Discontinuous Galerkin Immersed Finite Volume Element Method for Anisotropic Flow Models in Porous Medium

and Applied Analysis 3


Introduction
Let us consider the following elliptic interface problems in a convex domain Ω ⊂  2 : −∇ ⋅ (B∇) = , (, ) ∈ Ω,  = 0, (, ) ∈ Ω, where Ω is separated into two subdomains Ω + and Ω − by interface Γ ∈  2 , see Figure 1 for an illustration, and  ∈  2 (Ω);  satisfies the following homogenous jump conditions on the interface Γ: Equation ( 1) describes many real diffusion processes in fluid dynamics and engineering applications, such as the miscible displacement with discontinues conductivity due to complex strata or multiphase flux.It is significant to seek efficiently the numerical solution to the interface problems for better understanding of the mechanism of the flow process and conducting engineering practice.
When B() is a scale function, which corresponds to an isotropic flow case, two classes of numerical methods were developed to approximate (1) in terms of the meshes.One is the fitted finite element or fitted difference method [1][2][3], which restricts the mesh to be aligned with the smooth interface Γ. Consequently, the fitted methods are costly for more complicated time dependent problems in which the interface moves with time and repeated grid generation is called for.The other one is the immersed interface difference or finite element methods in which the Cartesian grid is naturally used even though it cannot match a curved interface.Although the immersed difference methods [4,5] were demonstrated to be very effective, convergence analysis of related finite difference methods is extremely difficult and is still open.On the other hand, the immersed finite element method (IFE) was developed, which allows the interface to go through the interior of the element; see the references [6][7][8][9] and the references therein.Numerical experiments demonstrated an optimal order of the errors.Once again, it is not easy to analyze this method.Further, to preserve the conservative characteristics of the interface model (1), [10] developed an immersed finite volume element (IFVE) method by combining the finite volume element method [11][12][13][14][15][16] and the immersed finite element method.
In realistic diffusion processes, the interface problem (1) displays much often its anisotropic type.That is, the conductivity B() becomes a tenser-formed function.The goals of this paper are as follows: (1) to develop a discontinuous Galerkin-immersed interface-finite volume element

The Construction of the Trial Function Space
In this section, we recall the definitions of IFE spaces discussed in [7].Let T ℎ = {} be a regular triangulation of Ω with the diameter size ℎ.We can separate the triangles on a partition into two classes: (1) interface element: the interface Γ passes through the interior of ; (2) noninterface element: the interface does not intersect with this triangle, or it intersects with this triangle but does not separate its interior into two nontrivial subsets.Let T  ℎ be the collection of all noninterface elements and let T  ℎ be the collection of all interface elements.We will use   = (  ,   ),  = 1, 2, 3 to denote the vertices of T, and we will use  to denote the line segment connecting the intersection of the interface and the edges of a triangle .This line segment  divides T into two parts  + and  − with  =  + ∪  − ∪  (see Figure 2).
By φ(  ) =   ,  = 1, 2, 3, we can construct the basis function φ(x) on an interface element  as follows: Satisfy where n  is the unit normal vector on the linesegment .
Abstract and Applied Analysis 3 By [17,18], we have the following conclusions.
Based on the above results, the finite element space Ŝℎ () on a typical interface element  ∈ T  ℎ is defined by Ŝℎ () = { φ : φ is piecewise linear and satisfies (9)} . ( We call Ŝℎ () the immersed interface element space.For any  ∈ H2 (Ω) and  ∈ T  ℎ , we define and we call Π ℎ  the interpolant of  in Ŝℎ ().Similar to [7], we have an estimate of the interpolant given in the following theorem.

DGIFVE Procedure
In this section, we will construct a dual grid T * ℎ based on T ℎ .Assume that the triangulation T ℎ is quasi-uniform.For a given triangle  ∈ T ℎ , we divide  ∈ T ℎ into three triangles by connecting the barycenter  and the three corners of the triangle as shown in Figure 4. Let T * ℎ consist of all these triangles .
For the T * ℎ , we define the test function space as follows: Analogous to the operator Π ℎ , we introduce the interpolation operator  ℎ :  ℎ =  ℎ (Ω)+ H2 (Ω)∩ 1 0 (Ω) →  * ℎ (Ω) defined by, for ∀V ∈  ℎ , Let  be an interior edge shared by two elements  1 and  2 in T ℎ .Define the unite normal vectors n 1 and n 2 on  pointing exterior to  1 and  2 , respectively.For scalar V function and vector function q, we define their average {⋅} and jump [⋅] on , as follows (see [19]): If  is an edge on the boundary of Ω, we define Let  ℎ denote the union of the boundaries of the triangle  of T ℎ and let  0 ℎ :=  ℎ \ Ω,  * ℎ be the union of the boundaries cutting by the Γ.A straightforward computation gives Figure 4: (a) K is a noninterface element and (b) K is an interface element.
We multiply (1) by V ℎ ∈  * ℎ (Ω); using [B∇]| Γ = 0 and Green's formula, we have where n is the unit outward normal vector on .Let   ∈ T * ℎ ( = 1, 2, 3) be three triangles in  ∈ T ℎ .Then, we have Using (18) and the fact that [B∇] = 0, (20) becomes By ( 19) and ( 21), we can get By the definition of  ℎ , the discontinuous Galerkin immersed finite volume element formulation is equivalent to finding  ℎ ∈  0ℎ (Ω) such that where is the bilinear formulation defined on  0ℎ (Ω) ×  0ℎ (Ω), and in addition to penalty term ∑ ∈ ℎ ( 0 /ℎ  ) ∫  [ ℎ ][ ℎ ], the penalty parameter  0 > 0. Since [ ℎ ]  = 0, it is easy to see that  satisfies the solution of (1) as follows: Let Then, Similarly, We find that due to the fact that B∇ ℎ is a constant vector on each edge and the definition of  ℎ .Thus, we can get following from ( 28) and (29).For  ∈ T  ℎ (see Figure 4(b)), it follows from the same arguments above and the Summarizing the results above, we have Thus, (24) can be written by (34)

Some Lemmas
We define a norm ||| ⋅ ||| for  ℎ as follows: In order to prove the existence and uniqueness of the solution to (24) and conduct its convergence analysis, we need the following lemmas.
Let  be a noninterface element; we have the conclusion (38) by [12].Therefore, we focus (38) on interface element  (Figure 4(b)).For ∀ ∈  ℎ (Ω), we have the following form: The jump conditions on  lead to (see [7]) or where n  = (n  , n  )  and  = ( − / + ).We know that where Since  is continuous on  1  2 , there exists a point  such that We suppose that  fall on  1 ; then, we have where we used  − () =  + ().Because  + (x) and  − (x) are linear polynomial, we have From these expansions of  and (41), we have Then, If  is on →  2 , similarly, we also have (48).Analogously, we can have the following inequality: This completes the proof of (38) by ( 48) and (49).
Lemma 6 (see [20]).Let T be a regular triangulation; then, there exists a constant  > 0 independent of ℎ  such that, for  ∈  1 () and  ∈ T ℎ , the following inequality holds:

Existence, Uniqueness, and Convergence of DIFVE Solution
In this section, we will prove the existence and uniqueness of the solution to (24) and conduct its convergence analysis in the broken ||| ⋅ ||| norm.