split it into ΩS=(0,1/2)×(0,1) and ΩD=(1/2,1)×(0,1). The interface boundary is Γ={0.5,y∣0<y<1}. We take ν=1, α=1, k~=1, and K=I and the right hand sides f, g such that the velocity and pressure solution in the two subdomains are given by (67)uS=y4ex,eycos2x,x,y∈ΩSuD=y4ex,4y3ex,x,y∈ΩDp=y4ex,x,y∈Ω.Note that for this problem forcing terms are needed to balance the equations; notably additional terms are added to the interface conditions in (6) and (7) as follows: (68)--pSI+2νDvSnS·nS=pD+g1,onΓ,vS·τ=-2k~αDvS·nS·τonΓ,where g1=-2y4ex, and g2=eycos(2x)+4y3ex-2eysin(2x).
The problem is solved using a Q1-Q1 velocity-pressure approximation with a two-level subgrid stabilization on a uniform mesh with δ=0.4. Rates of convergence for the velocity and pressure errors for h=1/8,1/16,1/32,1/64, and 1/128 are displayed in Tables 1 and 2.
Rates of convergence for velocity and pressure solution in the Stokes subdomain.
u-uh0,ΩS
∇u-uh0,ΩS
p-ph0,ΩS
h = 18
—
—
—
h = 116
1.9303
1.0284
0.8480
h = 132
1.9735
1.0208
0.9149
h = 164
1.9890
1.0119
0.9511
h = 1128
1.9951
1.0055
0.9725
Rates of convergence for velocity and pressure solution in the Darcy subdomain.
u-uh0,ΩD
divu-uh0,ΩD
p-ph0,ΩD
h = 18
—
—
—
h = 116
0.8813
0.8412
1.0416
h = 132
0.9534
0.9235
1.0318
h = 164
0.9642
0.9514
1.0167
h = 1128
0.9857
0.9657
1.0085
In Table 1, we see clearly that the velocity field in the Stokes subdomain is of second-order accuracy with respect to the L2-norm and first-order accuracy with respect to H1-seminorm, and the pressure is of first-order accuracy. In addition, In Table 2, we observe that the velocity field and its divergence are of first-order accuracy in the Darcy subdomain, and the pressure is of first-order accuracy with respect to the L2-norm. So, clearly these results are in agreement with the theoretical results of the previous section and 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 Sultan Qaboos University, under Contract IG/SCI/DOMS/14/07.
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