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

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 the H2-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 ofmain 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


Introduction
This paper considers the multiscale method for the following convection-diffusion-reaction problem where Ω ⊂ ℝ 2 (or ℝ 3 ) 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 the H 2 -norm of u ε .As coefficients vary on a scale of ε, the solution u ε 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 a 0 , b 0 , and c 0 so that 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][5][6][7][8][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 H 1 -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.

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.
Let T H be a quasi-uniform triangulation of the polygonal domain Ω.The barycenter dual decomposition T * H 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 K * P is the dual element with respect to P (referring to Figure 1).Let H be the maximum length of the edges, N H the set of all nodal points, K the vertices of K.
H be the interpolation operator from V H to the piecewise constant space on T * H : Then, the finite volume method [17] reads as finding where We next construct our multiscale method with respect to (1).

Complexity
By the numerical integration for the barycenter quadrature rule, ( 5) and ( 6) can be written as Thus, the barycenter quadrature approximation of the FVM can be read as finding u H ∈ V H such that To obtain error estimate, this paper considers the coefficients of scale separation.Assume that the multiscale coefficients a ε x , b ε x , and c ε x have the forms a x, x/ε , b x, x/ε , and c x, x/ε .Furthermore, assume that a x, y , b x, y , and c x, y are smooth in x and periodic in y with respect to the unit cube Y.
In the absence of explicit expressions of a H , b H , and c H , a H Q ∇u H , b H Q ∇u H , and c H Q can be approximated by where R u H is the solution of the following microcell problem and K δ Q is a cube of size δ centered at Q. Thus, the HMM-FVM can be summarized as the following variational problem: ∀v H ∈ V H , find u H ∈ V H satisfying (9)-(11).

A Priori Error Estimate
This section deduces the main result of the paper-optimal H 1 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 constant C independent of H, ε, and δ such that for a H , b H , where a 0 ij = 1/ Y Y a ik x, y δ kj + ∂x j /∂y k x, y dy b 0 i = 1/ Y Y b i x, y + a ik ∂η/∂y k x, y dy.Here, χ j and η are, respectively, the periodic solutions of −∇ y • a x, y ∇ y χ j x, y = ∇ y • a x, y e j , −∇ y • a x, y ∇ y η x, y = ∇ y • b x, y with zero mean (i.e., Y χ j dy = 0, Y ηdy = 0).
Mathematically, the finite element method for the homogenization problem (2) is equivalent to finding u H ∈ V H such that where Applying barycenter quadrature to (17), the bilinear form can be refined as Finally, to estimate the priori error estimate, we borrow inf-sup condition of bilinear form B given in [17].
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.
where C is a positive constant independent of ε, δ, and H.
Proof.The bilinear form can be split into three terms where

23
Denote by By the standard estimate of [21], we get Accordingly, the numerical quadrature error next utilize Lemma 1 and Lemma 2 to estimate the modeling error ε 2 u H , v H .By the definition of c H in (11) and Lemma 2, we see that the error between c H and c 0 can be treated as where c 0 = 1/|Y| Y c x, y dy and C is a positive constant independent of ε, δ, and H. Let By Lemma 1 and ( 27), Thus, the modeling error ε 2 u H , v H can be further estimated as It remains to estimate the term ε 3 u H , v H which concerns three components.We next start this work.By Green's formula, the first The first term of A FVM u H , v H can be derived similarly The combination of the two gives

33
Recall the interpolation operator Π * H (see Lemma 2.1 in [17]) satisfying From ( 29), (33), and (34), the last term ε 3 u H , v H can be estimated as By the estimates of With the preparation of Theorem 1, we summarize the main result in Theorem 2.
Theorem 2. Denote by u 0 and u H the solution of ( 2) and ( 9), respectively.Then for sufficient small H and ε/δ, we have where C is a positive constant independent of ε, δ, and H.
Proof.In order to estimate the error between u 0 and u H , we separate it into two parts where u 0 H 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 It remains to treat the second part of (38).By Lemma 3, we have According to Theorem 1, T 1 can be estimated as On the other hand, the numerator of T 2 in (40) is 5 Complexity Thus, We see (37) follows immediately by combining the results of (38)-(43).

Conclusion
This paper investigates the multiscale convection-diffusionreaction 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 in H 1 -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.

Figure 1 :
Figure 1: An element K (a) and its dual element (b) with respect to a node P.