# Error Estimates for the Heterogeneous Multiscale Finite Volume Method of Convection-Diffusion-Reaction Problem

Authors: Tao Yu; Peichang Ouyang; Haitao Cao
Journal: Complexity
 (2018)

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## Abstract

Based on the heterogeneous multiscale method, this paper presents a finite volume method to solve multiscale convection-diffusion-reaction problem. The paper constructs an algorithm of the optimal order convergence rate inH1-norm under periodic medias.


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## Body

## 1. Introduction

This paper considers the multiscale method for the following convection-diffusion-reaction problem(1)−∇aεx∇uεx+bεx∇uεx+cεxuεx=fx,x∈Ω,uεx=0,x∈∂Ω,where Ω⊂ℝ2 (or ℝ3) is a bounded convex polygonal domain with Lipschitz boundary ∂Ω, ε≪1 is a positive parameter which signifies the multiscale nature of (1). This problem is related to groundwater and solute transport in porous media [1].Optimal order convergence rate of classical finite element method based on piecewise linear polynomials relies on theH2-norm of uε. As coefficients vary on a scale of ε, the solution uε may also oscillate at the same scale. A direct numerical solution of this multiscale problem is very difficult to derive unless the mesh size is sufficiently smaller. However, it is not feasible in practice since the amount of computation will increase sharply as the amount of calculation increases. On the other hand, from an engineering point of view, the macroscopic features of the solution are often of main interest and importance. According to homogenization theory [2, 3], there is a homogenized equation which can capture the macroscopic properties. In other words, there exist homogenized coefficients a0, b0, and c0 so that
(2)−∇a0x∇u0x+b0x∇u0x+c0xu0x=fx,x∈Ω,u0x=0,x∈Ω.Though there exist many classical methods for solving (2), unfortunately, in general, there are no explicit formulae for the homogenized coefficients, except that there are many restrictive assumptions on the media. Thus, developing numerical methods that can capture the effect of small scale on the large scale is an attractive subject. To overcome the above difficulties, many strategies have been established to solve the problems on grids which are coarser than the scale of oscillation; see [4–9] and references therein.The multiscale finite volume (MSFV) method was first introduced by Jenny et al. for the elliptic problem of highly oscillatory coefficients [9]. Based on a special interpolation from the coarse mesh to the fine mesh, this method captured the effect of small scales on a coarse grid efficiently. It not only fitted into finite volume framework nicely but also kept both the conservation of coarse and fine scales. This method was widely applied in many situations, such as discrete fracture modeling [10], parabolic problem [11], and Maxwell’s equations [12]. Recently, Weinan E. and Engquist established a general efficient methodology—heterogeneous multiscale method (HMM) [13], for the multiscale problems, which consists of two components: (a) selecting a macroscopic solver on a coarse mesh and (b) estimating the missing macroscale data by solving local fine-scale problems. The careful selection of the macro solver and local fine problems is a key issue for the method. A different choice of macroscopic solver will lead to a different heterogeneous multiscale method; see some examples for finite element method [8, 14, 15] and discontinuous finite element method [4, 16]. It is well known that the finite volume method has many advantages, such as keeping conservation and applying to complex regions. Thus, in this paper, we choose the finite volume method (FVM) introduced in [17] as the macroscopic solver. For convenience, the heterogeneous multiscale method taking finite volume method as the macroscopic solver is denoted as HMM-FVM. We will show that our method has optimal order convergence rate in H1-norm for periodic medias.The rest of this paper is organized as follows. To solve (1), Section 2 first constructs a HMM-FVM. Then, Section 3 will study the approximate solution and its error estimates for periodic media associated with (1) in detail. We finally conclude the paper in Section 4.


## 2. HMM-FVM for Convection-Diffusion-Reaction Problem

In this section, we first concisely describe the finite volume method for convection-diffusion-reaction problem in [17]. Then, using the method of [17] as a macroscopic solver of HMM, we derive the HMM-FVM in detail.LetTH be a quasi-uniform triangulation of the polygonal domain Ω. The barycenter dual decomposition TH∗ is constructed by connecting the barycenter to the midpoints of edges of each triangle element. Suppose K is a triangle element, denote e and Q, respectively, as an edge and barycenter of K and R the midpoint of e. Assume P is a nodal point and KP∗ is the dual element with respect to P (referring to Figure 1). Let H be the maximum length of the edges, NH the set of all nodal points, K̇ the vertices of K.Figure 1
An elementK (a) and its dual element (b) with respect to a node P.
(a)
(b)LetV=H01Ω=u∈H1Ω∣u=0on∂Ω and VH⊂V be the piecewise linear finite element space on TH, ΠH∗ be the interpolation operator from VH to the piecewise constant space on TH∗:
(3)ΠH∗vHx=vHP,x∈KP∗,∀P∈NH,vH∈VH.Then, the finite volume method [17] reads as finding uH∈VH such that
(4)AuH,ΠH∗vH=f,ΠH∗vH,∀vH∈VH,where
(5)AuH,ΠH∗vH=∑P∈NH−∫∂KP∗n·aH∇uHΠH∗vHds+∫KP∗bH∇uH+cHuHΠH∗vHdx=∑K∈TH−∑P∈K̇∫K∩∂KP∗n·aH∇uHΠH∗vHds+∫KbH∇uH+cHuHΠH∗vHdx,(6)f,ΠH∗vH=∑P∈NH∫KP∗fΠH∗vHdx=∑K∈TH∫KfΠH∗vHdx.We next construct our multiscale method with respect to (1).By the numerical integration for the barycenter quadrature rule, (5) and (6) can be written as
(7)AFVMuH,ΠH∗vH=∑K∈TH−∑P∈K̇K∩∂KP∗n·aHQ∇uHΠH∗vH+KbHQ∇uH+cHQuHQΠH∗vH,(8)f,ΠH∗vHH=∑K∈THKfQΠH∗vH.Thus, the barycenter quadrature approximation of the FVM can be read as findinguH∈VH such that
(9)AFVMuH,ΠH∗vH=f,ΠH∗vHH,∀vH∈VH.To obtain error estimate, this paper considers the coefficients of scale separation. Assume that the multiscale coefficientsaεx, bεx, and cεx have the forms ax,x/ε, bx,x/ε, and cx,x/ε. Furthermore, assume that ax,y, bx,y, and cx,y are smooth in x and periodic in y with respect to the unit cube Y.In the absence of explicit expressions ofaH, bH, and cH, aHQ∇uH, bHQ∇uH, and cHQ can be approximated by
(10)aHQ∇uH≃1KδQ∫KδQaQ,xε∇RuHdx,bHQ∇uH≃1KδQ∫KδQbQ,xε∇RuHdx,(11)cHQ≃1KδQ∫KδQcQ,xεdx,where RuH is the solution of the following microcell problem
(12)−∇⋅aQ,xε∇RuH=0inKδQ,RuH=uH,on∂KδQ,and KδQ is a cube of size δ centered at Q.Thus, the HMM-FVM can be summarized as the following variational problem:∀vH∈VH, find uH∈VH satisfying (9)–(11).


## 3. A Priori Error Estimate

This section deduces the main result of the paper—optimalH1 error estimate (Theorem 2). To this end, we first introduce Lemma 1 and Lemma 2 to derive its modeling error. Then, by regarding the HMM-FVM as a perturbation of the linear finite element method (Theorem 1), we obtain the optimal error estimate of HMM-FVM.Lemma 1 ([14, 18, 19]).
There exists a constantC independent of H, ε, and δ such that for aH, bH,
(13)aijH−aij0≤Cεδ,i,j=1,2,(14)biH−bi0≤Cεδ,i=1,2,where aij0=1/Y∫Yaikx,yδkj+∂xj/∂ykx,ydy bi0=1/Y∫Ybix,y+aik∂η/∂ykx,ydy. Here, χj and η are, respectively, the periodic solutions of −∇y·ax,y∇yχjx,y=∇y·ax,yej, −∇y·ax,y∇yηx,y=∇y·bx,y with zero mean (i.e., ∫Yχjdy=0, ∫Yηdy=0).Lemma 2 ([18, 20]).
Given domainKδ with diamK=δ, let φs,y defined in Y be a Y−periodic function in y, where Y is a unit cube and s∈R is fixed. Then
(15)1Y∫Yφs,ydy−<φs,xε>K≤Cεδ,where C is independent of ε, δ, and s.Mathematically, the finite element method for the homogenization problem (2) is equivalent to finding uH∈VH such that
(16)BuH,vH=f,vH,∀vH∈VH,where
(17)BuH,vH=∑K∈TH∫Ka0∇uH·∇vH+b0∇uHvH+c0uHvHdx,(18)f,vH=∑K∈TH∫KfvHdx.Applying barycenter quadrature to (17), the bilinear form can be refined as
(19)B~uH,vH=∑K∈THKa0Q∇uH∙∇υH+b0Q∇uHυHQ+c0QuHQυHQ.Finally, to estimate the priori error estimate, we borrow inf-sup condition of bilinear formB given in [17].Lemma 3 [17].
For0<H<h0,
(20)uH1,Ω≤Csup0≠vH∈VHBuH,vHυH1,Ω,∀uH∈VH.The following theorem characterizes the difference between the bilinear form of HMM-FVM and FEM, which will play the key role in the subsequent analysis.Theorem 1.
∀uH,vH∈VH, we have
(21)BuH,vH−AFVMuH,vH≤CH+εδ∥uH∥1,Ω∥vH∥1,Ω,where C is a positive constant independent of ε, δ, and H.Proof.
The bilinear form can be split into three terms(22)BuH,vH−AFVMuH,vH≤∣BuH,vH−B~uH,vH∣+B~uH,vH−A~uH,vH+A~uH,vH−AFVMuH,vH≔ε1uH,vH+ε2uH,vH+ε3uH,vH,where
(23)A~uH,vH=∑K∈THKaHQ∇uH·∇vH+bHQ∇uHvHQ+cHQuHQvHQ.
Denote by(24)EKf=∫Kfxdx−KfQ.
By the standard estimate of [21], we get
(25)EKaxpxqx≤CH∥a∥1,∞∥p∥0,K∥q∥0,K.
Accordingly, the numerical quadrature errorε1uH,vH is
(26)ε1uH,vH≤∑K∈THEKa0∇uH·∇vH+EKb0uH∇vH+EKc0uHvH≤CHuH1,ΩvH1,Ω.
We next utilize Lemma1 and Lemma 2 to estimate the modeling error ε2uH,vH. By the definition of cH in (11) and Lemma 2, we see that the error between cH and c0 can be treated as
(27)cH−c0≤Cεδ,where c0=1/∣Y∣∫Ycx,ydy and C is a positive constant independent of ε, δ, and H. Let
(28)eHMM=maxx∈ΩaijH−aij0,biH−bi0,cH−c0.
By Lemma1 and (27),
(29)eHMM≤Cεδ.
Thus, the modeling errorε2uH,vH can be further estimated as
(30)ε2uH,vH≤CeHMMuH1,ΩvH1,Ω≤CεδuH1,ΩvH1,Ω.
It remains to estimate the termε3uH,vH which concerns three components. We next start this work.
By Green’s formula, the first term ofA~uH,vH is
(31)KaHQ∇uH∇vH=∫KaHQ∇uH∇vHdx=−∫K∇aHQ∇uHvHdx+∫∂Kn·aHQ∇uHvHds=∫∂Kn·aHQ∇uHvHds.
The first term ofAFVMuH,vH can be derived similarly
(32)−∑P∈K̇K∩∂KP∗n·aHQ∇uHΠH∗vH=−∑P∈K̇∫K∩∂KP∗n·aHQ∇uHΠH∗vHds=−∑P∈K̇∫∂K∩∂KP∗n·aHQ∇uHΠH∗vHds+∫∂Kn·aHQ∇uHΠH∗vHds=−∫K∇aHQ∇uHΠH∗vHdx+∫∂Kn·aHQ∇uHΠH∗vHds=∫∂Kn·aHQ∇uHΠH∗vHds.
The combination of the two gives(33)∑K∈THKaHQ∇uH∇vH−∑K∈TH−∑P∈K̇K∩∂KP∗n·aHQ∇uHΠH∗vH=∑K∈TH∑e∈∂K∫en·aHQ∇uHvH−ΠH∗vHds.
Recall the interpolation operatorΠH∗ (see Lemma 2.1 in [17]) satisfying
(34)vH−ΠH∗vH0,K≤CHvH1,K,∀vH∈VH.
From (29), (33), and (34), the last term ε3uH,vH can be estimated as
(35)ε3uH,vH=∑K∈TH∑e∈∂K∫en·aHQ∇uHvH−ΠH∗vHds+KbHQ∇uH+cHQuHvH−ΠH∗vH=∑K∈THKbHQ∇uH+cHQuHvH−ΠH∗vH≤∑K∈THKbH0,∞,K∇uH0,∞,K+cH0,∞,KuH0,∞,KvH−ΠH∗vH0,∞,K≤C∑K∈THbH0,∞,K∇uH0,2,K+cH0,∞,KuH0,2,KvH−ΠH∗vH0,2,K≤CbH0,∞,Ω+cH0,∞,ΩuH1,ΩvH−ΠH∗vH0,Ω≤CHbH0,∞,Ω+cH0,∞,ΩuH1,ΩvH1,Ω≤CHb00,∞,Ω+c00,∞,Ω+eHMMuH1,ΩvH1,Ω≤CH+eHMMuH1,ΩvH1,Ω≤CH+εδuH1,ΩvH1,Ω.
By the estimates ofε1uH,vH, ε2uH,vH, and ε3uH,vH, we finally get
(36)BuH,vH−AFVMuH,vH≤CH+εδuH1,ΩvH1,Ω.With the preparation of Theorem1, we summarize the main result in Theorem 2.Theorem 2.
Denote byu0 and uH the solution of (2) and (9), respectively. Then for sufficient small H and ε/δ, we have
(37)u0−uH1,Ω≤CH+εδ,where C is a positive constant independent of ε, δ, and H.Proof.
In order to estimate the error betweenu0 and uH, we separate it into two parts
(38)u0−uH1,Ω≤u0−uH01,Ω+uH0−uH1,Ω,where uH0 is the numerical solution of (16), which is FEM formula of the homogenized problem (2).
From the standard error estimate [21], the first part of (38) can be estimated as
(39)u0−uH01,Ω≤CHu01,Ω.
It remains to treat the second part of (38). By Lemma 3, we have
(40)uH0−uH1,Ω≤Csup0≠vH∈VHBuH0−uH,vHvH1,Ω≤Csup0≠vH∈VHBuH,vH−AFVMuH,ΠH∗vHvH1,Ω+f,vHH−f,ΠH∗vHHvH1,Ω≔T1+T2.
According to Theorem1, T1 can be estimated as
(41)T1≤CH+ϵδuH1,Ω≤CH+ϵδu01,Ω+uH−u01,Ω.
On the other hand, the numerator ofT2 in (40) is
(42)f,vHH−f,ΠH∗vHH=∑K∫KfvHdx−∑KKfQΠH∗vH=∑K∫KfvHdx−∑K∫KfQΠH∗vHdx=∑K∫KfvHdx−∑K∫KfQvHdx=∑K∫Kfx−fQvHdx≤CH∑K∇f0,2,KvH0,2,K≤CHf1,ΩvH0,Ω≤CHf1,ΩvH1,Ω.
Thus,(43)T2≤CHf1,Ω.
We see (37) follows immediately by combining the results of (38)–(43).


## 4. Conclusion

This paper investigates the multiscale convection-diffusion-reaction problem. Based on the heterogeneous multiscale method, we first choose a finite volume method as macroscopic solver elaborately which can keep conservation and can be applied to complex regions. Then, we construct a suitable HMM-FVM strategy to solve the problem. We show that this method possesses the optimal order convergence rate inH1-norm for periodic medias. As a matter of fact, this method can be applied in practice to multiscale problems without assumptions of periodic and scale separation.



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