JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 761242 10.1155/2012/761242 761242 Research Article A Discontinuous Finite Volume Method for the Darcy-Stokes Equations Yin Zhe Jiang Ziwen Xu Qiang Padra Claudio School of Mathematical Sciences Shandong Normal University Jinan, Shandong 250014 China sdnu.edu.cn 2012 27 12 2012 2012 12 06 2012 07 12 2012 2012 Copyright © 2012 Zhe Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper proposes a discontinuous finite volume method for the Darcy-Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh-dependent norm. First-order L2-error estimates are derived for the approximations of both velocity and pressure. Some numerical examples verifying the theoretical predictions are presented.

1. Introduction

The study of discontinuous Galerkin methods has been a very active research field since they were proposed by Reed and Hill  in 1973. Discontinuous Galerkin methods use discontinuous functions as finite element approximation and enforce the connections of the approximate solutions between elements by adding some penalty terms. The flexibility of discontinuous functions gives discontinuous Galerkin methods many advantages, such as high parallelizability and localizability. Arnold et al.  provided a framework for the analysis of a large class of discontinuous Galerkin methods for second-order elliptic problems.

Based on the advantages of using discontinuous functions for approximation in discontinuous Galerkin methods, it is natural to consider using discontinuous functions as trial functions in the finite volume method, which is called the discontinuous finite volume method. Such a method has the flexibility of the discontinuous Galerkin method and the simplicity and conservative properties of the finite volume method. Ye  developed a new discontinuous finite volume method and analyzed it for the second-order elliptic problem. Bi and Geng  proposed the semidiscrete and the backward Euler fully discrete discontinuous finite volume element methods for the second-order parabolic problems. Ye  considered the discontinuous finite volume method for solving the Stokes problems on both triangular and rectangular meshes and derived an optimal order error estimate for the approximation of velocity in a mesh-dependent norm and first-order L2-error estimates for the approximations of both velocity and pressure.

The Darcy-Stokes problem is interesting for a variety of reasons. Apart from being a modeling tool in its own right, it also appears, less obviously, in time-stepping methods for Stokes and for high Reynolds number flows (where of course the convective term causes additional difficulties). In , the nonconforming Crouzeix-Raviart element is stabilized for the Darcy-Stokes problem with terms motivated by a discontinuous Galerkin approach. In , a new stabilized mixed finite element method is presented for the Darcy-Stokes equations.

In this paper, we will extend the discontinuous finite volume methods to solve the Darcy-Stokes equations. In our methods, velocity is approximated by discontinuous piecewise linear functions on triangular meshes and by discontinuous piecewise rotated bilinear functions on rectangular meshes. Piecewise constant functions are used as the test functions for velocity in the discontinuous finite volume methods. We obtained an optimal error estimate for the approximation of velocity in a mesh-dependent norm. First-order L2-error estimates are derived for the approximations of both velocity and pressure. For the sake of simplicity and easy presentation of the main ideas of our method, we restrict ourselves to the model problem.

We consider the Darcy-Stokes equations(1.1a)σu-μΔu+p=f,xΩ,(1.1b)·u=0,xΩ,(1.1c)u=0,xΩ,where Ω is a bounded polygonal domain in R2 with boundary Ω. u=(u1,u2) is the velocity, p is the pressure, and f is a given force term. We assume σ=1,μ=1.

2. Discontinuous Finite Volume Formulation

Let h be a triangular or rectangular partition of Ω. The triangles or rectangles in h are divided into three or four subtriangles by connecting the barycenter of the triangle or the center of the rectangles to their corner nodes, respectively. Then we define the dual partition 𝒯h of the primal partition h to be the union of the triangles shown in Figures 1 and 2 for both triangular and rectangular meshes.

Element T𝒯h for triangular mesh.

Element T𝒯h for rectangular mesh.

Let Pk(T) consist of all the polynomials with degree less than or equal to k defined on T. We define the finite dimensional trial function space for velocity on a triangular partition by (2.1)Vh={vL2(Ω)2:v|KP1(K)2,Kh} and on rectangular partition by (2.2)Vh={vL2(Ω)2:v|KQ^1(K)2,Kh}, where Q^1 denotes the space of functions of the form a+bx1+cx2+d(x12-x22) on K.

Let Qh be the finite dimensional space for pressure (2.3)Qh={qL02(Ω):q|KP0(K),Kh}, where (2.4)L02(Ω)={qL2(Ω):Ωqdx=0}. Define the finite dimensional test function space Wh for velocity associated with the dual partition 𝒯h as (2.5)Wh={ξL2(Ω)2:ξ|TP0(T)2,T𝒯h}.

Multiplying (1.1a) and (1.1b) by ξWh and qQh, respectively, we have (2.6)(u,ξ)-T𝒯hTun·ξds+T𝒯hTpξ·nds=(f,ξ),KhK·uqdx=0, where n is the unit outward normal vector on T.

Let Tj𝒯h(j=1,,t) be the triangles in Kh, where t=3 for triangular meshes and t=4 for rectangular meshes, as shown as Figures 3 and 4. Then we have (2.7)T𝒯hTun·ξds=Khj=1tAj+1CAjun·ξds+KhKun·ξds, where At+1=A1.

Triangular partition and its dual.

Rectangular partition and its dual.

For vectors v=(v1,v2) and n=(n1,n2), let vn denote the matrix whose ijth component is vi·nj as in . For two matrix valued variables σ and τ, we define σ:τ=i,j=12σi,jτi,j. Let Γ=KhK,Γ0=ΓΩ. Let e be an interior edge shared by two elements K1 and K2 in h. We define the average {·} and jump [·] on e for scalar q, vector w, and matrix τ, respectively. If eΓ0, (2.8){q}=12(q|K1+q|K2),{w}=12(w|K1+w|K2),{τ}=12(τ|K1+τ|K2),[q]=q|K1n1+q|K2n2,[w]=w|K1·n1+w|K2·n2,[τ]=τ|K1·n1+τ|K2·n2, where n1 and n2 are unit normal vectors on e pointing exterior to K1 and K2, respectively. We also define a matrix valued jump · for a vector w as (2.9)w=w|K1n1+w|K2n2. If eΩ, define (2.10){q}=q,[w]=w·n,{τ}=τ,w=wn. A straightforward computation gives (2.11)KhKqv·nds=eΓ0e[q]·{v}ds+eΓe{q}[v]ds,(2.12)KhKv·τnds=eΓ0e[τ]·{v}ds+eΓe{τ}:vds. Let Γqds=eΓeqds. Using (2.7), (2.12), and the fact that [u]=0 for u(H01(Ω)H2(Ω))2 on Γ0, (2.7) becomes (2.13)T𝒯hTun·ξds=Khj=1tAj+1CAjun·ξds+Γξ:{u}ds. Since [p]=0 for pH1(Ω) on Γ0, we also have (2.14)T𝒯hTpξ·nds=Khj=1tAj+1CAjpξ·nds+Γ{p}[ξ]ds. Let V(h)=Vh+(H2(Ω)H01(Ω))2. Define a mapping γ:V(h)Wh, (2.15)γv|T=1heev|Tds,T𝒯h, where he is the length of the edge e.

We define two norms for V(h) as follows: (2.16)|v|12=v1,h2+eΓγve2,|v|2=|v|12+KhhK2|v|2,K2, where v1,h2=|v|0,h2+|v|1,h2,  |v|0,h2=Kh|v|0,K2,|v|1,h2=Kh|v|1,K2, and hK= diameter of K.

As in , the standard inverse inequality implies that there is a constant C such that (2.17)|v|  C|v|1,vVh.

Lemma 2.1.

There exists a positive constant C independent of h such that (2.18)h|v|Cv  ,      vC|v|,vVh.

Proof.

As in , (2.19)h|v|1,hCv,      vC|v|1,h,vVh, where |v|1,h2=|v|1,h2+eΓγve2+KhhK2|v|2,K2. Since |v|1,h|v|, we have vC|v|. Note that vVh is a piecewise linear function, and h2|v|2=h2|v|0,h2+h2|v|1,h2+h2eΓγve2=I1+I2+I3. By Lemma 3.6 in , I2Cv2,I3Cv2, we have h|v|Cv.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B1">4</xref>]).

The operator γ is self-adjoint with respect to the L2-inner product, (u,γv)=(v,γu),u,vVh. Define |v|0=(v,γv)1/2. Then |·|0 and · are equivalent; here the equivalence constants are independent of h. And γv=v,vVh.

Let (2.20)a0(v,ξ)=(v,ξ)-Khj=1tAj+1CAjvn·ξds-Γξ:{v}ds,c(ξ,q)=Khj=1tAj+1CAjqξ·nds+Γ{q}[ξ]ds,b0(v,q)=KhK·vqdx. It is clear that the solutions (u,p) of the Darcy-Stokes equations (1.1a)–(1.1c) satisfy the following: (2.21)a0(u,ξ)+c(ξ,p)=(f,ξ),ξWh,b0(u,q)=0,qQh. Define the following bilinear forms: (2.22)A0(v,w)=a0(v,γw),w,vV(h),B0(v,q)=b0(v,q),vV(h),qL02(Ω),C(v,q)=c(γv,q),vV(h),qL02(Ω). Then systems (2.21) are equivalent to (2.23)A0(u,v)+C(v,p)=(f,γv),vVh,B0(u,q)=0,qQh.

We propose two discontinuous finite volume formulations based on modification of the weak formulation (2.23) for Darcy-Stokes problem (1.1a)–(1.1c). Let us introduce the bilinear forms as follows: (2.24)A1(v,w)=A0(v,w)+αeΓγve:γwe,B(v,q)=B0(v,q)-Γ{q}[γv]ds, where α>0 is a parameter to be determined later. For the exact solution (u,p) of (1.1a)–(1.1c), we have (2.25)A0(u,v)=A1(u,v),vVh,B0(u,q)=B(u,q),qQh. Therefore, it follows from (2.23) that (2.26)A1(u,v)+C(v,p)=(f,γv),vVh,B(u,q)=0,qQh. The corresponding discontinuous finite volume scheme seeks (uh,ph)Vh×Qh, such that (2.27)A1(uh,v)+C(v,ph)=(f,γv),vVh,B(uh,q)=0,qQh.

Let e be an edge of element K. It is well known (see ) that there exists a constant C such that for any function gH2(K), (2.28)ge2C(hK-1gK2+hK|g|1,K2),(2.29)gne2C(hK-1|g|1,K2+hK|g|2,K2), where C depends only on the minimum angle of K.

Let hv and h·v be the functions whose restriction to each element Kh is equal to v and ·v, respectively.

Lemma 2.3.

For v,wV(h), there exists a positive constant C independent of h such that (2.30)A1(v,w)C|v||w|.

Proof.

Let A**(v,w)=(v,γw)+A*(v,w), (2.31)A*(v,w)=-Khj=1tAj+1CAjvn·γwds. By Lemma 3.1 in , (2.32)A*(v,w)=(hv,hw)+KhK(γw-w)vnds+Kh(Δv,w-γw)K.|A**(v,w)||(v,γw)|  +|(hv,hw)|+|KhK(γw-w)vnds|+|Kh(Δv,w-γw)K|C(|v|0,h|w|0,h+|v|1,h|w|1,h+Kh(hK-1w-γwK2+hK|w-γw|1,K2)1/2×(hK-1|v|1,K2+hK|v|2,K2)  1/2+KhhK|v|2,K|w|1,K)C(|v|0,h|w|0,h+|v|1,h|w|1,h+(Kh|w|1,K2)1/2×(|v|1,h+(KhhK2|v|2,K2)1/2)+(KhhK2|v|2,K2)1/2|w|1,h)C|v||w|,A1(v,w)=A**(v,w)-Γγw:{v}ds+αeΓγve:γwe,C(|v||w|+(Kh(|v|1,K2+hK2|v|2,K2)1/2)(eΓγwe2)1/2+α(eΓγve2)1/2(eΓγwe2)1/2)C|v||w|.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B3">5</xref>]).

For any (v,q)V(h)×Qh, one has (2.33)C(v,q)=-B(v,q).

Lemma 2.5 (see [<xref ref-type="bibr" rid="B3">5</xref>]).

For (v,q)V(h)×L02(Ω), there exists a positive constant M independent of h such that (2.34)C(v,q)M|v|(q+(KhhK2|q|1,K2)1/2). If (v,q)Vh×Qh, then (2.35)C(v,q)M|v|q.

Lemma 2.6.

For any vVh, there is a constant C0 independent of h such that for α large enough (2.36)A1(v,v)C0|v|2.

Proof.

Using the proof of Lemmas 3.1 and 3.5 in , for vVh, (2.37)Γγv:vdsC|v|1(eΓγve2)1/2,A*(v,w)=(hv,hw),v,wVh, we have (2.38)A1(v,v)=(v,γv)+(hv,hv)+αeΓγve2-Γγv:{v}ds,|v|0,h2+|v|1,h2+αeΓγve2-C|v|1(eΓγve2)1/2C|v|12C0|v|2, when α is large enough.

The value of α depends on the constant in the inverse inequality. Therefore, the value of α for which A1(·,·) is coercive is mesh dependent. We introduce a second discontinuous finite volume scheme which is parameter insensitive. Define a bilinear form as follows: (2.39)A2(v,w)=A1(v,w)+Γγv:{w}ds. Similar to the bilinear form A1(·,·), for the exact solution (u,p) of the Darcy-Stokes problem we have (2.40)A2(u,v)=A0(u,v),vVh. Consequently, the solution of the Darcy-Stokes problem satisfies the following variational equations: (2.41)A2(u,v)+C(v,p)=(f,γv),vVh,B(u,q)=0,qQh. Our second discontinuous finite volume scheme for (1.1a)–(1.1c) seeks (uh,ph)Vh×Qh, such that (2.42)A2(uh,v)+C(v,ph)=(f,γv),vVh,B(uh,q)=0,qQh.

For any value of α>0, we have (2.43)A2(v,v)=(v,γv)+(hv,hv)+αeΓγve2C|v|12C0|v|2,vVh. Similarly, we can prove that (2.44)A2(v,w)C|w||v|,v,wV(h). Let A(v,w)=A1(v,w) or A(v,w)=A2(v,w). In the rest of the paper, we assume that the following is true: (2.45)A(v,v)C0|v|2. If A(v,w)=A2(v,w), (2.45) holds for any α>0. If A(v,w)=A1(v,w), (2.45) holds for only α large enough.

3. Error Estimates

We will derive optimal error estimates for velocity in the norm |·| and for pressure in the L2-norm. A first-order error estimate for velocity in L2-norm will be obtained.

Let e be an interior edge shared by two elements K1 and K2 in h. If ev|K1ds=ev|K2ds, we say that v is continuous on e. We say that v is zero at eΩ if evds=0. Define a subspace V^h of Vh by (3.1)V^h={vL2(Ω)2:v|KQ^1(K)2Kh  is  continuous  at  eΓ0    and  is  zero  at  eΩ}   for rectangular meshes and by (3.2)V^h={vL2(Ω)2:v|KP1(K)2Kh  is  continuous  ateΓ0  and  is  zero  at  eΩ} for triangular mesh.

It has been proven in [8, 9] that the following discrete inf-sup condition is satisfied; that is, there exists a positive constant β0 such that (3.3)supvV^h(h·v,q)|v|1,hβ0q,qQh.

Lemma 3.1.

The bilinear form B(·,·) satisfies the discrete inf-sup condition (3.4)supvVhB(v,q)|v|βq,qQh, where β is a positive constant independent of the mesh size h.

Proof.

For vV^hVh and qQh, we have B(v,q)=(h·v,q), and |v|1=v1,h. By Poincare-Friedrichs v1,hC|v|1,h, with (3.3), and (2.17) we get for any qQh(3.5)β0qsupvV^h(·v,q)|v|1,hCsupvV^hB(v,q)|v|1C1supvVhB(v,q)|v|. With β=β0/C1, we have proven (3.4).

Define an operator πK:H1(K)P1(K) or Q^1(K). For all vH1(K), (3.6)eiπKvds=eivds,i=1,,t, where ei,i=1,,t, are the t sides of the element K. t=3 if K is a triangle and t=4 if K is a rectangle. It was proven in  that (3.7)|πKv-v|s,KCh2-s|v|2,K,s=0,1,2. For all v=(v1,v2)H01(Ω)2, define Π1v=(Π1v1,Π1v2)Vh by (3.8)Π1vi|K=πKvi,Kh,i=1,2. Using the definition of Π1 and integration by parts, we can show that (3.9)B(v-Π1v,q)=0,qQh. The Cauchy-Schwarz inequality implies (3.10)γve2=(1heevds)2(1he)2ev2dseds=e1hev2ds. Equations (2.28) and (3.8) imply that (3.11)eΓγ(u-Π1u)e2C(|u-Π1u|1,h2+Khh-2u-Π1uK2)Ch2u22. The definitions of the norm |·|, (3.7), and (3.11) give (3.12)|u-Π1u|2=|u-Π1u|0,h2+|u-Π1u|1,h2+eΓγ(u-Π1u)e2+Khh2|u-Π1u|2,K2Ch2u22.

Theorem 3.2.

Let (uh,ph)Vh×Qh be the solution of (2.27), and let (u,p)(H2(Ω)H01(Ω))2×(L02(Ω)H1(Ω)) be the solution of (1.1a)–(1.1c). Then there exists a constant C independent of h such that (3.13)|u-uh|+p-phCh(u2+p1),(3.14)u-uhCh(u2+p1).

Proof.

Let ε=u-Π1u,εh=uh-Π1u,η=p-Π2p,ηh=ph-Π2p, where Π2 is L2 projection from L02(Ω)Qh. Then u-uh=ε-εh,p-ph=η-ηh. Subtracting (2.26) from (2.27) and using Lemma 2.4, we get error equations(3.15a)A(εh,v)-B(v,ηh)=A(ε,v)+C(v,η),vVh,(3.15b)B(εh,q)=B(ε,q)=0,qQh.By letting v=εh in (3.15a) and q=ηh in (3.15b), the sum of (3.15a) and (3.15b) gives (3.16)A(εh,εh)=A(ε,εh)+C(εh,η). Thus, it follows from the coercivity (2.45), the boundedness (2.30), (2.44), and (2.34) that (3.17)|εh|2C(|ε||εh|  +(η+(KhhK2|η|1,K2)1/2)|εh|)  , which implies the following: (3.18)|εh|C(|ε|+η+(KhhK2|η|1,K2)1/2). The previous estimate can be rewritten as (3.19)|uh-Π1u|C(|u-Π1u|+p-Π2p+(KhhK2|p-Π2p|1,K2)1/2). Now using the triangle inequality, (3.7), the definition of Π2, and the inequality mentioned previously, we get (3.20)|u-uh|C(|u-Π1u|+|uh-Π1u|)Ch(u2+p1), which completes the estimate for the velocity approximation.

Discrete inf-sup condition (3.4), (3.15a), (3.15b), Lemmas 2.5, 2.4, and inverse inequality give (3.21)ph-Π2p1βsupvVhB(v,Π2p-ph)|v|1,h=1βsupvVhC(v,ph-Π2p)|v|1,h=1βsupvVhC(v,ph-p)+C(v,p-Π2p)|v|=1βsupvVhA(u-uh,v)+C(v,p-Π2p)|v|C(|u-uh|+p-Π2p+(KhhK2|p-Π2p|1,K2)1/2)Ch(u2+p1). Using the previous inequality and the triangle inequality, we have completed the proof of (3.13).

Using Lemma 2.1, (3.12), and (3.13), we have (3.22)uh-Π1uhC|uh-Π1uh|C(|u-uh|+|u-Π1uh|)Ch(u2+p1). Equations (3.22) and (3.7) and the triangle inequality imply (3.14). We have completed the proof.

4. Numerical Experiments

In this section, we present a numerical example for solving the problems (1.1a)–(1.1c) by using the discontinuous finite volume element method presented with (2.27) and (2.42). Let Ω=(0,1)×(0,1), h be the Delaunay triangulation generated by EasyMesh  over Ω with mesh size h as shown in Figure 5. We consider the case of σ=1,μ=1, the exact velocity u1(x,y)=-x2(x-1)2y(y-1)(2y-1), u2(x,y)=-u1(y,x) and the pressure p(x,y)=(x-0.5)(y-0.5). Denote the numerical solution as uh and ph with step hd which is used to generate the mesh data in the EasyMesh input file, and h=max{he:eΓ}. For α=2, the numerical results are presented in Tables 1 and 2. It is observed from the tables that the numerical results support our theory.

Error behavior for scheme (2.27).

h d h | u - u h | | u - u 2 h | | u - u h | u - u h u - u 2 h u - u h p - p h p - p 2 h p - p h
1 / 8 1.598 e - 1 2.082 e - 2 3.393 e - 4 1.068 e - 2
1 / 16 8.372 e - 2 1.031 e - 2 2.0 9.649 e - 5 3.5 5.345 e - 3 2.0
1 / 32 3.679 e - 2 5.185 e - 3 2.0 2.598 e - 5 3.7 2.650 e - 3 2.0
1 / 64 1.899 e - 2 2.611 e - 3 2.0 6.795 e - 6 3.8 1.323 e - 3 2.0
1 / 128 9.413 e - 3 1.307 e - 3 2.0 1.730 e - 6 3.9 6.598 e - 4 2.0

Error behavior for scheme (2.42).

h d h | u - u h | | u - u 2 h | | u - u h | u - u h u - u 2 h u - u h p - p h p - p 2 h p - p h
1 / 8 1.598 e - 1 2.071 e - 2 3.280 e - 4 1.079 e - 2
1 / 16 8.372 e - 2 1.027 e - 2 2.0 9.204 e - 5 3.5 5.380 e - 3 2.0
1 / 32 3.679 e - 2 5.175 e - 3 2.0 2.476 e - 5 3.7 2.659 e - 3 2.0
1 / 64 1.899 e - 2 2.608 e - 3 2.0 6.361 e - 6 3.8 1.325 e - 3 2.0
1 / 128 9.413 e - 3 1.306 e - 3 2.0 1.613 e - 6 3.9 6.603 e - 4 2.0

Triangular and its dual partition of (0,1)×(0,1).

Acknowledgments

This paper is supported by the Excellent Young and Middle-Aged Scientists Research Fund of Shandong Province (2008BS09026), National Natural Science Foundation of China (11171193), and National Natural Science Foundation of Shandong Province (ZR2011AM016).

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