A Discontinuous Finite Volume Method for the Darcy-Stokes Equations

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 -error estimates are derived for the approximations of both velocity and pressure. Some numerical examples verifying the theoretical predictions are presented.


Introduction
The study of discontinuous Galerkin methods has been a very active research field since they were proposed by Reed and Hill 1 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. 2 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 3 developed a new discontinuous finite volume method and analyzed it for the second-order elliptic problem.Bi and Geng 4 proposed the semidiscrete and the backward Euler fully discrete discontinuous finite volume element methods for the second-order parabolic problems.Ye 5 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 L 2 -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 6 , the nonconforming Crouzeix-Raviart element is stabilized for the Darcy-Stokes problem with terms motivated by a discontinuous Galerkin approach.In 7 , 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 L 2error 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 where Ω is a bounded polygonal domain in R 2 with boundary ∂Ω.u u 1 , u 2 is the velocity, p is the pressure, and f is a given force term.We assume σ 1, μ 1.

Discontinuous Finite Volume Formulation
Let R h be a triangular or rectangular partition of Ω.The triangles or rectangles in R 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 T h of the primal partition R h to be the union of the triangles shown in Figures 1 and 2 for both triangular and rectangular meshes.
Let P k 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 and on rectangular partition by where Q 1 denotes the space of functions of the form a bx 1 cx 2 d x 2 1 − x 2 2 on K.  Let Q h be the finite dimensional space for pressure where Define the finite dimensional test function space W h for velocity associated with the dual partition T h as Multiplying 1.1a and 1.1b by ξ ∈ W h and q ∈ Q h , respectively, we have where n is the unit outward normal vector on ∂T .Let T j ∈ T h j 1, . . ., t be the triangles in K ∈ R h , where t 3 for triangular meshes and t 4 for rectangular meshes, as shown as Figures 3 and 4. Then we have where For two matrix valued variables σ and τ, we define σ : Let e be an interior edge shared by two elements K 1 and K 2 in R h .We define the average {•} and jump • on e for scalar q, vector w, and matrix τ, respectively.If e ∈ Γ 0 , where n 1 and n 2 are unit normal vectors on e pointing exterior to K 1 and K 2 , respectively.We also define a matrix valued jump • for a vector w as

2.13
Since p 0 for p ∈ H 1 Ω on Γ 0 , we also have where h e is the length of the edge e.We define two norms for V h as follows: ,K , and h K diameter of K.As in 5 , the standard inverse inequality implies that there is a constant C such that 2.17

2.20
It is clear that the solutions u, p of the Darcy-Stokes equations 1.1a -1.1c satisfy the following:

2.21
Define the following bilinear forms:

2.22
Then systems 2.21 are equivalent to

2.23
Journal of Applied Mathematics 7 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: 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
Therefore, it follows from 2.23 that

2.26
The corresponding discontinuous finite volume scheme seeks u h , p h ∈ V h × Q h , such that

2.27
Let e be an edge of element K.It is well known see 2 that there exists a constant C such that for any function g ∈ H 2 K , where C depends only on the minimum angle of K.
Let ∇ h v and ∇ h • v be the functions whose restriction to each element ∀K ∈ R h is equal to ∇v and ∇ • v, respectively.Lemma 2.3.For v, w ∈ V h , there exists a positive constant C independent of h such that • γw ds.
Lemma 2.5 see 5 .For v, q ∈ V h × L 2 0 Ω , there exists a positive constant M independent of h such that Lemma 2.6.For any v ∈ V h , there is a constant C 0 independent of h such that for α large enough

2.36
Proof.Using the proof of Lemmas 3.1 and 3.5 in 5 , for v ∈ V h ,

2.37
Journal of Applied Mathematics we have when α is large enough.
The value of α depends on the constant in the inverse inequality.Therefore, the value of α for which A 1 •, • 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
Similar to the bilinear form A 1 •, • , for the exact solution u, p of the Darcy-Stokes problem we have Consequently, the solution of the Darcy-Stokes problem satisfies the following variational equations:

2.41
Our second discontinuous finite volume scheme for 1.1a -1.1c seeks

2.42
For any value of α > 0, we have

2.43
Similarly, we can prove that

2.44
Let A v, w A 1 v, w or A v, w A 2 v, w .In the rest of the paper, we assume that the following is true:

2.45
If A v, w A 2 v, w , 2.45 holds for any α > 0. If A v, w A 1 v, w , 2.45 holds for only α large enough.

Error Estimates
We will derive optimal error estimates for velocity in the norm | • | and for pressure in the L 2 -norm.A first-order error estimate for velocity in L 2 -norm will be obtained.
Let e be an interior edge shared by two elements K 1 and K 2 in R h .If e v| K 1 ds e v| K 2 ds, we say that v is continuous on e.We say that v is zero at e ∈ ∂Ω if e v ds 0. Define a subspace V h of V h by for rectangular meshes and by 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 β 0 such that where β is a positive constant independent of the mesh size h.
h , with 3.3 , and 2.17 we get for any q ∈ Q h With β β 0 /C 1 , we have proven 3.4 .

Journal of Applied Mathematics
Define an operator π K : v ds, i 1, . . ., t, where e i , 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 8 that Using the definition of Π 1 and integration by parts, we can show that 3.9 The Cauchy-Schwarz inequality implies

Proof. Let
26 from 2.27 and using Lemma 2.4, we get error equations

3.15b
By letting v ε h in 3.15a and q η h in 3.15b , the sum of 3.15a and 3.15b gives

3.16
Thus, it follows from the coercivity 2.45 , the boundedness 2.30 , 2.44 , and 2.34 that 3.17 which implies the following:

Journal of Applied Mathematics
The previous estimate can be rewritten as

3.19
Now using the triangle inequality, 3.7 , the definition of Π 2 , and the inequality mentioned previously, we get 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
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
Equations 3.22 and 3.7 and the triangle inequality imply 3.14 .We have completed the proof.

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 , R h be the Delaunay triangulation generated by EasyMesh 10 over Ω with mesh size h as shown in Figure 5.We consider the case of σ 1, μ 1, the exact velocity u 1 x, y −x 2 x − 1 2 y y − 1 2y − 1 , u 2 x, y −u 1 y, x and the pressure p x, y x − 0.5 y − 0.5 .Denote the numerical solution as u h and p h with step h d which is used to generate the mesh data in the EasyMesh input file, and h max{h e : e ∈ Γ}.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.

Figure 1 :
Figure 1: Element T ∈ T h for triangular mesh.

Figure 2 :
Figure 2: Element T ∈ T h for rectangular mesh.
2. Let u h , p h ∈ V h ×Q h be the solution of 2.27 , and let u, p ∈ H 2 Ω ∩H 1

Table 1 :
Error behavior for scheme 2.27 .

Table 2 :
Error behavior for scheme 2.42 .