# A New Finite Element Method for  Darcy-Stokes-Brinkman Equations

Authors: Bishnu P. Lamichhane
Journal: ISRN Computational Mathematics
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

Recently developing an efficient finite element method for Darcy-Stokes-Brinkman problem has become an active area of research [1–5]. This mathematical model is often called a generalized Stokes problem [6] and is also a special case of Oseen's equations when the convective term is absent [1]. 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 [4]. 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 [7] and the grad-div stabilization proposed earlier for the Stokes equations [8]. 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 [8] 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 Section3. 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. LetΩ in ℝd, 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 g∈L2(Ω), the Darcy-Stokes-Brinkman equations with homogeneous Dirichlet boundary condition in Γ read
(1)-νΔu+∇p+σu=finΩ,divu=ginΩ,
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 ν≥0 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 x∈Ω. 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 notationsL2(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={q∈L2(Ω):∫Ωqdx=0}.The weak formulation of the Darcy-Stokes-Brinkman problem is to find(u,p)∈W×M such that
(3)ν∫Ω∇u:∇vdx+∫Ωdivvpdx+σ∫Ωuvdx=ℓ(v),v∈W,∫Ω‍divuqdx=g(q),q∈M,
where ℓ(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 viscosityν→0, the appropriate space to seek a solution of the variational problem (8) is H0(div,Ω) defined as
(4)H0(div,Ω)={σ∈H(div,Ω):σ·n=0on∂Ω},
where
(5)H(div,Ω)={u∈[L2(Ω)]d:divu∈L2(Ω)}
(see [3]). A standard norm on the space H(div,Ω) is
(6)∥u∥div,Ω2=∥u∥0,Ω2+∥divu∥0,Ω2.With this change of the character of the solution, when the viscosityν vanishes, we define an appropriate norm for the problem (8) as
(7)∥v∥a2:=ν∥v∥1,Ω2+σ∥v∥div,Ω2,v∈[νW∩σH(div,Ω)].Therefore now we use the spaceV:=νW∩σH(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)ν∫Ω∇u:∇vdx+∫Ωdivvpdx+σ∫Ωuvdx=ℓ(v),v∈W,∫Ωdivuqdx=g(q),q∈M.


## 3. Finite Element Discretizations

We consider a quasiuniform triangulation𝒯h 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 integern and an element K∈𝒯h, 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)∈ℝ2:x>0,y>0,x+y<1},
and the reference tetrahedron is
(10)K^:={(x,y,z)∈ℝ3:x>0,y>0,z>0,x+y+z<1}.A typical elementK∈𝒯h 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 elementK are obtained by maps of linear functions from the reference element:
(11)Sh:={vh∈H01(Ω),vh∣K=v^h∘FK-1,v^h∈𝒫1(K^),K∈𝒯h}.Let the set of all vertices in𝒯h be denoted by 𝒱h:={xi}i=1N and 𝒩h:={1,2,…,N}.A dual mesh𝒯h* is introduced based on the primal mesh 𝒯h so that the elements of 𝒯h* are called control volumes. Each control volume element Vi∈𝒯h* is associated with a vertex xi∈𝒱h. 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 xi∈𝒱h let 𝒯hi and ℰhi be the set of elements and edges touching xi, respectively. Then the volume element Vi corresponding to the vertex xi∈𝒱h is the polygonal domain joining centroids of all elements in 𝒯hi and centroids of all edges in ℰhi. 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.Figure 1
Primal and dual meshes for a triangular grid.In the following we use a generic constantC, 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 mesh𝒯h* regular or quasiuniform if there exists a positive constant C>0 such that
(12)Chd≤|Vi|≤hd,Vi∈𝒯h*,
where h is the maximum diameter of all elements T∈𝒯h. It can be shown that, if 𝒯h is locally regular, that is, there is a constant C such that
(13)ChTd≤|T|≤hTd,T∈𝒯h
with diam(T)=hT for all elements T∈𝒯h, 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*:={p∈L02(Ω):p∣V∈𝒫0(V),V∈𝒯h*}.Nowph∈Sh* and uh∈Sh 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 elementK^, we define a bubble function(15)bK^={27xy(1-x-y),ifK^isthereferencetriangle,256xyz(1-x-y-z),ifK^isthereferencetetrahedron.Then defining the space of bubble functionsBh as
(16)Bh:={bh∈H01(Ω),bh∣k=bK^∘FK-1,bK^isabubblefunctiononK^,K∈𝒯h},
we introduce our finite element space for velocity as Vh=[Sh⊕Bh]d.With this definition of the spacesVh and Sh* we have the following lemma [13].Lemma 1.
There exists a constantβ>0 independent of the mesh-size h such that
(17)infqh∈Sh*∖{0}supvh∈Vh∖{0}∫Ωdivvhqh∥vh∥1,Ω∥qh∥0,Ω≥β.We now consider the limiting case whenν→0. The main difficulty in this case using the conforming approximation is to get the coercivity of the bilinear form
(18)a~(uh,vh)=σ∫Ω‍uh·vhdx
on the kernel space
(19)Kh:={vh∈Vh:∫Ωdivvhqhdx=0,qh∈Sh*}.Whenν→0, the coercivity of the bilinear form a~(·,·) on Kh demands the existence of a constant C>0 such that
(20)a~(vh,vh)≥C∥vh∥a2=Cσ∥vh∥div,Ω2.Assuming thatσ>0 this condition requires the existence of a constant C>0 such that
(21)∥vh∥0,Ω≤C∥divvh∥0,Ω.Since this condition is not satisfied for the spaceVh, we add a stabilising term to the bilinear form a~(·,·) to obtain another bilinear form a(·,·)(22)a(uh,vh):=a~(uh,vh)+σ∫Ω(divuh-Qhdivuh)‍×(divvh-Qhdivvh)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)=ℓ(vh),vh∈Vh,b(uh,qh)=g(qh),qh∈Sh*.Remark 2.
Due to the fact that the basis functions for the spaceSh* 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 [14] at least for a simplicial grid, where computation of the operator Qh will be more difficult, expensive, and nonlocal.Remark 3.
If the source termg 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 [8].We now show that the bilinear forma(·,·) is uniformly coercive on Kh with respect to the norm ∥·∥a.Lemma 4.
One has(24)a(vh,vh)=ν∥vh∥1,Ω2+σ∥vh∥div,Ω2,vh∈Kh.Proof.
Note that(25)a(vh,vh)=ν∥vh∥1,Ω2+σ∥vh∥0,Ω2+σ∥divvh-Qhdivvh∥0,Ω2=ν∥vh∥1,Ω2+σ∥vh∥0,Ω2+σ∥divvh∥0,Ω2=ν∥vh∥1,Ω2+σ∥vh∥div,Ω2,
as Qhdivvh=0 for vh∈Kh.Clearly, we see that the bilinear formsa(·,·) and b(·,·), and the linear form ℓ(·) are also continuous on the spaces where they are defined. Moreover, since for a constant C>0 we have
(26)∥vh∥a≤C∥vh∥1,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)∥uh∥a+∥ph∥0,Ω≤C∥ℓ∥0,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-uh∥a+∥p-ph∥0,Ω≤C(infvh∈Vh∥u-vh∥a+infqh∈Sh*∥p-qh∥0,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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