ISRN.COMPUTATIONAL.MATHEMATICS ISRN Computational Mathematics 2090-7842 Hindawi Publishing Corporation 798059 10.1155/2013/798059 798059 Research Article A New Finite Element Method for Darcy-Stokes-Brinkman Equations Lamichhane Bishnu P. Upadhyay R. K. Weng W. G. 1 School of Mathematical & Physical Sciences Mathematics Building University of Newcastle University Drive Callaghan NSW 2308 Australia ncl.ac.uk 2013 24 11 2013 2013 19 08 2013 15 09 2013 2013 Copyright © 2013 Bishnu P. Lamichhane. 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.

We present a new finite element method for Darcy-Stokes-Brinkman equations using primal and dual meshes for the velocity and the pressure, respectively. Using an orthogonal basis for the discrete space for the pressure, we use an efficiently computable stabilization to obtain a uniform convergence of the finite element approximation for both limiting cases.

1. Introduction

Recently developing an efficient finite element method for Darcy-Stokes-Brinkman problem has become an active area of research . This mathematical model is often called a generalized Stokes problem  and is also a special case of Oseen's equations when the convective term is absent . This model incorporates both Darcy and Stokes problem. If the viscosity vanishes, the model becomes the Darcy problem, and if another parameter is set to zero, it reduces to Stokes problem. Therefore, this is important when studying the coupling between Stokes and Darcy equations. Some applications of Darcy-Stokes-Brinkman problem are given in . This model also arises from the time discretization of time-dependent Stokes equations.

In this paper we propose a new finite element method for this problem using stabilization based on a local projection. Our technique is the combination of the local projection stabilization for Stokes equations  and the grad-div stabilization proposed earlier for the Stokes equations . We note that we mainly focus on the Darcy-Stokes-Brinkman model so that the divergence of the velocity field is not identically zero but is given as a function in L2-space as in [3, 4]. Using a standard continuous linear, bilinear, or trilinear finite element space for the velocity, the grad-div stabilization  is to be modified in this case. Since the modification involves projecting the divergence of the velocity field onto the finite element space for the pressure, we first use a discontinuous pressure space in the finite element scheme. This leads to an efficient realization of the stabilization. The finite element space for the pressure is the piecewise constant function space on the dual mesh. The inf-sup condition is satisfied using an element-wise defined bubble function in each element.

This paper is organized as follows. We present the boundary value problem for the Darcy-Stokes-Brinkman problem in the next section. We present our finite element method, propose stabilization, and analyze the discrete problem in Section 3. We prove an optimal a priori error estimate for both cases in this section. Finally, a conclusion is drawn in the last section.

2. Darcy-Stokes-Brinkman Problem

This section is devoted to the introduction of the boundary value problem of the Darcy-Stokes-Brinkman flow. Let Ω in d, d{2,3}, be a bounded domain with polygonal or polyhedral boundary Γ. For a prescribed body force f[L2(Ω)]d and another function gL2(Ω), the Darcy-Stokes-Brinkman equations with homogeneous Dirichlet boundary condition in Γ read (1)-νΔu+p+σu=fin  Ω,divu=gin  Ω, with u=  0 on Γ, where u is the velocity, p is the pressure, and ν denotes the viscosity of the fluid. For simplicity we assume that ν0 and σ are constant. However, the theory also can be easily extended to cover the case when   ν and σ depend on the spatial coordinates xΩ. When the viscosity of the fluid ν tends to zero, this problem reduces to the Darcy problem, and when the parameter σ approaches zero, this problem reduces to the Stokes problem. Here we aim at developing a finite element method which is stable in both limit cases.

Here we use standard notations L2(Ω), H1(Ω), and H01(Ω) for Sobolev spaces; see [9, 10] for details. Let W:=[H01(Ω)]d be the vector Sobolev space with inner product (·,·)1,Ω and norm ·1,Ω defined in the standard way: (u,v)1,Ω:=i=1d(ui,vi)1,Ω, with the norm being induced by this inner product. We also define another subspace M of L2(Ω) as (2)M={qL2(Ω):Ωqdx=0}.

The weak formulation of the Darcy-Stokes-Brinkman problem is to find (u,p)W×M such that (3)νΩu:vdx+Ωdivvpdx+σΩuvdx=(v),vW,Ωdivuqdx=g(q),qM, where (v)=Ωf·vdx. It is well known that the weak formulation of this Darcy-Stokes-Brinkman problem is well-posed [3, 11].

We note that, when the viscosity ν0, the appropriate space to seek a solution of the variational problem (8) is H0(div,Ω) defined as (4)H0(div,Ω)={σH(div,Ω):σ·n=0onΩ}, where (5)H(div,Ω)={u[L2(Ω)]d:divuL2(Ω)} (see ). A standard norm on the space H(div,Ω) is (6)udiv,Ω2=u0,Ω2+divu0,Ω2.

With this change of the character of the solution, when the viscosity ν vanishes, we define an appropriate norm for the problem (8) as (7)va2:=νv1,Ω2+σvdiv,Ω2,v[νWσH(div,Ω)].

Therefore now we use the space V:=νWσH(div,Ω) as the space for the velocity u. Due to the occurrences of the parameters in the problem it is appropriate to use the suitable norm defined above although it is clear that H(div,Ω)[H1(Ω)]d. Thus we consider the problem of finding (u,p)V×M such that (8)νΩu:vdx+Ωdivvpdx+σΩuvdx=(v),vW,Ωdivuqdx=g(q),qM.

3. Finite Element Discretizations

We consider a quasiuniform triangulation 𝒯h of the polygonal or polyhedral domain Ω, where 𝒯h consists of simplices (triangles or tetrahedra), where h denotes the mesh-size. Note that 𝒯h denotes the set of elements.

For nonnegative integer n and an element K𝒯h, we let 𝒫n(K) denote the space of polynomials in two variables of total degree less than or equal to n, where K is a triangle or tetrahedron. Let K^ be a reference triangle or tetrahedron, where the reference triangle is defined as (9)K^:={(x,y)2:x>0,y>0,x+y<1}, and the reference tetrahedron is (10)K^:={(x,y,z)3:x>0,y>0,z>0,x+y+z<1}.

A typical element K𝒯h is generated by an affine map FK from the reference element K^, in which FK is defined using the basis functions corresponding to 𝒫1.

The finite element space of displacement is taken to be the space of continuous functions whose restrictions to an element K are obtained by maps of linear functions from the reference element: (11)Sh:={vhH01(Ω),vhK=v^hFK-1,v^h𝒫1(K^),K𝒯h}.

Let the set of all vertices in 𝒯h be denoted by 𝒱h:={xi}i=1N and 𝒩h:={1,2,,N}.

A dual mesh 𝒯h* is introduced based on the primal mesh 𝒯h so that the elements of 𝒯h* are called control volumes. Each control volume element Vi𝒯h* is associated with a vertex xi𝒱h. For simplicity we explain the construction of the dual mesh for triangular meshes [12, 13]. The construction is easily extended to a tetrahedral mesh. For a vertex xi𝒱h let 𝒯hi and hi be the set of elements and edges touching xi, respectively. Then the volume element Vi corresponding to the vertex xi𝒱h is the polygonal domain joining centroids of all elements in 𝒯hi and centroids of all edges in hi. The set of all volume elements in 𝒯h* will form a nonoverlapping decomposition of the polygonal domain Ω; see also [12, 13]. A dual mesh for a triangular mesh is shown in Figure 1.

Primal and dual meshes for a triangular grid.

In the following we use a generic constant C, which takes different values at different places but is always independent of the mesh-size h and the parameters ν and σ.

We call the control volume mesh 𝒯h* regular or quasiuniform if there exists a positive constant C>0 such that (12)Chd|Vi|hd,Vi𝒯h*, where h is the maximum diameter of all elements T𝒯h. It can be shown that, if 𝒯h is locally regular, that is, there is a constant C such that (13)ChTd|T|hTd,T𝒯h with diam(T)=hT for all elements T𝒯h, then this dual mesh 𝒯h* is also locally regular. The dual volume element space Sh* to discretize the pressure is now defined by (14)Sh*:={pL02(Ω):pV𝒫0(V),V𝒯h*}.

Now phSh* and uhSh can be written as uh=i=1Nuiϕi and ph=i=1Npiχi, where ϕi are the standard nodal basis functions associated with the vertex xi, and χi are the characteristic functions of the volume Vi.

For the reference element K^, we define a bubble function(15)bK^={27xy(1-x-y),if  K^  is  the  reference  triangle,256xyz(1-x-y-z),if  K^  is  the  reference  tetrahedron.

Then defining the space of bubble functions Bh as (16)Bh:={bhH01(Ω),bhk=bK^FK-1,bK^isabubblefunctiononK^,K𝒯h}, we introduce our finite element space for velocity as Vh=[ShBh]d.

With this definition of the spaces Vh and Sh* we have the following lemma .

Lemma 1.

There exists a constant β>0 independent of the mesh-size h such that (17)infqhSh*{0}supvh  Vh{0}Ωdivvhqhvh1,Ωqh0,Ωβ.

We now consider the limiting case when ν0. The main difficulty in this case using the conforming approximation is to get the coercivity of the bilinear form (18)a~(uh,vh)=σΩ  uh·vhdx on the kernel space (19)Kh:={vhVh:Ωdivvhqhdx=0,qhSh*}.

When ν0, the coercivity of the bilinear form a~(·,·) on Kh demands the existence of a constant C>0 such that (20)a~(vh,vh)Cvha2=Cσvhdiv,Ω2.

Assuming that σ>0 this condition requires the existence of a constant C>0 such that (21)vh0,ΩCdivvh0,Ω.

Since this condition is not satisfied for the space Vh, we add a stabilising term to the bilinear form a~(·,·) to obtain another bilinear form a(·,·)(22)a(uh,vh):=a~(uh,vh)+σΩ(divuh-Qhdivuh)×(divvh-Qh divvh)dx, where Qh is an orthogonal projection onto the space Sh*. With this definition of the bilinear form, our discrete problem is to find [uh,ph]Vh×Sh* such that (23)a(uh,vh)+b(vh,ph)=(vh),vhVh,b(uh,qh)=g(qh),qh  Sh*.

Remark 2.

Due to the fact that the basis functions for the space Sh* are orthogonal with respect to L2-inner product, computing the action of the operator Qh is very easy and efficient. Since the basis functions of Sh* are piecewise constant, the stabilization is also local. In fact, similar stabilization can be applied using mini type finite element method  at least for a simplicial grid, where computation of the operator Qh will be more difficult, expensive, and nonlocal.

Remark 3.

If the source term g is identically zero, the stabilization is simplified as Qhdivvh=0. In fact, such stabilization is called grad-div stabilization and was used before for Stokes equations in .

We now show that the bilinear form a(·,·) is uniformly coercive on Kh with respect to the norm ·a.

Lemma 4.

One has (24)a(vh,vh)=νvh1,Ω2+σvhdiv,Ω2,vhKh.

Proof.

Note that (25)a(vh,vh)=νvh1,Ω2+σvh0,Ω2+σdivvh-Qhdivvh0,Ω2=νvh1,Ω2+σvh0,Ω2+σdivvh0,Ω2=νvh1,Ω2+σvhdiv,  Ω2, as Qhdivvh=0 for vhKh.

Clearly, we see that the bilinear forms a(·,·) and b(·,·), and the linear form (·) are also continuous on the spaces where they are defined. Moreover, since for a constant C>0 we have (26)vhaCvh1,Ω, we have the inf-sup condition from Lemma 1 when the ·1,Ω is replaced by ·a. Thus we have the following theorem.

Theorem 5.

The discrete saddle point problem (23) has a unique solution [uh,ph]Vh×Sh* such that (27)uha+ph0,ΩC0,Ω.

The error estimate is given in the following theorem.

Theorem 6.

Let [u,p]V×M be the unique solution of (8) and [uh,ph]Vh×Sh* that of the discrete (23). One has the following error estimate uniformly with respect to σ and ν: (28)u-uha+p-ph0,ΩC(infvhVhu-vha+infqhSh*p-qh0,Ω).

4. Conclusion

We have presented a very efficient finite element technique for approximating the solution of Darcy-Stokes-Brinkman equations. Using local projection stabilization for the limiting case of vanishing viscosity we have shown that the error estimate is uniform for both limiting cases.

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