ANALYSIS OF THE HETEROGENEOUS MULTISCALE METHOD FOR ELLIPTIC HOMOGENIZATION PROBLEMS

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1 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S XX ANALYSIS OF THE HETEROGENEOUS MULTISCALE METHOD FOR ELLIPTIC HOMOGENIZATION PROBLEMS WEINAN E, PINGBING MING, AND PINGWEN ZHANG Contents 1. Introduction and main results General methodology Heterogeneous multiscale method Main results Recovering the microstructural information Some technical background 7 2. Generalities 9 3. Estimating ehmm Reconstruction and compression Reconstruction procedure Compression operator Nonlinear homogenization problems Algorithms and main results Estimating ehmm 23 Appendix A. Estimating ehmm for problems with random coefficients 28 References 33 Received by the editors January 2, Mathematics Subject Classification. Primary 65N30, 74Q05; Secondary 74Q15, 65C30. Key words and phrases. Heterogeneous Multiscale Method, Homogenization Problems. The work of the first author was partially supported by ONR grant N and National Natural Science Foundation of China through a Class B Award for Distinguished Young Scholars The work of the second author was partially supported by the Special Funds for the Major State Basic Research Projects G and was also supported by the National Natural Science Foundation of China The work of the third author was partially supported by the Special Funds for the Major State Research Projects G and the National Natural Science Foundation of China for Distinguished Young Scholars We thank Bjorn Engquist for inspiring discussions on the topic studied here. 1 c 1997 American Mathematical Society

2 2 W. E, P.-B. MING, AND P.-W. ZHANG 1. Introduction and main results 1.1. General methodology. Consider the classical elliptic problem { div a x u x = fx x D R d, 1.1 u x = 0 x D. Here is a small parameter that signifies explicitly the multiscale nature of the coefficient a x. Several classical multiscale methodologies have been developed for the numerical solution of this elliptic problem, the most well known among which is the multigrid technique [8]. These classical multiscale methods are designed to resolve the details of the fine scale problem 1.1 and are applicable for general problems, i.e., no special assumptions are required for the coefficient a x. In contrast modern multiscale methods are designed specifically for recovering partial information about u at a sublinear cost, i.e., the total cost grows sublinearly with the cost of solving the fine scale problem [18]. This is only possible by exploring the special features that a x might have, such as scale separation. The simplest example is when 1.2 a x = a x, x, where ax, y can either be periodic in y, in which case we assume the period to be I = [ 1/2, 1/2] d, or random but stationary under shifts in y, for each fixed x D. In both cases, it has been shown that [5, 36] 1.3 u x Ux L 2 D 0, where Ux is the solution of a homogenized equation: { div Ax Ux = fx x D, 1.4 Ux = 0 x D. The homogenized coefficient Ax can be obtained from the solutions of the socalled cell problem. In general, there are no explicit formulas for Ax, except in one dimension. Several numerical methods have been developed to deal specifically with the case when ax, y is periodic in y. References [3, 4, 7] propose to solve the homogenized equations as well as the equations for the correctors. Schwab et al. [29, 38] use multiscale test functions of the form ϕx, x/ where ϕx, y is periodic in y to extract the leading order behavior of u x, extending an idea that was used analytically in the work of [2, 15, 34, 44] for the homogenization problems. These methods have the feature that their cost is independent of, hence sublinear as 0, but so far they are restricted to the periodic homogenization problem. An alternative proposal for more general problems but with much higher cost is found in [20, 25] Heterogeneous multiscale method. HMM [16, 17, 18] is a general methodology for designing sublinear algorithms by exploiting scale separation and other special features of the problem. It consists of two components: selection of a macroscopic solver and estimating the missing macroscale data by solving locally the fine scale problem. For 1.1 the macroscopic solver can be chosen as a conventional P k finite element method on a triangulation T H of element size H which should resolve the macroscale features of a x. The missing data is the effective stiffness matrix at

3 HETEROGENEOUS MULTISCALE METHOD 3 this scale. This stiffness matrix can be estimated as follows. Assuming that the effective coefficient at this scale is A H x, if we knew A H x explicitly, we could have evaluated the quadratic form V x A H x V x dx D by numerical quadrature: For any V X H, the finite element space, 1.5 A H V, V ω l V AH V x l, K x l K where {x l } and {ω l } are the quadrature points and weights in K, K is the volume of K. In the absence of explicit knowledge of A H x, we approximate V AH V x l by solving the problem: 1.6 { div a x v l x = 0 x I x l, v l x = V l x x I x l, where I x l is a cube of size centered at x l, and V l is the linear approximation of V at x l. We then let 1.7 V AH V x l 1 d vl x a x vl x dx. I x l 1.5 and 1.7 together give the needed approximate stiffness matrix at the scale H. For convenience, we will define the corresponding bilinear form: For any V, W X H A H V, W : = vl x a x wl x dx, K d ω l x l K I x l where wl is defined for W X H in the same way that vl in 1.6 was defined for V. In order to reduce the effect of the imposed boundary condition on I x l, we may replace 1.7 by 1.8a V AH V x l 1 d vl x a x vl x dx, I x l where <. For example, we may choose = /2. In 1.6, we used the Dirichlet boundary condition. Other boundary conditions are possible, such as Neumann and periodic boundary conditions. In the case when a x = ax, x/ and ax, y is periodic in y, one can take I x l to be x l + I, i.e., = and use the boundary condition that vl x V lx is periodic on I. So far the algorithm is completely general. The savings compared with solving the full fine scale problem comes from the fact that we can choose I x l to be smaller than K. The size of I x l is determined by many factors, including the accuracy and cost requirement, the degree of scale separation, and the microstructure in a x. One purpose for the error estimates that we present below is to give guidelines on how to select I x l. As mentioned already, if a x = ax, x/ and ax, y is periodic in y, we can simply choose I x l to be x l + I, i.e., =. If ax, y is random, then should be a few times larger than the local correlation

4 4 W. E, P.-B. MING, AND P.-W. ZHANG length of a. In the former case, the total cost is independent of. In the latter case, the total cost depends only weakly on see [31]. The final problem is to solve min V X H 2 A HV, V f, V. K Figure 1. Illustration of HMM for solving 1.1. The dots are the quadrature points. The little squares are the microcell I x l. To summarize, HMM has the following features: 1 It gives a framework that allows us to maximally take advantage of the special features of the problem such as scale separation. For periodic homogenization problems, the cost of HMM is comparable to the special techniques discussed in [3, 7, 29, 35]. However HMM is also applicable for random problems and for problems whose coefficient a x does not has the structure of ax, x/. For problems without scale separation, we may consider other possible special features of the problem such as local self-similarity, which is considered in [19]. 2 For problems without any special features, HMM becomes a fine scale solver by choosing an H that resolves the fine scales and letting A H x = a x. Some related ideas exist in the literature. Durlofsky [14] proposed an up-scaling method, which directly solves some local problems for obtaining the effective coefficients [33, 40, 41]. Oden and Vemaganti [35] proposed a method that aims at recovering the oscillations in u locally by solving a local problem with some given approximation to the macroscopic state U as the boundary condition. This idea is sometimes used in HMM to recover the microstructural information. Other numerical methods that use local microscale solvers to help extract macroscale behavior are found in [26, 27]. The numerical performance of HMM including comparison with other methods is discussed in [31]. This paper will focus on the analysis of HMM. We will estimate the error between the numerical solutions of HMM and the solutions of 1.4. We will also discuss how to construct better approximations of u from the HMM solutions. Our basic strategy is as follows. First we will prove a general statement that the error between

5 HETEROGENEOUS MULTISCALE METHOD 5 the HMM solution and the solution of 1.4 is controlled by the standard error in the macroscale solver plus a new term, called ehmm, due to the error in estimating the stiffness matrix. We then estimate ehmm. This second part is only done for either periodic or random homogenization problems, since concrete results are only possible if the behavior of u is well understood. We believe that this overall strategy will be useful for analyzing other multiscale methods. We will always assume that a x is smooth, symmetric and uniformly elliptic: 1.9 λi a ΛI for some λ, Λ > 0. We will use the summation convention and standard notation for Sobolev spaces see [1]. We will use to denote the absolute value of a scalar quantity, the Euclidean norm of a vector and the volume of a set K. For the quadrature formula 1.5, we will assume the following accuracy conditions for kth-order numerical quadrature scheme [11]: 1.10 px dx: = 1 K K K px dx = L ω l px l for all px P 2k 2. Here ω l > 0, for l = 1,, L. For k = 1, we assume the above formula to be exact for p P Main results. Our main results for the linear problem are as follows. Theorem 1.1. Denote by U 0 and U HMM the solution of 1.4 and the HMM solution, respectively. Let l=1 ehmm = max x l K Ax l A H x l, where is the Euclidean norm. If U 0 is sufficiently smooth and 1.10 holds, then there exists a constant C independent of, and H, such that U 0 U HMM 1 C H k + ehmm, U 0 U HMM 0 C H k+1 + ehmm. If there exits a constant C 0 such that ehmm ln H < C 0, then there exists a constant H 0 such that for all H H 0, 1.13 U 0 U HMM 1, C H k + ehmm ln H. At this stage, no assumption on the form of a x is necessary. U 0 can be the solution of an arbitrary macroscopic equation with the same right-hand side as in 1.1. Of course for U HMM to converge to U 0, i.e., ehmm 0, U 0 must be chosen as the solution of the homogenized equation, which we now assume exists. To obtain quantitative estimates on ehmm, we must restrict ourselves to more specific cases. Theorem 1.2. For the periodic homogenization problem, we have { C if I x l = x l + I, ehmm C + otherwise.

6 6 W. E, P.-B. MING, AND P.-W. ZHANG In the first case, we replace the boundary condition in 1.6 by a periodic boundary condition: v l V l is periodic with period I. For the second result we do not need to assume that the period of ax, is a cube: In fact it can be of arbitrary shape as long as its translation tiles up the whole space. Another important case for which a specific estimate on ehmm can be obtained is the random homogenization. In this case, using results in [43], we have Theorem 1.3. For the random homogenization problem, assuming that A in the Appendix holds see [43], we have κ, Cκ d = 3 E ehmm remains open, d = 2 1/2, Cκ d = 1 where κ = 6 12γ 25 8γ for any 0 < γ < 1/2. By choosing γ small, κ can be arbitrarily close to 6/25. The probabilistic set-up will be given in the Appendix. To prove this result, we assume that 1.8a is used with = / Recovering the microstructural information. In many applications, the microstructure information in u x is very important. U HMM by itself does not give this information. However, this information can be recovered using a simple post-processing technique. For the general case, having U HMM, one can obtain locally the microstructural information using an idea in [35]. Assume that we are interested in recovering u and u only in the subdomain Ω D. Consider the following auxiliary problem: 1.14 { div a x ũ x = fx x Ω η, ũ x = U HMM x x Ω η, where Ω η satisfies Ω Ω η D and dist Ω, Ω η = η. We then have Theorem 1.4. There exists a constant C such that 1/ u ũ 2 C dx U 0 U HMM L Ω η Ω η + u U 0 L Ω η. For the random problem, the last term was estimated in [43]. A much simpler procedure exists for the periodic homogenization problem. Consider the case when k = 1 and choose I = x K + I, where x K is the barycenter of K. Here we have assumed that the quadrature point is at x K. Let ũ be defined piecewise as follows: 1 ũ I = vk, where v K is the solution of 1.6 with the boundary condition that vk U HMM is periodic with period I and I ũ U HMM x dx = 0. 2 ũ U HMM K is periodic with period I. For this case, we can prove

7 HETEROGENEOUS MULTISCALE METHOD 7 Theorem 1.5. Let ũ be defined as above. Then 1/ u ũ 2 0,K C + H. Similar results with some modification hold for nonlinear problems. The details are given in Some technical background. In this subsection, we will list some general results that will be frequently referred to later on. Given a triangulation T H, it is called regular if there is a constant σ such that H K ρ K σ for all K T H and if the quantity H = max H K approaches zero, where H K is the diameter of K and ρ K is the diameter of the largest ball inscribed in K. T H satisfies an inverse assumption if there exists a constant ν such that H ν for all K T H. H K A regular family of triangulation of T H satisfying the inverse assumption is called quasi-uniform. The following interpolation result for Lagrange finite element is adapted from [10]. Here and in what follows, for any k 2, k v is understood in a piecewise manner. Theorem 1.6 [10]. Let Π be kth-order Lagrange interpolate operator, and assume that the following inclusions hold: 1.17 W k+1,p ˆK C 0 ˆK and W k+1,p ˆK W m,q ˆK. Then 1.18 v Πv m,q,k C K If T H is regular, we have the global estimate 1/q 1/p Hk+1 K ρ m K v k+1,p,k v Πv m,q,d CH k+1 m+min{0,d1/q 1/p} v k+1,p,d. Inequality 1.18 is proven in [10, Theorem 3.1.6], and 1.19 is a direct consequence of 1.18 and the inverse inequality below. Using 1.19 with p = q = 2 and m = 2, k = 1, we have v Πv 2,D C v 2,D. Hence 1.20 Πv 2,D v Πv 2,D + v 2,D C v 2,D. We will also need the following form of the inverse inequality. Theorem 1.7. [10, Theorem 3.2.6] Assume that T H is regular, and assume also that the two pairs l, r and m, q with l, m 0 and r, q [0, ] satisfy l m and P k ˆK W l,r ˆK W m,q ˆK. Then there exists a constant C = Cσ, ν, l, r, m, q such that 1.21 v m,q,k CH l m+d1/q 1/r K v l,r,k

8 8 W. E, P.-B. MING, AND P.-W. ZHANG for any v P k ˆK. If in addition T H satisfies the inverse assumption, then there exists a constant C = Cσ, ν, l, r, m, q such that 1/q 1/r 1.22 CH l m+min{0,d1/q 1/r} v q m,q,k for any v X H and r, q <, with max v m,,k max v l,,k replacing replacing v q m,q,k v r l,r,k The following simple result will be used repeatedly. v r l,r,k 1/q, if q =, 1/r, if r =. Lemma 1.8. Let A 1 x and A 2 x be symmetric matrices satisfying 1.9. Let ϕ 1 be the solution of 1.23 div A 1 x ϕ 1 x = div à 1 x F 1 x x Ω, with either the Dirichlet or periodic boundary condition on Ω. Let ϕ 2 be a solution of 1.23 with A 1, Ã1 and F 1 replaced by A 2, Ã2 and F 2, respectively, and let ϕ 2 satisfy the same boundary condition as ϕ 1. Then λ ϕ 1 ϕ 2 0,Ω max x Ω Ã1 Ã2x F 1 0,Ω + max x Ω A 1 A 2 x ϕ 2 0,Ω max x Ω Ã2x F 1 F 2 0,Ω. Proof. Inequality 1.24 is a direct consequence of λ ϕ 1 ϕ 2 2 0,Ω ϕ 1 ϕ 2 Ã2 Ã1 F 1 +A 2 A 1 ϕ 2 +Ã2 F 2 F 1. Ω The following simple result underlies the stability of HMM for problem 1.1. Lemma 1.9. Let ϕ be the solution of { div a ϕ = 0 in Ω R d, 1.25 ϕ = V l on Ω, where V l is a linear function and a = a ij satisfies Then we have 1.26 V l 0,Ω ϕ 0,Ω and λi a ΛI. Ω 1/2 ϕ a ϕ Ω V l a V l 1/2. Proof. Notice that ϕ = V l on the edges of Ω, using the fact that V l is a constant in Ω, and integration by parts leads to ϕ V l x V l x dx = 0, Ω

9 HETEROGENEOUS MULTISCALE METHOD 9 which implies ϕx 2 dx = Ω Ω V l x 2 dx + Ω ϕ V l x 2 dx. This gives the first result in Multiplying 1.25 by ϕx V l x and integrating by parts, we obtain ϕx a ϕx dx + ϕ V l x a ϕ V l x dx Ω This gives the second part of Ω = V l x a V l x dx. Ω Remark For this result, the coefficient a = a ij may depend on the solution, i.e., 1.25 may be nonlinear. Remark The same result holds if we use instead a periodic boundary condition: ϕ V l is periodic with period Ω. 2. Generalities Here we prove Theorem 1.1. We will let U H = U HMM for convenience. Since U H is the numerical solution associated with the quadratic form A H, U 0 is the exact solution associated with the quadratic form A, defined for any V H0 1 D as AV, V = V x Ax V x dx. D To estimate U 0 U H, we view A H as an approximation to A, and we use Strang s first lemma [10]. Using 1.26 with Ω = I x l and 1.9, for any V X H, we have 2.1 A H V, V λ K = λ K = λ V 2 0. x l K x l K ω l V l x 2 dx I x l ω l V x l 2 Similarly, for any V, W X H, we obtain A H V, W 1 1 K ω l V l a 2 V l W l a 2 W l x l K I x l I x l Λ ω l V x l W x l 2.2 = Λ K K x l K Λ V 0 W 0. V x W x dx The existence and the uniqueness of the solutions to 1.8 follow from 2.1 and 2.2 via the Lax-Milgram lemma and the Poinceré inequality.

10 10 W. E, P.-B. MING, AND P.-W. ZHANG To streamline the proof of Theorem 1.1, we introduce the following auxiliary bilinear form ÂH.  H V, W =  K V, W with  K V, W = K ω l W A V x l. x l K Classical results on numerical integration [11, Theorem 6] give for any V, W X H, 2.3  K V, W W A V dx CHm V m,k W 0,K 1 m k. K Moreover, for any V, W X H, if V k+1 and W 2 are bounded, we have [11, Theorem 8], 2.4 ÂHV, W AV, W CH k+1 V k+1 W 2. Proof of Theorem 1.1. Using the first Strang lemma [10, Theorem 4.1.1], we have A H V, W AV, W U 0 U H 1 C inf U 0 V 1 + sup. V X H W X H W 1 Let V = ΠU 0 and using 1.19 with m = 1, p = q = 2, we have 2.5 inf V X H U 0 V 1 U 0 ΠU 0 1 CH k. It remains to estimate A H V, W AV, W for V = ΠU 0 and W X H. Using 2.3, we get 2.6 A H V, W AV, W A H V, W ÂHV, W + ÂHV, W AV, W ehmm V 0 + CH k V k W 0. This gives 1.11 To get the L 2 estimate, we use the Aubin-Nitsche dual argument [10]. To this end, consider the following auxiliary problem: Find w H 1 0 D such that 2.7 Av, w = U 0 U H, v for all v H 1 0 D. The standard regularity result reads [24] 2.8 w 2 C U 0 U H Putting v = U 0 U H into the right-hand side of 2.7, we obtain U 0 U H 2 0 = AU 0 U H, w Πw + A H U H, Πw AU H, Πw = AU 0 U H, w Πw + A H U H ΠU 0, Πw AU H ΠU 0, Πw + A H ΠU 0, Πw AΠU 0, Πw. Using 2.6 with k = 1, we bound the first two terms in the right-hand side of the above identity as and AU 0 U H, w Πw C U 0 U H 1 w Πw 1 CH U 0 U H 1 w 2 A H U H ΠU 0, Πw AU H ΠU 0, Πw ehmm+ch U 0 U H 1 Πw 1.

11 HETEROGENEOUS MULTISCALE METHOD 11 The last term in the right-hand side of 2.9 may be decomposed into A H ΠU 0, Πw AΠU 0, Πw = A H ΠU 0, Πw ÂHΠU 0, Πw It follows from 2.4 that + Â H ΠU 0, Πw AΠU 0, Πw. ÂHΠU 0, Πw AΠU 0, Πw CH k+1 U 0 k+1 w 2. By definition of ehmm and using 1.20, we get A H ΠU 0, Πw ÂHΠU 0, Πw CeHMM ΠU 0 0 w 2. Combining the above estimates and using 2.8 lead to It remains to prove As in [37], for any point z D, we define the regularized Green s function G z H0 1D and the discrete Green s function Gz H X H as 2.10 AG z, V = z, V for all V H 1 0 D, AG z H, V = z, V for all V X H, where z is the regularized Dirac- function defined in [37]. It is well known that 2.11 G z G z H 1,1 C and G z H 1,1 C ln H. A proof for 2.11 can be obtained by using the weighted-norm technique [37]. We refer to [9, Chapter 7] for details. Using the definition of G z and G z H, a simple manipulation gives U 0 U H z = AG z, U 0 ΠU 0 + AG z, ΠU 0 U H Using 2.11, we obtain = AG z G z H, U 0 ΠU 0 + AG z H, U 0 U H = AG z G z H, U 0 ΠU 0 + A H U H, G z H AU H, G z H = AG z G z H, U 0 ΠU 0 + A H ΠU 0, G z H AΠU 0, G z H + A H U H ΠU 0, G z H AU H ΠU 0, G z H. U 0 U H 1, C U 0 ΠU 0 1, + AΠU 0, G z H A H ΠU 0, G z H Using 2.6, we get + AU H ΠU 0, G z H A HU H ΠU 0, G z H. AΠU 0, G z H A HΠU 0,G z H ehmm + CH k ΠU 0 k,k G z H 0,K C ehmm + H k ΠU 0 k,,k G z H L 1 K C ehmm + H k ln H U 0 k+1,, where we have used the inverse inequality 1.21.

12 12 W. E, P.-B. MING, AND P.-W. ZHANG Similarly, we have AU H ΠU 0, G z H A HU H ΠU 0, G z H ehmm + CH U H ΠU 0 1,K G z H 0,K C ehmm + H U H ΠU 0 1,,K G z H 0,1,K C ehmm + H ln H U 0 U H 1, + C ehmm + H ln H H k U 0 k+1,. A combination of the above three estimates yields U 0 U H 1, CH k + C ehmm + H + C ln H U 0 U H 1, ehmm + H k ln H U 0 k+1,. If ehmm ln H < C 0 : = 1/2C, then there exits a constant H 0 such that for all H H 0, C ehmm + H ln H 1/2 + CH ln H < 1. We thus obtain 1.13 and this completes the proof. Combining the foregoing proof for the L 2 and W 1, estimates, using the Green s function defined in [39], we obtain Remark 2.1. Under the same condition for the W 1, estimate in Theorem 1.1, we have U 0 U H L C ehmm + H k+1 ln H Estimating ehmm In this section, we estimate ehmm for problems with locally periodic coefficients. The estimate of ehmm for problems with random coefficients can be found in the Appendix. We assume that a x = ax, x/, where a is smooth in x and periodic in y with period I. Define κ = /, and we introduce ˆV l as 3.1 ˆVl x = V l x + χ k x l, x Vl x, x k where {χ j } d j=1 is defined as: For j = 1,, d, χj x, y is periodic in y with period I and satisfies 3.2 χ j a ik x, y = a ij x, y in I, χ j x, y dy = 0. y i y k y i Given {χ j } d j=1, the homogenized coefficient A = A ij x is given by χ A ij x = a j ij + a ik x, y dy. I y k Note that {χ j } d j=1 is smooth and bounded in all norms. First let us consider the case when I x l = x l + I, and 1.6 is solved with the periodic boundary condition. Denote by ˆv l the solution of 1.6 with the I

13 HETEROGENEOUS MULTISCALE METHOD 13 coefficients a x replaced by ax l, x/. ˆv l. Using Lemma 1.8, we get v l 3.3 v l ˆv l 0,I C V l 0,I. Observe that ˆv l = ˆV l. A direct calculation yields W AH A V [ x l = wl I a x, x Using 3.3, we get 3.4 ehmm C. may be viewed as a perturbation of a x l, x ] vl dx + wl I a x l, x vl ˆv l dx. Next we consider the more general case when I is a cube of size not necessarily equal to. The following analysis applies equally well to the case when the period of ax, is of general and even non-polygonal shape. This situation arises in some examples of composite materials [30]. We will show that if is much larger than, then the averaged energy density for the solution of 1.6 closely approximates the energy density of the homogenized problem. We begin with the following observation: W A V xl = W l x Ax l V l x = W l a x l, x 3.5 I κx l ˆV l dx. We first establish some estimates on the solution of the cell problem 1.6. We will write I instead of I x l if there is no risk of confusion. Lemma 3.1. There exists a constant C independent of and such that for each l, 1/2 3.6 vl 0,I \I κ C + V l 0,I. Proof. We still denote by ˆv l the solution of 1.6 with the coefficient a x replaced by ax l, x/. Using Lemma 1.8, we get 3.7 v l ˆv l 0,I C V l 0,I. Define θl = ˆv l ˆV l, which obviously satisfies div a x l, x θl x = 0 x I x l, 3.8 θl x = χ k x l, x Vl x x I x l. x k Note that θl is simply the boundary layer correction for the cell problem 1.6 [5]. It is proved in [45, 1.51 in 1.4], using the rescaling x = x/ over I and = /. 1/2 Vl 3.9 θl 0,I C 0,I. This together with 3.7 gives 3.10 v l ˆV l 0,I C 1/2 + V l 0,I.

14 14 W. E, P.-B. MING, AND P.-W. ZHANG A straightforward calculation gives 3.11 ˆV 1/2 Vl l 0,I \I κ C 0,I, which together with 3.10 leads to vl 0,I \I κ ˆV l 0,I \I κ + vl ˆV l 0,I \I κ ˆV l 0,I \I κ + vl ˆV l 0,I 1/2 C + V l 0,I. This gives 3.6. As in 3.6, we also have 3.12 v l 0,I \I κ 2 C Theorem ehmm C 1/2 + V l 0,I. +. Proof. Note that vl = v l ˆv l + θ l + ˆV l. We have W AH A V x l = :I 1 + I 2 + I 3, where I 1 = wl I a x, x I 3 = wl I a x, x Using 3.7 and 2.2, we bound I 1 as vl ˆv l dx, I 2 = wl I a x, x θl dx, ˆV l dx W l Ax l V l. I 1 Λ d v l ˆv l 0,I w l 0,I C 1 d V l 0,I W l 0,I = C V l W l. Using the symmetry of a, I 2 = I θl x, a x wl dx and I 2 = θl I + χ k x l, x Vl 1 ρ a x, x wl dx x k χ k x l, x I where ρ x C 0 I, ρ C/, and 3.14 ρ x = Vl x k 1 ρ { 1 if distx, I 2, 0 if distx, I. a x, x wl dx, Using 1.6 and θl x + χk l, x Vl x k 1 ρ H0 1 I, integrating by parts makes the first term in the right-hand side of I 2 vanish; therefore we write I 2 as I 2 = I a ij x, x w l χ k V l 1 ρ dx + x i y j x k I a ij x, x w l x i χ k V l x k ρ x j dx.

15 HETEROGENEOUS MULTISCALE METHOD 15 Using 3.12, we bound I 2 as I 2 C d w l 0,I \I κ 2 V l 0,I \I κ 2 C + 2 W l V l. Using 3.2 and integrating by parts, we obtain wl I a x l, x ˆV l dx = W l a x l, x I ˆV l dx, which together with 3.5 gives [ I 3 = wl I a x, x a x l, x + 1 d I \I κ ] ˆV l dx W l a x l, x ˆV l dx + κ/ d 1 W l Ax l V l. The last term of I 3 is bounded by κ/ d 1 Wl Ax l V l C V l W l, where we have used κ/ d 1 C/. Using 3.11, we get d W l a x l, x ˆV l dx C 1 d ˆV l 0,I \I κ W l 0,I \I κ I \I κ Consequently, we obtain C + 2 V l W l. I 3 C 1 d wl 0,I ˆV l 0,I + C + 2 V l W l C + V l W l. Combining the estimates for I 1, I 2 and I 3 gives the desired result Remark 3.3. An explicit expression for vl is available in one dimension, from which we may show that the bound for ehmm is sharp. 4. Reconstruction and compression 4.1. Reconstruction procedure. Next we consider how to construct better approximations to u from U H. We will restrict ourselves to the case when k = 1. Proof of Theorem 1.4. Subtracting 1.1 from 1.14, we obtain { div a x ũ u x = 0 x Ω η, ũ u x = U H x u x x Ω η. Using classical interior estimates for elliptic equation [24], we have ũ u 0,Ω C η ũ u 0,Ωη.

16 16 W. E, P.-B. MING, AND P.-W. ZHANG Using the Hopf maximum principle, we get 1 η 2 ũ Ω u x 2 dx C η η 2 ũ u 2 L Ω C η η 2 u U H 2 L Ω η C U η 2 0 U H 2 L Ω + η u U 0 2 L Ω η. A combination of the above two results implies Theorem 1.4. Proof of Theorem 1.5. Denote I x K = x K + I and define û as the solution of 4.1 div a x K, x û x = 0 in I x K, with the boundary condition that û U H is periodic on I x K and û U H dx = 0, I x K where x K is the barycenter of K. It is easy to verify that û takes the explicit form 4.2 û x = U H x + χ k x K, x UH x. x k Note that the periodic extension of û U H is still χ k x K, x UH x k x. This means that û is also well defined for the whole of K and takes the same explicit form as 4.2. Using I χk x K, y dy = 0 for k = 1,, d and that U H is a piecewise constant on K, we obtain 4.3 û U H x dx = χ k x K, x U H x dx = 0. x k I x K As in 3.7, we have I x K ũ û 0,Ix K C U H 0,Ix K. From the construction of ũ, we have for any x 1 K, ũ û 0,Ix 1 = ũ û 0,Ix K. Since U H is constant over K, we get 4.4 ũ û 0,K C U H 0,K. Adding up for all K T H and using the a priori estimate U H 0 C f 0, we obtain 1/2 4.5 ũ û 2 0,K C UH 0 C. Using 4.2, a straightforward calculation gives û = U H x i x i Define the first order approximation of u as + χk y i x K, x u 1 x = U 0 + χ k x, x UH x k. U0 x k,

17 HETEROGENEOUS MULTISCALE METHOD 17 where {χ k } d k=1 is the solutions of 3.2. Obviously, u 1 = U 0 + χk + x, χk x x i x i x i y i A combination of the above estimates leads to U0 x i + χ k x, x 2 U 0 x i x k. û u 1 0,K C U H U 0 0,K + C U 0 1,K + χ k x, x x χk K, x U0 y i y i x k + C U 0 2,K 0,K C U 0 U H 1,K + C + H U 0 2,K. Summing up for all K T H and using Theorem 1.1 for k = 1 and Theorem 1.2 for the case I = x K + I, we get 1/2 û u 1 2 0,K C + H, which together with 4.5 and the classical estimate for u u 1 [5, 32, 45], i.e., gives u ũ 2 0,K Corollary 4.1. u u 1 1 C, 1/2 u u u 1 û 2 0,K 1/2 + û ũ 2 0,K C + H. 4.6 ũ u 0 C + H 2. Proof. Using the definition of ũ, we have I ũ x K U H x dx = 0. Together with 4.3, we have ũ û x dx = 0. I x K An application of the Poincaré inequality gives ũ û 0,Ix K C ũ û 0,Ix K C 2 U H 0,Ix K. As before for any I x 1, ũ û 0,Ix 1 = ũ û 0,Ix K, note that U H is a constant on K. We obtain 4.7 ũ û 0,K C 2 U H 0,K. On each element K, we have û U H 0,K C U H 0,K. Combining the above and summing up for all K T H, we get ũ U H 0 C U H 0 C, 1/2

18 18 W. E, P.-B. MING, AND P.-W. ZHANG which together with u U H 0 u U U 0 U H 0 C + H 2 leads to 4.6, where we have used the estimate for U 0 [5, 32, 45], i.e., u U 0 0 C Compression operator. The compression operator denoted by Q maps the microvariables to the macrovariables [16]. It plays an important role in the general framework of HMM, even though for the present problem HMM can be formulated without explicitly specifying the compression operator beforehand. Typically the compression operator is some spatial/temporal averaging, or projection to some slow manifolds. It is of interest to consider the error bound for Qu U H. We first list some natural properties of the compression operator. For any φ X, Qφ X H. There exists a constant C such that Qφ 0 C φ 0. For any k 1, if φ H k+1 Ω H0 1 Ω, then φ Qφ 0 CH k+1 φ k+1. Theorem 4.2. Assume that Q satisfies all three requirements and U 0 H k+1 D for any k 1. Then 4.8 Qu U H 0 C + H k+1. Moreover, if T H is quasi-uniform, then 4.9 Qu U H 1 C H + Hk. Proof. We decompose Qu U H into 4.10 Qu U H = Qu U 0 + QU 0 U 0 + U 0 U H. Using the fact that Q is bounded in L 2 norm, we obtain Using the third property of Q, we have Qu U 0 0 C u U 0 0 C. QU 0 U 0 0 CH k+1. Using Theorem 1.1 and the first estimate in Theorem 1.2, we have U 0 U H 0 C + H k+1. A combination of these three estimates implies 4.8, which together with the inverse inequality cf. Theorem 1.7 leads to 4.9. It remains to give some examples of the compression operator. The following two types operators meet all three requirements: the L 2 -projection operator onto X H, the Clément-type interpolation operator [12].

19 HETEROGENEOUS MULTISCALE METHOD 19 Remark 4.3. Notice that in one dimension, the standard Lagrange interpolant does not meet the second requirement. However, it is still possible to derive 4.9 via another approach. Moreover, a careful study of one dimensional examples shows that the term /H in 4.9 is sharp. 5. Nonlinear homogenization problems 5.1. Algorithms and main results. We consider the following nonlinear problem which has been discussed in [6, 23]: { div a x, u x u x = fx x D, 5.1 u x = 0 x D. In this section, we define X: = W 1,p 0 D with p > 1 and X H is defined as the P k finite element subspace of X. We assume that a x, u satisfies λ ξ 2 a ijξ i ξ j Λ ξ 2 for all ξ R d, with 0 < λ Λ. Moreover, we assume that a x, z is equi-continuous in z uniformly with respect to x and. The homogenized problem, if it exists, is of the following form: { LU0 : = div A x, U 0 x U 0 x = fx x D, If we let then U 0 x = 0 Av, w = Ax, v v, w for all v, w X, 5.2 AU 0, v = f, v for all v X, x D. where X is the dual space of X. The linearized operator of L at U 0 is defined for any v H 1 0 D by L lin U 0 v = div Ax, U 0 v + A p x, U 0 U 0 v, where A p x, z: = z Ax, z. L lin induces a bilinear form through Âu; v, w: = Ax, u v, w + A p x, u u v, w for all v, w H 1 0 D. Our basic assumption is that the linearized operator L lin is an isomorphism from H0 1D to H 1 D, so U 0 must be an isolated solution of 5.2. To formulate HMM, for each quadrature point x l, define vl to be the solutions of { div a x, vl v l x = 0 x I x l, 5.3 We can define w l similarly. For any V, W X H, define and W x l A H xl, V x l V x l = A H V, W : = K v l x = V l x x I x l. wl x a x, vl x vl x dx I x l ω l W x l A H xl, V x l V x l. x l K

20 20 W. E, P.-B. MING, AND P.-W. ZHANG The HMM solution is given by the problem: Problem 5.1. Find U H X H such that 5.4 A H U H, V = f, V for all V X H. For any v, v H, w X, define 5.5 Rv, v H, w: = Av H, w Av, w Âv; v H v, w. It is easy to see that for any v and v H satisfying v 1, + v H 1, M, 5.6 Rv, v H, w CM e H 2 0,2p + e H e H 0,p w 0,q for e H : = v v H and 1 p + 1 q = 1, p, q 1 see [42, Lemma 3.1] for a similar result. Therefore we have Lemma 5.2. U H X H is the solution of Problem 5.1 if and only if 5.7 ÂU 0 ; U 0 U H, V = RU 0, U H, V For any V, W X H, define + A H U H, V AU H, V for all V X H. 5.8 EV, W : = W x l A H A x l, V x l V x l. Define ehmm as 5.9 ehmm = EV, W max x l K, V l W l. V X H W 1, D, W X H The existence and uniqueness of the solution of Problem 5.1 are proved in the following lemma. Lemma 5.3. Assume that U 0 W 2,p D with p > d and L lin is an isomorphism from H 1 0 D to H 1 D. If ehmm is uniformly bounded and there exist constants H 0 > 0 and M 1 > 0 such that for 0 < H H 0 and 5.10 ehmm 1/2 ln H M 1, then Problem 5.1 has a solution U H satisfying 5.11 where P H U 0 X H is defined as U H P H U 0 1, ehmm 1/2 + H 1 d/p, U 0 U H 1, C ehmm 1/2 + H 1 d/p, 5.12 ÂU 0 ; P H U 0, V = ÂU 0; U 0, V for all V X H. Moreover, if there exists a constant ηm with 0 < ηm < 1 such that 5.13 ω l EV, Z EW, Z ηm V W 1 Z 1 K x l K for all V, W X H W 1, D and Z X H, satisfying V 1, + W 1, M, then there exists a constant H 1 > 0 such that for 0 < H H 1, the HMM solution U H satisfying is locally unique.

21 HETEROGENEOUS MULTISCALE METHOD 21 Proof. Since L lin is an isomorphism from H 1 0 D to H 1 D, there exists a constant C such that ÂU 0 ; V, W sup C V 1 for all V H W H0 1D 0 1 W D. 1 Using [42, Lemma 2.2], we conclude that there exists a constant H 2 > 0 such that for 0 < H H 2, 5.14 sup W X H ÂU 0 ; V, W W 1 C V 1 for all V X H. Therefore there is a unique solution P H U 0 X H satisfying 5.12 and 5.15 U 0 P H U 0 1, CH 1 d/p. Moreover, let Ĝz H be the finite element approximation of the regularized Green s function associated with ÂU 0;,. Using [42, equation 2.11], or using 5.14, similarly to 2.11, we have 5.16 Ĝz H 1,1 C ln H. Define a nonlinear mapping T : X H X H by ÂU 0 ; T V, W = ÂU 0; U 0, W RU 0, V, W + AV, W A H V, W, for any W X H. Obviously T is continuous due to 5.14 and 5.6. Let B: = { V X H V P H U 0 1, ehmm 1/2 + H 1 d/p }. We next prove that there exists a constant H 0 > 0 such that for all 0 < H H 0, T B B. Notice that ÂU 0 ; T V P H U 0, W = RU 0, V, W + AV, W A H V, W. Taking W = Ĝz H V 1, M, in the above equation, using 5.16 and 5.6, we obtain, for T V P H U 0 1, CM U 0 V 2 1, ln H + C ehmm + H ln H V 1, CM U 0 P H U 0 2 1, + P H U 0 V 2 1, ln H + CM ehmm + H ln H CM ehmm + H 2 2d/p ln H + CM ehmm + H ln H. Since V B and ehmm is uniformly bounded, e.g., ehmm M 1, we have 5.17 V 1, V P H U 0 1, + P H U 0 1, CU 0 + ehmm 1/2 CU 0 + M 1/2 1 = :M 0. Combining the above two estimates, we obtain T V P H U 0 1, CM 0 ehmm + H 2 2d/p + H ln H. Define M 1 : = 1/CM 0. Using 5.10, we obtain T V P H U 0 1, ehmm 1/2 + CM 0 H 2 2d/p + H ln H. Therefore there exits a constant H 3 such that for 0 < H H 3, we have T V P H U 0 1, ehmm 1/2 + H 1 d/p.

22 22 W. E, P.-B. MING, AND P.-W. ZHANG Let H 0 : = minh 2, H 3. Then for 0 < H H 0, we have T B B. An application of Brouwer s fixed point theorem gives the existence of a U H B such that T U H = U H. By definition, U H satisfies , which together with 5.15 yields To prove uniqueness, assume that both U H and ÛH are solutions of 5.4 satisfying Using 5.14, we obtain C U H ÛH 1 sup W X H where U t H = 1 tûh + tu H. Note that 1 0 ÂU t H ; U H ÛH, W dt W 1 AU H, W AÛH, W sup, W X H W 1 AU H, W AÛH, W = AU H, W A H U H, W AÛH, W A H ÛH, W. Since both U H and ÛH sit in the set B, we can use 5.17 to get U H 1, + ÛH 1, 2M 0. Using 5.13, we obtain If we choose H 1 such that U H ÛH 1 η2m 0 + C 1 H U H ÛH 1. η2m 0 + C 1 H 1 < 1, then if H < H 1, we have U H = ÛH. Therefore the HMM solution is locally unique. From here on, when we talk about the HMM solution, we are referring to this particular solution that satisfies the condition in Lemma 5.3. Based on the above lemma, we prove a nonlinear analog of Theorem 1.1. Theorem 5.4. Under the assumptions in Lemma 5.3, let U 0 and U H be solutions of 5.2 and 5.4, respectively. Assume in addition that U 0 W k+1, D. Then there exist constants H 0 and M 1 such that if 0 < H < H 0 and M 1 < M 1, then U 0 U H 1 C H k + ehmm, U 0 U H 1, C H k + ehmm ln H. Proof. Note that U 0 W k+1, D and from 5.14, we have 5.20 U 0 P H U 0 1 CH k, U 0 P H U 0 1, CH k. Using 5.7 with V = P H U 0 U H and 5.14 and 2.3, we obtain 5.21 P H U 0 U H 1 C U 0 U H 2 1,4 + C H k + ehmm. Using the interpolation inequality together with 5.20 and 5.17 gives U 0 U H 2 1,4 U 0 U H 1 U 0 U H 1, P H U 0 U H 1 C 1 ehmm 1/2 + H 1 d/p P H U 0 U H 1 + C H k + ehmm.

23 HETEROGENEOUS MULTISCALE METHOD 23 Let V = Ĝz H in 5.7. Using 5.16, we obtain P H U 0 U H 1, C 2 ehmm + H ln H PH U 0 U H 1, + C U 0 U H 2 1, + ehmm + H k ln H. Since 5.10 holds, using and , we obtain P H U 0 U H 1, C 2 ehmm 1/2 + H 1 d/p ln H P H U 0 U H 1, + C ehmm + H k ln H. Now we choose ln H M 1 1: = min, 2C 1 2C 2 and H 0 such that ehmm 1/2 ln H M1, and therefore Thus we obtain C 1 ehmm 1/2 + H 1 d/p C 1H 1 d/p 0 < 1, C 2 ehmm 1/2 + H 1 d/p ln H C 2H 1 d/p ln H < 1. P H U 0 U H 1 C H k + ehmm, P H U 0 U H 1, C H k + ehmm ln H. Using 5.20 once again gives 5.18 and Estimating ehmm. It remains to estimate ehmm and verify assumptions 5.10 and We assume that a x, u = a ij x, x/, u, and for 1 i, j d, the coefficients aij x, y, z are smooth in x, z and periodic in y with period I. These types of problems, among others, have been considered in [5, 6, 23]. The homogenized coefficient A = A ij x, p is given for any p R by χ A ij x, p = a j ij + a ik x, y, p dy, I y k where {χ k } d k=1 is defined for any p R by 5.22 χ j a ik x, y, p = a ij x, y, p, y i y k y i with the periodic boundary condition in y and I χk x, y, p dy = 0. It is clear that Ax, p is also smooth in x and p and satisfies [6, Proposition 3.5] 5.23 λi A Λ2 λ I. Using Lemma 1.9 and Remark 1.10, for the solution of 5.3, we have 5.24 V l 0,I v l 0,I Λ/λ V l 0,I. This gives a bound for A H : λi A H Λ2 λ I, which together with 5.23 implies 5.25 ehmm 2Λ 2 /λ. This shows that ehmm is uniformly bounded.

24 24 W. E, P.-B. MING, AND P.-W. ZHANG To simplify the presentation, we will show how to estimate ehmm when 5.3 is changed slightly to { div a x, V x l vl x = 0 x I x l, 5.26 and A H V, W is changed to A H V, W = K x l K v lx = V l x x I x l ω l wl x a x, V x l vl x dx. I x l If =, we replace the Dirichlet boundary condition in 5.26 by the periodic boundary condition, i.e., v l x V lx is periodic on I x l. Theorem 5.5. If / + ln H is sufficiently small, then 5.10 and 5.13 hold and 1/ ehmm C +. In the case of =, if ln H is sufficiently small, then 5.10 and 5.13 hold and 5.28 ehmm C. In what follows, we concentrate on the first case. The second case when = will be commented on. Let us first fix more notation. Denote by ˆv l the solutions of 5.26 with the coefficient a x, x/, V x l replaced by a x l, x/, V x l. Similarly we define wl to be the solution of 5.26 with V replaced by W X H. Also, ŵl can be defined in the same way, and wl and ŵ l can be viewed as the perturbations of v l and ˆv l, respectively. Moreover, we define a V x l = a x l, x, V x l, a W x l = a x l, x, W x l, Observe that ˆv l a V x = a x, x, V x l, a W x = a x, x, W x l. and ŵ l also satisfy 5.24, and using Lemma 1.8, we have 5.29 v l ˆv l 0,I C V l 0,I, w l ŵ l 0,I C W l 0,I. Lemma 5.6. We have 5.30 ˆv l ŵ l 0,I C V W x l V l 0,I + W l 0,I Proof. Observe that + V l W l 0,I. div a V x l ˆv l V l = div a V x l V l, div a W x l ŵ l W l = div a W x l W l. Both ˆv l V l and ŵl W l vanish on I x l. Using Lemma 1.8, we obtain λ ˆv l V l ŵl + W l 0,I Λ V l W l 0,I Using 5.24 for ŵl, we obtain + max x I a V a W x l V l 0,I + ŵ l W l 0,I. ŵl W l 0,I ŵl 0,I + W l 0,I C1 + Λ/λ W l 0,I, which together with max x I a V a W x l C V W x l gives 5.30.

25 HETEROGENEOUS MULTISCALE METHOD 25 Next we establish the estimate for vl ˆv l w l ŵ l. Let ψ l : = v l ˆv l and ψ l : = w l ŵ l. Clearly, ψ l, ψ l vanish on I x l and satisfy div a V x ψl = div av x a V x l ˆv l x I x l, div a W x ψ l = div aw x a W x l ŵl x I x l. Lemma 5.7. We have 5.31 ψ l ψ l 0,I C V W x l V l 0,I + W l 0,I Proof. Using Lemma 1.8, we have Using λ ψ l ψ l 0,I + V l W l 0,I. max x I a V x a V x l a W x + a W x l ˆv l 0,I + max a V x a W x ψ l 0,I x I + max a W x a W x l ˆv l ŵ l 0,I. x I 5.32 max x I a V x a V x l a W x + a W x l C V W x l, from , we have ψ l 0,I C W l 0,I. Collecting the above estimates and using 5.30, we obtain Define ˆV l x = V l x + χ k l Ŵ l x = W l x + χ k l x l, x, V x Vl l x l, x, W x l x, x k Wl x, x k where {χ k l }d k=1 are the solutions of 5.22 with coefficient replaced by a ijx l, y, p. Denote θl = ˆv l ˆV l and θ l = ŵ l Ŵl. Observe that div a V x l θl x = 0 x I x l, θl x = χk l xl, x, V x l V l x I x l, x k and div a W x l θ l x = 0 x I x l, θ l x = χk l xl, x, W x l W l x I x l. x k Similarly to 3.9, we have 1/2 Vl 1/2 Wl 5.33 θl 0,I C 0,I, θ l 0,I C 0,I. Let ρ be defined as in 3.14 and define ϕ l = θl + χ k l xl, x, V x l V l x k 1 ρ, ϕ l = θ l + χk l xl, x, W x l W l x k 1 ρ.

26 26 W. E, P.-B. MING, AND P.-W. ZHANG Observe that ϕ l and ϕ l vanish on I x l and satisfy div a V x l ϕ lx = div a V x l ϕ l θ l x x I x l, div a W x l ϕ l x = div a W x l ϕ l θ l x x I x l. Lemma 5.8. We have θl θ 1/2 l 0,I C V l W l 0,I V W x l V l 0,I + W l 0,I. Proof. Using Lemma 1.8, we obtain λ ϕ l ϕ l 0,I A direct computation gives that ϕ l θl 0,I C max x I a V x l a W x l ϕ l θ l 0,I + ϕ l 0,I + max x I a W x l ϕ l θ l ϕ l + θ l 0,I. 1 2 V l 0,I and ϕ l θ l 0,I C 1 2 W l 0,I, which together with 5.33 gives ϕ l 0,I θ l 0,I + ϕ l θ 1/2 Wl l 0,I C 0,I. Note that ϕ l θ l ϕ l + θ l = 1 ρ χ k l xl, x, V x l χ k l xl, x, W x l V l x k + 1 ρ χ k l xl, x, W x l V l W l. x k Using the continuity of {χ k l }d k=1, a direct computation gives ϕ l θ l ϕ l + θ l 0,I C Adding these up, we obtain /2 V W x l V l 0,I + V l W l 0,I. In the next lemma, we shall prove that EV, W has certain continuity with respect to V. Lemma 5.9. For any V, W, Z X H satisfying V 1, + W 1, M, there exists a constant CM such that 5.13 holds with ηm = CM / +. Proof. Using the definition of vl obtain and noticing that v l = v l ˆv l + θ l + ˆV l, we I z l a V x v l dx = I Z l a V x v l dx = Z l I a V x v l ˆv l + θ l + ˆV l dx which together with Iκ Z l a V x l ˆV l dx = Zx l Ax l, V x l V x l gives EV, Z EW, Z = :I 1 + I 2 + I 3,,

27 HETEROGENEOUS MULTISCALE METHOD 27 with and [ ] I 1 = Z l av x vl ˆv l a W x wl ŵ l dx, I I 2 = Z l [ I a V x θ l a W x θ l dx ], [ I 3 = Z l a V x I ˆV l dx a V x l I ˆV l dx κ ] a W x Ŵl dx + a W x l Ŵl dx. I I κ I 1 can be decomposed into [ I 1 = Z l av x a W x vl ˆv l dx + a W x ψl I I ψ ] l dx. Using 5.29 and 5.31, we bound I 1 as I 1 C V l W l + V W x l V l + W l Z l. Similarly, using Lemma 5.8, we bound I 2 as 1/2 I 2 C V l W l + V W x l V l + W l Z l. I 3 can be rewritten as av I 3 = Z l x a V x l ˆV l a W x a W x l Ŵl dx Z l I + 1 κ d Z l 1 κ d 1 d I \I κ I av x l ˆV l a W x l Ŵl dx av x l ˆV l a W x l Ŵl dx. As in I 1 and using 5.32, we bound I 3 as I 3 C + V l W l + V W x l V l + W l Z l. Adding these up, we get 1/2 EV, Z EW, Z C + + V l W l V W x l V l + W l Z l. Consequently we obtain K 1/2 ω l EV, Z EW, Z C + V W 0,K Z 0,K x l K 1/2 + C + V W L K V 0,K + W 0,K Z 0,K.

28 28 W. E, P.-B. MING, AND P.-W. ZHANG Using the inverse inequality 1.21 on each element, we obtain V W L K V 0,K CH d/2 K V W 0,KH d/2 K V L K = C V W 0,K V L K. Similarly V W L K W 0,K C V W 0,K W L K. A combination of the above estimates gives 5.13 with ηm = CM / +. Proof of the first case in Theorem 5.5. Let W = 0 in 5.35 and note that V W 1, D. We obtain Therefore if /+ ln H is sufficiently small, then ehmm 1/2 ln H can be smaller than any given threshold; this verify Next let ηm = CM / +, then if / + ln H is sufficiently small, we have ηm < 1; this verifies Remark Compared to the linear case, the upper bound for ehmm for the case when / Z degrades to /. This is due to the fact that A H is nonsymmetric. In the case of =, note that ˆv l = ˆV l and ŵl = Ŵl. So a direct calculation gives Lemma 5.6 for this case. Lemma 5.7 is also valid with replaced by. We also have θl = 0 and θ l = 0. Observing that for any V, Z X H, zl a V x l ˆv l dx = Z l Ax l, V x l V l dx, I x l K we may rewrite EV, Z as EV, Z = zl a V x vl ˆv l dx + zl a V x a V x l ˆv l dx. I I Consequently, 5.13 holds with ηm = CM. Proof of the second case in Theorem 5.5. Let W = 0 in 5.35 and note that V W 1, D. We obtain Therefore if ln H is sufficiently small, ehmm 1/2 ln H can be smaller than any given threshold; this verify Next let ηm = CM. Then if ln H is sufficiently small, we have ηm < 1; this proves Appendix A. Estimating ehmm for problems with random coefficients Here we estimate ehmm for the random case. Our basic strategy follows that of [43]. Denote by Ω, F, P a probability space and let ay, ω = a ij y, ω be a random field whose statistics is invariant under integer shifts and which satisfies the uniform ellipticity condition that there exist constants λ and Λ such that λ ξ 2 a ij y, ωξ i ξ j Λ ξ 2, for almost all y R d and ω Ω. For j = 1,, d, denote by ϕ j y, ω the solutions of the cell problem: A.1 L y ϕ j : = div y aij y, ω y ϕ j = divy a ij e j,

29 HETEROGENEOUS MULTISCALE METHOD 29 where {e j } d j=1 are the standard basis in Rd. ϕ j is required to be stationary under integer shift. The existence of ϕ j is proved in [28, 36]. In general ϕ j is not stationary. The homogenized coefficient A [28, 36] is given by A = ai + ϕ. Here and in the following, we use the notation f = E fy dy [0,1] d and [f; m] = 1 m d [0,m] d fy dy, where E denotes the expectation in the probability space Ω, F, P. As in [43], we will consider the following auxiliary problem: A.2 Lu + ρu = f, for any ρ > 0, where f is of the form f = with g j, h ρ G which is defined as d D j g j + h, j=1 ρ G : = {ψ ψ 2 G 2 }, and ψ is a random field whose statistics is stationary with respect to integer shifts. The solution of A.2 can be expressed with the help of a diffusion process η generated by the operator L. For each fixed realization of {ay, }, denote by η x the diffusion process generated by L 1 and starting from x at t = 0, and denote by M x the expectation with respect to η x. Let Γτ = τ 0 fηte ρt dt. Then it is well known [21] that the solution of A.2 is given by ux = M x Γ. The following results are either standard or proved in [28, 43]. Lemma A.1. If u is the solution of A.2, then there exists a constant C independent of ρ such that u 2 + ρ u 2 C g A.3 ρ h2, A.4 A.5 M x Γ 2 1/2 C G2 ρ, M x Γ Γs 2 C G2 ρ e 2sρ.

30 30 W. E, P.-B. MING, AND P.-W. ZHANG Because of the lowest order term ρu, the Green s function associated with the operator L 1 + ρi decays exponentially with rate O ρ. To make this statement precise, we define the norm x : = max i x i, and Q ρ : = { x R d x 1 ρ 1 2 [ln1/ρ]1/2 }. Let τ be the first exit time of Q ρ starting at x Q ρ. Let ˆϕ ρ x = M x Γτ. Lemma A.2 [43]. If ρ is sufficiently small, then A.6 E ϕ ρ x ˆϕ ρ x 2 dx CG 2 e C ln1/ρ 2, A.7 E x 10 ϕ ρ x ˆϕ ρ x 2 dx CG 2 e C ln1/ρ 2. x 1 To prepare for the discussion on the consequence of the mixing condition, we mention Lemma A.3 [43]. Let {a ij, g j } and {ã ij, g j } be two sets of data such that {a ij y, g j y} = {ã ij y, g j y} for y / B, where B is a domain in R d, and let ϕ ρ and ˆϕ ρ be the solutions of A.2 associated with {a ij, g j } and {ã ij, g j }, respectively with h = 0. Then A.8 ϕ ρ x ϕ ρ x 2 C dx G 2 + ϕ ρ x 2 I B x dx, ρ R d R d where I B is the indicator function of the domain B. Now we introduce the crucial mixing condition [22]. Let B be a domain in R d. Denote by FB the σ-algebra generated by {a ij y, ω, y B}. Let ξ, η be two random variables that are measurable with respect to FB 1 and FB 2, respectively. Then A Eξη EξEη Eξ 2 1/2 Eη 2 1/2 e λq, where q = distb 1, B 2, λ > 0 is a fixed constant. Lemma A.4. Under condition A, we have G A.9 E[ϕ ρ ; m] 2 2 ln1/ρ 2 d C + e cln1/ρ 2. ρ ρ 1/2 m Proof. For l = l 1,, l d Z d [0, m d : = Z d m, denote by I l the cube of size 1 centered at l = l ,, l d + 1 2, and let ϕl = ϕ I l ρ x dx. Then [ϕ ρ ; m] = 1 m d ϕ l. l Z d m We first estimate Eϕ l ϕ k. If l k Cρ 1/2 ln1/ρ 2, then Eϕ l ϕ k ϕ 2 ρx 1/2 ϕ 2 ρx 1/2 C G2 ρ.

31 HETEROGENEOUS MULTISCALE METHOD 31 If l k Cρ 1/2 ln1/ρ 2, then let B 1 : = l + Bρ 1/2 ln1/ρ 2 and B 2 : = k + Bρ 1/2 ln1/ρ 2, where Bs is a ball of size s in the norm. Denote by ϕ 1 x the solution of A.2 in which the coefficient a ij y, ω is modified in R d \B 1 such that it is measurable with respect to FB 1, and similarly denote by ϕ 2 x the solution of A.2 in which the coefficient a ij y, ω is modified in R d \B 2 such that it is measurable with respect to FB 2. The modified coefficients ã ij y, ω should still satisfy the condition on a ij listed in the beginning of this subsection. From ϕ 1 and ϕ 2, we can similarly define ϕ l 1 and ϕ l 2. Using A.6, we have E ϕ l 1 ϕl 2 CG 2 e C ln1/ρ 2, Since and we thus have Hence E[ϕ ρ ; m] 2 = 1 m 2d E ϕ k 1 ϕ k 2 CG 2 e C ln1/ρ 2. Eϕ l ϕ k = E ϕ l ϕ k + Eϕ l ϕ l 1 ϕk 2 + E ϕl 1 ϕk ϕ k 2 + Eϕ l ϕ l 1 ϕk ϕ k 2 E ϕ l 1, E ϕ k 2 CGe C ln1/ρ 2, G E ϕ l 1 ϕ k 2 2 C ρ e C l k + G 2 e C ln1/ρ 2, G Eϕ l ϕ k 2 C ρ e C l k + G 2 e C ln1/ρ 2. l,k Z d m Eϕ l ϕ k = 1 m 2d Eϕ l ϕ k + Eϕ l ϕ k l k ρ 1/2 ln1/ρ 2 l k >ρ 1/2 ln1/ρ 2 1 m 2d G 2 ρ md ρ 1/2 ln1/ρ 2 d + m 2d G 2 e C ln1/ρ 2 G2 ρ C G2 ρ l k ρ 1/2 ln1/ρ 2 e C l k ln1/ρ ρ 1/2 m d + e C ln1/ρ 2. Proceeding along the same line as in [43, Theorem 2.1], using condition A, we have Lemma A.5. For any 0 < γ < 1/2, under condition A, there exists a constant C such that A.10 A ai + χ ρ Cρ d 2 2γ 4+d, where χ ρ = {χ k,ρ } d k=1, and χ k,ρ is the solution of A.1 with g = a k1,, a kd.