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

Authors: Zhong-yan Liu; Huan-zhen Chen
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

By choosing the trial function space to the immersed finite element space and the test function space to be piecewise constant function space, we develop a discontinuous Galerkin immersed finite volume element method to solve numerically a kind of anisotropic diffusion models governed by the elliptic interface problems with discontinuous tensor-conductivity. The existence and uniqueness of the discrete scheme are proved, and an optimal-order energy-norm estimate andL2-norm estimate for the numerical solution are derived.


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## Body

## 1. Introduction

Let us consider the following elliptic interface problems in a convex domainΩ⊂R2:
(1)-∇·(B∇u)=f,(x,y)∈Ω,u=0,(x,y)∈∂Ω,
where Ω is separated into two subdomains Ω+ and Ω- by interface Γ∈C2, see Figure 1 for an illustration, and f∈L2(Ω); u satisfies the following homogenous jump conditions on the interface Γ:
(2)[u],[B∂u∂n]=0.Figure 1Equation (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.WhenB(x) 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 [1–3], 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 [6–9] 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 [11–16] 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(x) becomes a tenser-formed function. The goals of this paper are as follows: (1) to develop a discontinuous Galerkin-immersed interface-finite volume element (DGIFVE) method for the second-order elliptic problems with tensor-formed conductivity B(x) defined by
(3)B(x)=Bl(x)=(mlslslnl),
where x∈Ωl,ml>0,nl>0,mlnl>(sl)2,l=+,-; by doing so, we can use the ability of the penalty term in discontinuous Galerkin method to control the integrals on an element boundary, in order to prove the solvability of the scheme and derive easily an optimal-order error analysis, and we can use the conservation characteristics of the finite volume element method to construct a conservation-preserved numerical method; (2) to prove the existence and uniqueness of the proposed discontinuous Galerkin-immersed interface-finite volume element procedure based on the nonconforming interface finite element space for anisotropic flow model [17]; (3) to establish its optimal-order energy-norm estimate and L2-norm estimate.This paper is organized as follows. In the next section, we will introduce the trial function space and its approximation properties on primal triangulation. In Section3, we will formulate the DGIFVE procedure. In Section 4, we will introduce some important lemmas. In Section 5, we will prove the existence and uniqueness of the solution of the discrete scheme. In Section 6, we will derive the convergence analysis.Throughout this paper, the symbolC will be used as a generic positive constant independent of h and may have different values at different places.


## 2. The Construction of the Trial Function Space

In this section, we recall the definitions of IFE spaces discussed in [7]. Let Th={K} 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 interfaceΓ passes through the interior of K;(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. LetThn be the collection of all noninterface elements and let Thm be the collection of all interface elements. We will use Ai=(xi,yi),  i=1,2,3 to denote the vertices of T, and we will use DE¯ to denote the line segment connecting the intersection of the interface and the edges of a triangle K. This line segment DE¯ divides T into two parts K+ and K- with K=K+∪K-∪DE¯ (see Figure 2).Figure 2For the analysis, we introduce the spaces(4)H~2(Ω)={u∈H01(Ω):u∈H2(Ωs),s=+,-},H~2(K)={u∈H1(K):u∈H2(K∩Ωs),s=+,-}
equipped with the norms
(5)∥u∥H~2(Ω)2=∥u∥H2(Ω+)2+∥u∥H2(Ω-)2,∥u∥H~2(K)2=∥u∥H2(K∩Ω+)2+∥u∥H2(K∩Ω-)2,
where Hm=W2m(m=1,2) is the standard Sobolev spaces. In order to define the bilinear formulation, we introduce the broken Sobolev space H1(Th)={u∈L2(Ω):∀K∈Th,u|K∈H1(K) and ∀e⊂∂K∩∂Ω≠∅,u|e=0}.For a noninterface elementK, we use the standard linear shape functions on K whose degrees of freedom are functional values on the vertices of K, and we use S-h(K) to denote the linear spaces spanned by the three nodal basis functions on K as follows:
(6)S-h(K)=span⁡{ϕi∈P1(K):i=1,2,3},
where P1(K) is the linear space on K. For this space, we have the following estimate of the interpolant:
(7)∥u-Πhu∥L2(K)+h∥u-Πhu∥H1(K)⩽Ch2∥u∥H2(K),
where Πh:H2(K)→S-h(K) is the interpolation operator. We use S-h(Ω) to denote the space of the conforming piecewise linear polynomials on the domain Thn.For an interface elementK whose geometric configuration is given in Figure 3 in which A1=(0,0)T,A2=(h1,0)T,A3=(0,h2)T, the interface points D=(bh1,0)T and E=(0,ah2)T, where 0<a⩽1 and 0<b⩽1. Let ϕi,  i=1,2,3, denote the usual Lagrange nodal basis function associated with the vertex Ai,  i=1,2,3, respectively. Here we assume that the ratio r=h1/h2 is bounded below and above by some constants.Figure 3Byϕ^(Ai)=Vi,  i=1,2,3, we can construct the basis function ϕ^(x) on an interface element K as follows:
(8)ϕ^(x)={ϕ^-(x)=V1ϕ1+C1ϕ2+C2ϕ2,x=(x,y)∈K-,ϕ^+(x)=C3ϕ1+V2ϕ2+V3ϕ2,x=(x,y)∈K+.
Satisfy
(9)ϕ^+(D)=ϕ^-(D),ϕ^+(E)=ϕ^-(E),B+∂ϕ^+∂nDE¯=B-∂ϕ^-∂nDE¯,
where nDE¯ is the unit normal vector on the line segment DE¯.By [17, 18], we have the following conclusions.Lemma 1.
Whensl⩾0,l=+,-, the piecewise linear function ϕ^(x,y) defined by (8) is uniquely decided by three conditions in (9).Remark 2.
By [17], the condition sl≥0,l=+,- is necessary in Lemma 1. For some specially selected entries of Bl and the intersection points of the interface with the edges on K, ϕ^(x) satisfying (9) is uniquely undetermined by ϕ^(Ai)=Vi,i=1,2,3.
Based on the above results, the finite element spaceS^h(K) on a typical interface element K∈Thm is defined by
(10)S^h(K)={ϕ^:ϕ^ispiecewiselinearandsatisfies(9)}.
We call S^h(K) the immersed interface element space. For any u∈H~2(Ω) and K∈Thm, we define (Πhu)|K=Πh(u|K)∈S^h(K) by
(11)Πhu(Ai)=u(Ai),i=1,2,3,
and we call Πhu the interpolant of u in S^h(K). Similar to [7], we have an estimate of the interpolant given in the following theorem.Theorem 3.
For∀K∈Thm, there exists a constant C>0 such that the interpolation operator Πh:H~2(K)→S^h(K) satisfies
(12)∥u-Πhu∥L2(K)+h∥u-Πhu∥H1(K)⩽Ch2∥u∥H~2(K)bbbbbbbbbbbbbbbbbbbbbbbbbbbbb∀u∈H~2(K).Finally, we define trial function spaceSh(Ω) as the collection of functions such that
(13)ϕ|K∈S-h(K),Kisanoninterfaceelement,ϕ|K∈S^h(K),Kisaninterfaceelement.
The space Sh(Ω)⊂L2(Ω) is a subspace of H1(Th). We also use the space S0h(Ω)={vh∈Sh(Ω),vh|∂Ω=0}.


## 3. DGIFVE Procedure

In this section, we will construct a dual gridTh* based on Th. Assume that the triangulation Th is quasi-uniform. For a given triangle K∈Th, we divide K∈Th into three triangles by connecting the barycenter Q and the three corners of the triangle as shown in Figure 4. Let Th* consist of all these triangles T.(a)K is a noninterface element and (b) K is an interface element.
(a)
(b)For theTh*, we define the test function space as follows:
(14)Sh*(Ω)={ϕ(x)|ϕ(x)|T=constant,∀T∈Th*}.
Analogous to the operator Πh, we introduce the interpolation operator γh:Hh=Sh(Ω)+H~2(Ω)∩H01(Ω)→Sh*(Ω) defined by, for ∀v∈Hh,
(15)γhv|T=1|e|∫e‍v(x)dx,x∈T.Lete be an interior edge shared by two elements K1 and K2 in Th. Define the unite normal vectors n1 and  n2 on e pointing exterior to K1 and K2, respectively. For scalar v function and vector function q, we define their average {·} and jump [·] on e, as follows (see [19]):
(16){v}|e=12(v|∂K1+v|∂K2),[v]|e=v|∂K1·n1+v|∂K2·n2,{q}|e=12(q|∂K1+q|∂K2),[q]|e=q|∂K1·n1+q|∂K2·n2.
If e is an edge on the boundary of Ω, we define
(17){v}|e=v,[q]|e=q·n.
Let εh denote the union of the boundaries of the triangle K of Th and let εh0∶=εh∖∂Ω, εh* be the union of the boundaries cutting by the Γ. A straightforward computation gives
(18)∑‍K∈Th∫∂K‍vq·nds=∑‍e∈εh∫e‍[v]{q}ds+∑‍e∈εh0∫e‍{v}[q]ds.We multiply (1) by vh∈Sh*(Ω); using [B∇u]|Γ=0 and Green's formula, we have
(19)-∑‍T∈Th*∫∂T‍B∇u·nvhds=(f,vh),
where n is the unit outward normal vector on ∂T. Let Tj∈Th*(j=1,2,3) be three triangles in K∈Th. Then, we have
(20)∑‍T∈Th*∫∂TB∇u·nvhds=∑‍K∈Th∑‍j=1∫∂TjB∇u·nvhds=∑‍K∈Th∑‍j=1∫Aj+1QAjB∇u·nvhds+∑‍K∈Th∫∂KB∇u·nvhds.
Using (18) and the fact that [B∇u]=0, (20) becomes
(21)∑‍T∈Th*∫∂T‍B∇u·nvhds=∑‍K∈Th∑‍j=1∫Aj+1QAj‍B∇u·nvhds+∑‍e∈εh∫e‍{B∇u}[vh]ds.
By (19) and (21), we can get
(22)-∑‍K∈Th∑‍j=1∫Aj+1QAj‍B∇u·nvhds-∑‍e∈εh∫e‍{B∇u}[vh]ds=(f,vh).By the definition ofγh, the discontinuous Galerkin immersed finite volume element formulation is equivalent to finding uh∈S0h(Ω) such that
(23)ah*(uh,γhωh)=(f,γhωh),ωh∈S0h(Ω),
where
(24)ah*(uh,γhωh)=-∑‍K∈Th∑‍j=1∫Aj+1QAj‍B∇uh·nγhωhds-∑‍e∈εh∫e‍{B∇uh}[γhωh]ds+∑‍e∈εhσ0he∫e‍[uh][ωh]ds
is the bilinear formulation defined on S0h(Ω)×S0h(Ω), and in addition to penalty term ∑e∈εh(σ0/he)∫e‍[uh][ωh]ds, the penalty parameter σ0>0. Since [γhu]e=0, it is easy to see that u satisfies the solution of (1) as follows:
(25)ah*(u,γhωh)=f(u,γhωh).Let(26)A(uh,ωh)=-∑‍K∈Th∑‍j=13∫Aj+1QAj‍B∇uh·nγhωhds.
If K∈Thn, we have by ∇·(B∇uh)=0,uh∈S0h(Ω)(27)0=∫K‍∇·(B∇uh)γhωhdx=∑‍j=13∫Aj+1QAj‍B∇uh·nγhωhds+∑‍j=13∫AjAj+1‍B∇uh·nγhωhds.
Then,
(28)-∑‍j=13∫Aj+1QAj‍B∇uh·nγhωhds=∑‍j=13∫AjAj+1‍B∇uh·nγhωhds.
Similarly,
(29)0=-∫K‍∇·(B∇uh)ωhdx=-∑‍j=13∫AjAj+1‍B∇uh·nωhds+∫K‍B∇uh·∇ωhdx.
We find that
(30)∫∂K‍B∇uh·n(ωh-γhωh)ds=0,
due to the fact that B∇uh is a constant vector on each edge and the definition of γh. Thus, we can get
(31)∫K‍B∇uh∇ωhdx=-∑‍j=13∫Aj+1QAj‍B∇uh·nγhωhds
following from (28) and (29). For K∈Thm (see Figure 4(b)), it follows from the same arguments above and the [B∇uh]|DE¯=0 that
(32)∫K‍B∇uh∇ωhdx=-∑‍j=13∫Aj+1QAj‍B∇uh·nγhωhds+{∫A1D¯‍B+∇uh+·n(γhωh-ωh+)ds+∫DA2¯‍B-∇uh-·n(γhωh-ωh-)ds+∫A2E¯‍B-∇uh-·n(γhωh-ωh-)ds+∫EA3¯‍B+∇uh+·n(γhωh-ωh+)ds}.
Summarizing the results above, we have
(33)A(uh,ωh)=-∑‍K∑‍j=13∫Aj+1QAj‍B∇uh·nγhωhds=∑‍K∈Th∫K‍B∇uh∇ωhdx-∑‍K∈Thm∑‍e∈εh*∫e‍B∇uh·n(ωh-γhωh)ds=∑‍K∈Th∫K‍B∇uh∇ωhdx-∑‍e∈εh*∫e‍[B∇uh]{ωh-γhωh}ds-∑‍e∈εh*∫e‍{B∇uh}[ωh-γhωh]ds.
Thus, (24) can be written by
(34)ah*(uh,γhωh)=A(uh,ωh)-∑‍e∈εh∫e‍{B∇uh}[γhωh]ds+∑‍e∈εhσ0|e|α0∫e‍[uh][ωh]ds.


## 4. Some Lemmas

We define a norm|||·||| for Hh as follows:
(35)|||v|||2=|v|1,h2+∑‍e1he∫e‍[v]2ds+∑‍e∈εh*he∫e‍[β∇v]2ds.
In order to prove the existence and uniqueness of the solution to (24) and conduct its convergence analysis, we need the following lemmas.Lemma 4.
The operatorγh in (15) has the following properties:
(36)∫e‍(ω-γhω)ds=0,∀ω∈Hh,∀e∈εh;(37)[ω]|e=0⟹[γhω]|e=0,∀ω∈Hh;(38)∥γhω-ω∥0,K⩽ChK|ω|1,K,∀K∈Th,ω∈Hh.Proof.
Obviously, (36) and (37) follow from the definition of γ in (15). We only prove (38) below.
LetK be a noninterface element; we have the conclusion (38) by [12]. Therefore, we focus (38) on interface element K (Figure 4(b)). For ∀ω∈Sh(Ω), we have the following form:
(39)ω={ω+=a0+b0x+c0y,x=(x,y)∈K+,ω-=a1+b1x+c1y,x=(x,y)∈K-.
The jump conditions on DE¯ lead to (see [7])
(40)∇ω+=(ny2+ρnx2(ρ-1)nxny(ρ-1)nxnynx2+ρny2)∇ω-,
or
(41)∇ω+=NDE¯-∇ω-,
where nDE¯=(nx,ny)T and ρ=(β-/β+). We know that
(42)∥ω-γhω∥0,K2=∑‍j=13∫Tj‍(ω-γhω)2dx,
where
(43)∫Ti‍(ω-γhω)2dx=∫Ti+‍(ω+-γhω)2dx+∫Ti-‍(ω--γhω)2dx,i=1,2.
Since ω is continuous on A1A2¯, there exists a point ξ such that
(44)γhω|A1A2¯=ω(ξ).
We suppose that ξ fall on A1D¯; then, we have
(45)∫T1‍(ω-γhω)2dx=∫T1+‍(ω+-ω+(ξ))2dx+∫T1-‍(ω--ω+(ξ))2dx=∫T1+‍(ω+-ω+(ξ))2dx+∫T1-‍(ω--ω-(D)+ωh+(D)-ω+(ξ))2dx,
where we used ω-(D)=ω+(D). Because ω+(x) and ω-(x) are linear polynomial, we have
(46)ω+(x)=ω+(ξ)+∇ω+(x-ξ),ω-(x)=ω-(D)+∇ω-(x-D).
From these expansions of ω and (41), we have
(47)∫T1‍(ω-γhω)2dx=∫T1+‍(∇ω+(x-ξ))2dx+∫T1-‍(∇ω-(x-D)+NDE¯-∇ω-(D-ξ))2dx.
Then,
(48)∫T1‍(ω-γhω)2dx⩽Ch2|ω|1,T12.
If ξ is on DA2→, similarly, we also have (48). Analogously, we can have the following inequality:
(49)∫Ti‍(ω-γhω)2dx⩽Ch2|ω|1,Ti2,i=2,3.
This completes the proof of (38) by (48) and (49).Lemma 5.
For anyω∈Hh and e∈εh, one has
(50)∥[γhω]∥L2(e)⩽∥[ω]∥L2(e);(51)∥[ω-γhω]∥L2(e)⩽2∥[ω]∥L2(e).Proof.
Ife is the common side of T1, T2∈Th*, and T1⊂K1, T2⊂K2, by the definition of γh, we have
(52)∥[γhω]∥L2(e)2=∥γhω|∂T1·n1+γhω|∂T2·n2∥L2(e)2=∥he-1(∫e‍ω|T1ds·n1+∫e‍ω|T2ds·n2)∥L2(e)2=he-2∥∫e‍[ω]ds∥L2(e)2=he-1(∫e‍[ω]ds)2.
Using the Hölder inequality, we can get
(53)∥[γhω]∥L2(e)2⩽∫e‍[ω]2ds=∥[ω]∥L2(e)2.
If e∈∂Ω, there exists a T∈Th*, such that e∈∂T and T⊂K; we have
(54)∥[γhω]∥L2(e)2=∥γhω·n∥L2(e)2=∥he-1∫e‍ω|Tds·n∥L2(e)2=he-1(∫e‍[ω]ds)2⩽∥[ω]∥L2(e)2.
Thus, (50) is valid.
For (51), we have
(55)∥[ω-γhω]∥L2(e)=∥[ω]-[γhω]∥L2(e)⩽2∥[ω]∥L2(e)
following from (50).Lemma 6 (see [20]).
LetT be a regular triangulation; then, there exists a constant C>0 independent of hK such that, for w∈H1(K) and K∈Th, the following inequality holds:
(56)∫∂K‍|w|2ds⩽C{hK-1∥w∥0,K2+hK|w|1,K2}.


## 5. 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.Lemma 7.
There is a constantC independent of  h such that for σ0 large enough and ϵ small enough
(57)ah*(uh,γhuh)⩾C|||uh|||2.Proof.
By Cauchy-Schwarz's inequality, we have(58)∑‍e∈εh*∫e‍[B∇uh·n]{uh-γhuh}ds⩽(∑‍e∈εh*∥[B∇uh·n]∥L2(e)2)1/2(∑e∈εh*∥{uh-γhuh}∥L2(e)2)1/2.
Using the trace inequality (56), we have
(59)∥{uh-γhuh}∥L2(e)2⩽12(∥uh-γhuh|K1e∥L2(e)2+∥uh-γhuh|K2e∥L2(e)2)⩽C(hK1e|uh|1,K12+hK2e|uh|1,K22)⩽Ch(|uh|1,K12+|uh|1,K22),
where edge e is shared by the elements K1e and K2e. Therefore,
(60)(∑‍e∈εh*∥{uh-γhuh}∥L2(e)2)1/2⩽Ch1/2(∑‍K∈Thm|uh|1,k2)1/2.
By Young's inequality, we have, for ε>0,
(61)∑e∈εh*∫e‍[B∇uh·n]{uh-γhuh}ds⩽Ch1/2(∑K∈Thm|uh|1,k2)1/2(∑e∈εh*∥[B∇uh·n]∥L2(e)2)1/2⩽C0∑‍e∈εh*h∥[B∇uh·n]∥L2(e)2+ε3∑‍K∈Thm|uh|1,k2.
Similarly, we obtain
(62)∑‍e∈εh*∫e‍{B∇uh·n}[uh-γhuh]ds⩽(∑‍e∈εh*∥{B∇uh·n}∥L2(e)2)1/2×(∑‍e∈εh*∥[uh-γhuh]∥L2(e)2)1/2.
On the one hand, we have
(63)∥{B∇uh·n}∥L2(e)2⩽Ch-1(|uh|1,K12+|uh|1,K22),
and thus
(64)(∑‍e∈εh*∥{B∇uh·n}∥L2(e)2)1/2⩽Ch-1/2(∑‍K∈Thm|uh|1,k2)1/2.
On the other hand, we get
(65)(∑‍e∈εh*∥[uh-γhuh]∥L2(e)2)1/2⩽2(∑‍e∈εh*∥[uh]∥L2(e)2)1/2
following from (51). Therefore,
(66)∑e∈εh*∫e‍{B∇uh·n}[uh-γhuh]ds⩽Ch-1/2(∑K∈Thm|uh|1,K2)1/2(∑e∈εh*∥[uh]∥L2(e)2)1/2⩽Ch(α0-1)/2(∑‍K∈Thm|uh|1,K2)1/2×(∑‍e∈εh*1|e|α0∥[uh]∥L2(e)2)1/2,
where α0>1. By ε-inequality, we have
(67)∑‍e∈εh*∫e‍{B∇uh·n}[uh-γhuh]ds⩽ϵ3∑‍K∈Thm|uh|1,K2+C1∑‍e∈εh*1|e|α0∥[uh]∥L2(e)2.
Similarly, we can get
(68)∑‍e∈εh∫e‍{B∇uh·n}[γhuh]ds⩽ϵ3∑‍K∈Th|uh|1,K2+C2∑‍e∈εh1|e|α0∥[uh]∥L2(e)2.
Combining (61), (67), and (68), we obtain
(69)ah*(uh,γhuh)⩾(C-ϵ3)∑‍K∈Thn|uh|1,K2+(C-ϵ)∑‍K∈Thm|uh|1,K2+∑‍e∈εh*σ0-C1-C2|e|α0∥[uh]∥L2(e)2+∑‍e∈εh∖εh*σ0-C2|e|α0∥[uh]∥L2(e)2+∑‍e∈εh*C0h∥[B∇uh·n]∥L2(e)2.
Choosing σ0 large enough and ϵ small enough, we have
(70)ah*(uh,γhuh)⩾C|||uh|||2.Lemma 8.
Foru, ω∈Hh, one has
(71)ah*(u,γhω)⩽C+(|||u|||+(∑‍K∈Thh2|u|H~2(K)2)1/2)|||ω|||.
If uh, ωh∈Sh(Ω), then
(72)ah*(uh,γhωh)⩽C|||uh||||||ωh|||.Proof.
By (38) and the trace inequality (56), we have, for any es∈εh*,s=+,-,
(73)∫es‍B∇u·n(ω-γhω)ds⩽C(h-1|u|1,Ks2+h|u|2,Ks2)1/2(h-1|ω-γhω|L2(Ks)2bbbb+h|ω-γhω|1,Ks2)1/2⩽C(|u|1,Ks2+h2|u|2,Ks2)1/2|ω|1,Ks⩽C(|u|1,K2+h2|u|H~2(K)2)1/2|ω|1,K.
By the Cauchy-Schwarz inequality, we obtain
(74)|A(u,ω)|⩽|∑‍K∈Th∫K‍B∇u∇ωdx|+|∑‍K∈Thm∑‍e∈εh*∫e‍B∇u·n(ω-γhω)ds|⩽C(|u|1,h+(∑K∈Th(|u|1,K2+h2|u|H~2(K)2))1/2)|ω|1,h.
The definition of ah(u,γhω) and the inequality above imply that
(75)|ah(u,γhω)|⩽C{|u|1,h|ω|1,h}+(∑‍K∈Th(|u|1,K2+h2|u|H~2(K)2))1/2|ω|1,h+(∑‍K∈Th(|u|1,K2+h2|u|H~2(K)2))1/2×(∑‍e∈εh∥[ω]∥L2(e)2)1/2+(∑‍e∈εh∥[u]∥L2(e)2)1/2(∑‍e∈εh∥[ω]∥L2(e)2)1/2⩽C(|||u|||+(∑‍K∈Thh2|u|H~2(K)2)1/2)|||ω|||.
This completes the proof of (71). For (72), we can get the following from (71) and |uh|H~2(K)=0, ∀uh∈Sh(Ω).Lemmas7 and 8 guarantee the existence and uniqueness of the discontinuous immersed finite volume element solution to (23) when choosing σ0 large enough.


## 6. Error Estimates in the Energy Norm

We will derive an optimal-order error estimate in the norm|||·||| defined in (35) and a first order error estimate in L2-norm. We start with the following lemmas.Lemma 9.
Letu be the solutions of (1); one has the conclusion
(76)|||u-Πhu|||⩽Ch∥u∥H~2(Ω).Proof.
By the definition of norm|||·|||, we have
(77)|||u-Πhu|||2=|u-Πhu|1,h2+∑‍e∈εh1|e|α0∥[u-Πhu]∥L2(e)2+∑‍e∈εh*h∥[∇(u-Πhu)·n]∥L2(e)2.
Using trace inequality and (12), we have
(78)∥[u-Πhu]∥L2(e)2⩽C(∥u-Πhu|K1e∥L2(e)2+∥u-Πhu|K2e∥L2(e)2)⩽C(h-1∥u-Πhu∥L2(K1)2+h|u-Πhu|1,K12+h-1∥u-Πhu∥L2(K2)2+h|u-Πhu|1,K22)⩽Ch2(∥u∥H~2(K1)2+∥u∥H~2(K2)2),
where e is shared by the elements K1 and K2. Thus, we obtain
(79)∑‍e∈εh∥[u-Πhu]∥L2(e)2⩽Ch2∥u∥H~2(Ω)2.
Analogously, we can get
(80)∑‍e∈εh*h∥[∇(u-Πhu)·n]∥L2(e)2⩽Ch2∥u∥H~2(Ω)2.
By (12), (79), and (80), we obtain (76).Lemma 10 (see [21]).
There exists a constantC independent of h such that
(81)∥ω∥⩽C|||ω|||,∀ω∈Sh(Ω).Theorem 11.
Letuh∈S0h(Ω) and u∈H~2(Ω)∩H01(Ω) be the solutions of (23) and (1), respectively; then, there exists a constant C independent of h such that
(82)|||u-uh|||⩽Ch∥u∥H~2(Ω),(83)∥u-uh∥⩽Ch∥u∥H~2(Ω).Proof.
Subtracting (25) from (23) gives
(84)ah*(u-uh,γhωh)=0,∀ωh∈S0h(Ω).
Using (71), (76), and (84), we have
(85)|||uh-Πhu|||2⩽Cah*(uh-Πhuh,γh(uh-Πhuh))=Cah*(u-Πhu,γh(uh-Πhuh))⩽C(|||u-Πhu|||+(∑‍K∈Thh2|u-Πhu|H~2(K)2)1/2)×|||uh-Πhu|||⩽Ch2∥u∥H~2(Ω)2.
Thus, we can get
(86)|||u-uh|||⩽Ch∥u∥H~2(Ω)
following from triangle inequality, (76), and (85).
For (83), we can get by (12), (81) and triangle inequality. We have completed the proof.



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*Source: 520404-2014-06-01.xml*