Analysis of the heterogeneous multiscale method for elliptic homogenization problems

Abstract

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

1. Introduction and main results

1.1. General methodology

Consider the classical elliptic problem

$$\begin{equation} \left\{\begin{aligned} -\operatorname {div}\bigl (a^{\,\varepsilon }(x)\nabla u^{\varepsilon }(x)\bigr )&=f(x)\qquad && x\in D\subset \mathbb{R}^d,\\ u^{\varepsilon }(x)&=0\qquad &&x\in \partial D. \end{aligned}\right. {\tag{1.1}} \end{equation}$$

Here $\varepsilon$ is a small parameter that signifies explicitly the multiscale nature of the coefficient $a^{\,\varepsilon }(x)$. 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 $a^{\,\varepsilon }(x)$. In contrast modern multiscale methods are designed specifically for recovering partial information about $u^{\varepsilon }$ 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 $a^{\,\varepsilon }(x)$ might have, such as scale separation. The simplest example is when

$$\begin{equation} a^{\,\varepsilon }(x)=a\Bigl (x,\frac{x}{\varepsilon }\Bigr ), {\tag{1.2}} \end{equation}$$

where $a(x,y)$ 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 $x\in D$. In both cases, it has been shown that 536

$$\begin{equation} \|u^{\varepsilon }(x)-U(x)\|_{L^2(D)}\rightarrow 0, {\tag{1.3}} \end{equation}$$

where $U(x)$ is the solution of a homogenized equation:

$$\begin{equation} \left\{\begin{aligned} -\operatorname {div}\bigl (\mathcal{A}(x)\nabla U(x)\bigr )&=f(x)\qquad && x\in D,\\ U(x)&=0\qquad &&x\in \partial D. \end{aligned}\right. {\tag{1.4}} \end{equation}$$

The homogenized coefficient $\mathcal{A}(x)$ can be obtained from the solutions of the so-called cell problem. In general, there are no explicit formulas for $\mathcal{A}(x)$, except in one dimension.

Several numerical methods have been developed to deal specifically with the case when $a(x,y)$ is periodic in $y$. References 347 propose to solve the homogenized equations as well as the equations for the correctors. Schwab et al. 2938 use multiscale test functions of the form $\varphi (x,x/\varepsilon )$ where $\varphi (x,y)$ is periodic in $y$ to extract the leading order behavior of $u^{\varepsilon }(x)$, extending an idea that was used analytically in the work of 2153444 for the homogenization problems. These methods have the feature that their cost is independent of $\varepsilon$, hence sublinear as $\varepsilon \rightarrow 0$, 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 2025.

1.2. Heterogeneous multiscale method

HMM 161718 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 $\mathcal{P}_k$ finite element method on a triangulation $\mathcal{T}_H$ of element size $H$ which should resolve the macroscale features of $a^{\,\varepsilon }(x)$. The missing data is the effective stiffness matrix at this scale. This stiffness matrix can be estimated as follows. Assuming that the effective coefficient at this scale is $\mathcal{A}_H(x)$, if we knew $\mathcal{A}_H(x)$ explicitly, we could have evaluated the quadratic form

$$\begin{equation*} \int \limits _D\nabla V(x)\cdot \mathcal{A}_H(x)\nabla V(x)\,dx \end{equation*}$$

by numerical quadrature: For any $V\in X_H$, the finite element space,

$$\begin{equation} A_H(V,V)\simeq \sum _{K\in \mathcal{T}_H}\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell \bigl (\nabla V\cdot \mathcal{A}_H\nabla V\bigr )(x_\ell ), {\tag{1.5}} \end{equation}$$

where $\{x_\ell \}$ and $\{\omega _\ell \}$ are the quadrature points and weights in $K$, $\lvert K\rvert$ is the volume of $K$. In the absence of explicit knowledge of $\mathcal{A}_H(x)$, we approximate $\bigl (\nabla V\cdot \mathcal{A}_H\nabla V\bigr )(x_\ell )$ by solving the problem:

$$\begin{equation} \left\{\begin{aligned} -\operatorname {div}\bigl (a^{\,\varepsilon }(x)\nabla v^{\varepsilon }_\ell (x)\bigr )&=0\qquad &&x\in I_\delta (x_\ell ),\\ v^{\varepsilon }_\ell (x)&=V_\ell (x)\qquad &&x\in \partial I_\delta (x_\ell ), \end{aligned}\right. {\tag{1.6}} \end{equation}$$

where $I_\delta (x_\ell )$ is a cube of size $\delta$ centered at $x_\ell$, and $V_\ell$ is the linear approximation of $V$ at $x_\ell$. We then let

$$\begin{equation} \bigl (\nabla V\cdot \mathcal{A}_H\nabla V\bigr )(x_\ell )\simeq \frac{1}{\delta ^d}\int \limits _{I_\delta (x_\ell )}\nabla v^{\varepsilon }_\ell (x)\cdot a^{\,\varepsilon }(x)\nabla v^{\varepsilon }_\ell (x)\,dx. {\tag{1.7}} \end{equation}$$

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 $V,W\in X_H$

$$\begin{equation*} A_H(V,W){:}=\sum _{K\in \mathcal{T}_H}\dfrac{\lvert K\rvert }{\delta ^d}\sum _{x_\ell \in K}\omega _\ell \int \limits _{I_\delta (x_\ell )}\nabla v^{\varepsilon }_\ell (x)\cdot a^{\,\varepsilon }(x)\nabla w^{\varepsilon }_\ell (x)\,dx, \end{equation*}$$

where $w^{\varepsilon }_\ell$ is defined for $W\in X_H$ in the same way that $v^{\varepsilon }_\ell$ in 1.6 was defined for $V$.

In order to reduce the effect of the imposed boundary condition on $\partial I_\delta (x_\ell )$, we may replace 1.7 by

$$\begin{equation} \bigl (\nabla V\cdot \mathcal{A}_H\nabla V\bigr )(x_\ell )\simeq \frac{1}{{(\delta ^{\,'})}^d}\int \limits _{I_{\delta ^{\,'}}(x_\ell )} \nabla v^{\varepsilon }_\ell (x)\cdot a^{\,\varepsilon }(x)\nabla v^{\varepsilon }_\ell (x)\,dx, {\tag{1.8a}} \end{equation}$$

where $\delta '<\delta$. For example, we may choose $\delta ^{\,'}=\delta /2$. 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 $a^{\,\varepsilon }(x)=a(x,x/\varepsilon )$ and $a(x,y)$ is periodic in $y$, one can take $I_\delta (x_\ell )$ to be $x_\ell +\varepsilon I$, i.e., $\delta =\varepsilon$ and use the boundary condition that $v^{\varepsilon }_\ell (x)-V_\ell (x)$ is periodic on $I_\delta$.

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 $I_\delta (x_\ell )$ to be smaller than $K$. The size of $I_\delta (x_\ell )$ is determined by many factors, including the accuracy and cost requirement, the degree of scale separation, and the microstructure in $a^{\,\varepsilon }(x)$. One purpose for the error estimates that we present below is to give guidelines on how to select $I_\delta (x_\ell )$. As mentioned already, if $a^{\,\varepsilon }(x)=a(x,x/\varepsilon )$ and $a(x,y)$ is periodic in $y$, we can simply choose $I_\delta (x_\ell )$ to be $x_\ell +\varepsilon I$, i.e., $\delta =\varepsilon$. If $a(x,y)$ is random, then $\delta$ should be a few times larger than the local correlation length of $a^\varepsilon$. In the former case, the total cost is independent of $\varepsilon$. In the latter case, the total cost depends only weakly on $\varepsilon$ (see 31).

The final problem is to solve

$$\begin{equation} \min _{V\in X_H}\frac{1}{2}A_H(V,V)-(f,V). {\tag{1.8}} \end{equation}$$

Figure 1.

Illustration of HMM for solving 1.1. The dots are the quadrature points. The little squares are the microcell $I_\delta (x_\ell )$.

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 372935. However HMM is also applicable for random problems and for problems whose coefficient $a^{\,\varepsilon }(x)$ does not has the structure of $a(x,x/\varepsilon )$. 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 $\mathcal{A}_H(x)=a^{\,\varepsilon }(x)$.

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 334041. Oden and Vemaganti 35 proposed a method that aims at recovering the oscillations in $\nabla u^{\varepsilon }$ 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 2627.

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 $u^{\varepsilon }$ from the HMM solutions. Our basic strategy is as follows. First we will prove a general statement that the error between the HMM solution and the solution of 1.4 is controlled by the standard error in the macroscale solver plus a new term, called $e(\mathrm{HMM})$, due to the error in estimating the stiffness matrix. We then estimate $e(\mathrm{HMM})$. This second part is only done for either periodic or random homogenization problems, since concrete results are only possible if the behavior of $u^{\varepsilon }$ is well understood. We believe that this overall strategy will be useful for analyzing other multiscale methods.

We will always assume that $a^{\,\varepsilon }(x)$ is smooth, symmetric and uniformly elliptic:

$$\begin{equation} \lambda I\le a^{\,\varepsilon }\le \varLambda I {\tag{1.9}} \end{equation}$$

for some $\lambda ,\varLambda >0$. We will use the summation convention and standard notation for Sobolev spaces (see 1). We will use $\lvert \cdot \rvert$ 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 $k$th-order numerical quadrature scheme 11:

$$\begin{equation} \intbar _K p(x)\,dx{:}=\dfrac{1}{\lvert K\rvert }\int \limits _K p(x)\,dx=\sum _{\ell =1}^L\omega _\ell p(x_\ell )\quad \text{for all } p(x)\in \mathcal{P}_{2k-2}. {\tag{1.10}} \end{equation}$$

Here $\omega _\ell >0$, for $\ell =1$, …, $L$. For $k=1$, we assume the above formula to be exact for $p\in \mathcal{P}_1$.

1.3. Main results

Our main results for the linear problem are as follows.

Theorem 1.1.

Denote by $U_0$ and $U_\mathrm{HMM}$ the solution of 1.4 and the $\mathrm{HMM}$ solution, respectively. Let

$$\begin{equation*} e(\mathrm{HMM})=\max _{\substack{x_\ell \in K\\K\in \mathcal{T}_H}}\|\mathcal{A}(x_\ell )-\mathcal{A}_H(x_\ell )\|, \end{equation*}$$

where $\|\cdot \|$ is the Euclidean norm. If $U_0$ is sufficiently smooth and 1.10 holds, then there exists a constant $C$ independent of $\varepsilon ,\delta$ and $H$, such that

$$\begin{align} \|U_0-U_\mathrm{HMM}\|_1&\le C\bigl (H^k+ e(\mathrm{HMM})\bigr ),{\tag{1.11}}\\ \|U_0-U_\mathrm{HMM}\|_0&\le C\bigl (H^{k+1}+ e(\mathrm{HMM})\bigr ). {\tag{1.12}} \end{align}$$

If there exits a constant $C_0$ such that $e(\mathrm{HMM})\lvert \ln H\rvert < C_0$, then there exists a constant $H_0$ such that for all $H\le H_0$,

$$\begin{equation} \|U_0-U_\mathrm{HMM}\|_{1,\infty }\le C\bigl (H^k+ e(\mathrm{HMM})\bigr )\lvert \ln H\rvert . {\tag{1.13}} \end{equation}$$

At this stage, no assumption on the form of $a^{\,\varepsilon }(x)$ 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 $U_\mathrm{HMM}$ to converge to $U_0$, i.e., $e(\mathrm{HMM})\rightarrow 0$, $U_0$ must be chosen as the solution of the homogenized equation, which we now assume exists. To obtain quantitative estimates on $e(\mathrm{HMM})$, we must restrict ourselves to more specific cases.

Theorem 1.2.

For the periodic homogenization problem, we have

$$\begin{equation*} e(\mathrm{HMM})\le \begin{cases} C\varepsilon &\quad \text{if  } I_\delta (x_{\ell })=x_\ell +\varepsilon I, \\ C\Bigl (\dfrac{\varepsilon }{\delta }+\delta \Bigr )&\quad \text{otherwise.  } \end{cases} \end{equation*}$$

In the first case, we replace the boundary condition in 1.6 by a periodic boundary condition: $v^{\varepsilon }_\ell -V_\ell$ is periodic with period $\varepsilon I$. For the second result we do not need to assume that the period of $a(x,\cdot )$ 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 $e(\mathrm{HMM})$ 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 $\mathbf{(A)}$ in the Appendix holds (see 43), we have

$$\begin{equation*} \mathbb{E}\, e(\mathrm{HMM})\le \begin{cases} C(\kappa )\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{\kappa }\qquad &d= 3,\\ \text{remains open}\qquad &d=2,\\ C(\kappa )\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{1/2}\qquad &d=1, \end{cases} \end{equation*}$$

where

$$\begin{equation*} \kappa =\dfrac{6-12\gamma }{25-8\gamma } \end{equation*}$$

for any $0<\gamma <1/2$. By choosing $\gamma$ small, $\kappa$ 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 $\delta {\,'}=\delta /2$.

1.4. Recovering the microstructural information

In many applications, the microstructure information in $u^{\varepsilon }(x)$ is very important. $U_\mathrm{HMM}$ by itself does not give this information. However, this information can be recovered using a simple post-processing technique. For the general case, having $U_\mathrm{HMM}$, one can obtain locally the microstructural information using an idea in 35. Assume that we are interested in recovering $u^{\varepsilon }$ and $\nabla u^{\varepsilon }$ only in the subdomain $\varOmega \subset D$. Consider the following auxiliary problem:

$$\begin{equation} \left\{\begin{aligned} -\operatorname {div}\bigl (a^{\,\varepsilon }(x)\nabla \tilde{u}^{\,\varepsilon }(x)\bigr )&=f(x)\qquad && x\in \varOmega _\eta ,\\ \tilde{u}^{\,\varepsilon }(x)&=U_\mathrm{HMM}(x)\qquad &&x\in \partial \varOmega _\eta , \end{aligned}\right. {\tag{1.14}} \end{equation}$$

where $\varOmega _\eta$ satisfies $\varOmega \subset \varOmega _\eta \subset D$ and $\text{dist}(\partial \varOmega ,\partial \varOmega _\eta )=\eta$. We then have

Theorem 1.4.

There exists a constant $C$ such that

$$\begin{equation} \Bigl (\intbar _{\varOmega }\lvert \nabla (u^{\varepsilon }-\tilde{u}^{\,\varepsilon })\rvert ^2\,dx\Bigr )^{1/2}\le \frac{C}{\eta }\Bigl (\|U_0-U_\mathrm{HMM}\|_{L^{\infty }(\varOmega _\eta )} +\|u^{\varepsilon }-U_0\|_{L^\infty (\varOmega _\eta )}\Bigr ). {\tag{1.15}} \end{equation}$$

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 $I_\delta = x_K + \varepsilon I$, where $x_K$ is the barycenter of $K$. Here we have assumed that the quadrature point is at $x_K$.

Let $\tilde{u}^{\,\varepsilon }$ be defined piecewise as follows:

(1)

${\tilde{u}^{\,\varepsilon }}\lvert _{I_\delta }=v^{\varepsilon }_K$, where $v^{\varepsilon }_K$ is the solution of 1.6 with the boundary condition that $v^{\varepsilon }_K-U_\mathrm{HMM}$ is periodic with period $\varepsilon I$ and $\int _{I_\delta }(\tilde{u}^{\,\varepsilon }-U_\mathrm{HMM})(x)\,dx=0$.

(2)

$\bigl (\tilde{u}^{\,\varepsilon }-U_\mathrm{HMM}\bigr )\lvert _K$ is periodic with period $\varepsilon I$.

For this case, we can prove

Theorem 1.5.

Let $\tilde{u}^{\,\varepsilon }$ be defined as above. Then

$$\begin{equation} \Bigl (\sum _{K\in \mathcal{T}_H}\|\nabla (u^{\varepsilon }-\tilde{u}^{\,\varepsilon })\|_{0,K}^2\Bigr )^{1/2}\le C\bigl (\sqrt \varepsilon +H\bigr ). {\tag{1.16}} \end{equation}$$

Similar results with some modification hold for nonlinear problems. The details are given in §5.

1.5. Some technical background

In this subsection, we will list some general results that will be frequently referred to later on.

Given a triangulation $\mathcal{T}_H$, it is called regular if there is a constant $\sigma$ such that

$$\begin{equation*} \dfrac{H_K}{\rho _K}\le \sigma \qquad \text{for all } K\in \mathcal{T}_H \end{equation*}$$

and if the quantity

$$\begin{equation*} H=\max _{K\in \mathcal{T}_H} H_K \end{equation*}$$

approaches zero, where $H_K$ is the diameter of $K$ and $\rho _K$ is the diameter of the largest ball inscribed in $K$. $\mathcal{T}_H$ satisfies an inverse assumption if there exists a constant $\nu$ such that

$$\begin{equation*} \dfrac{H}{H_K}\le \nu \qquad \text{for all } K\in \mathcal{T}_H. \end{equation*}$$

A regular family of triangulation of $\mathcal{T}_H$ satisfying the inverse assumption is called quasi-uniform.

The following interpolation result for the Lagrange finite element is adapted from 10. Here and in what follows, for any $k\ge 2,\nabla ^k v$ is understood in a piecewise manner.

Theorem 1.6 (10).

Let $\varPi$ be $k$th-order Lagrange interpolate operator, and assume that the following inclusions hold:

$$\begin{equation} W^{k+1,p}(\hat{K})\hookrightarrow \mathcal{C}^0(\hat{K})\quad \text{and}\quad W^{k+1,p}(\hat{K})\hookrightarrow W^{m,q}(\hat{K}). {\tag{1.17}} \end{equation}$$

Then

$$\begin{equation} \lvert v-\varPi v\rvert _{m,q,K}\le C\lvert K\rvert ^{1/q-1/p}\dfrac{H_K^{k+1}}{\rho _K^m}\lvert v\rvert _{k+1,p,K}. {\tag{1.18}} \end{equation}$$

If $\mathcal{T}_H$ is regular, we have the global estimate

$$\begin{equation} \lvert v-\varPi v\rvert _{m,q,D}\le CH^{k+1-m+\min \{0,d(1/q-1/p)\}}\lvert v\rvert _{k+1,p,D}. {\tag{1.19}} \end{equation}$$

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 $\|v-\varPi v\|_{2,D}\le C\|v\|_{2,D}$. Hence

$$\begin{equation} \|\varPi v\|_{2,D}\le \|v-\varPi v\|_{2,D}+\|v\|_{2,D}\le C\|v\|_{2,D}. {\tag{1.20}} \end{equation}$$

We will also need the following form of the inverse inequality.

Theorem 1.7 (10, Theorem 3.2.6).

Assume that $\mathcal{T}_H$ is regular, and assume also that the two pairs $(l,r)$ and $(m,q)$ with $l,m\ge 0$ and $r,q\in [0,\infty ]$ satisfy

$$\begin{equation*} l\le m\quad \text{and}\qquad \mathcal{P}_k(\hat{K})\subset W^{l,r}(\hat{K})\cap W^{m,q}(\hat{K}). \end{equation*}$$

Then there exists a constant $C=C(\sigma ,\nu ,l,r,m,q)$ such that

$$\begin{equation} \lvert v\rvert _{m,q,K}\le CH_K^{l-m+d(1/q-1/r)}\lvert v\rvert _{l,r,K} {\tag{1.21}} \end{equation}$$

for any $v\in \mathcal{P}_k(\hat{K})$.

If in addition $\mathcal{T}_H$ satisfies the inverse assumption, then there exists a constant $C=C(\sigma ,\nu ,l,r,m,q)$ such that

$$\begin{equation} \Bigl (\sum _{K\in \mathcal{T}_H}\lvert v\rvert _{m,q,K}^q\Bigr )^{1/q}\le CH^{l-m+\min \{0,d(1/q-1/r)\}}\Bigl (\sum _{K\in \mathcal{T}_H}\lvert v\rvert _{l,r,K}^r\Bigr )^{1/r} {\tag{1.22}} \end{equation}$$

for any $v\in X_H$ and $r,q<\infty$, with

$$\begin{align*} \max _{K\in \mathcal{T}_H}\lvert v\rvert _{m,\infty ,K}\quad &\text{replacing }\quad \Bigl (\sum _{K\in \mathcal{T}_H}\lvert v\rvert _{m,q,K}^q\Bigr )^{1/q},\quad \text{if  }q=\infty ,\\ \max _{K\in \mathcal{T}_H}\lvert v\rvert _{l,\infty ,K}\quad &\text{replacing }\quad \Bigl (\sum _{K\in \mathcal{T}_H}\lvert v\rvert _{l,r,K}^r\Bigr )^{1/r},\quad \text{if  }r=\infty . \end{align*}$$

The following simple result will be used repeatedly.

Lemma 1.8.

Let $A_1(x)$ and $A_2(x)$ be symmetric matrices satisfying 1.9. Let $\varphi _1$ be the solution of

$$\begin{equation} -\operatorname {div}\bigl (A_1(x)\nabla \varphi _1(x)\bigr )=\operatorname {div}\bigl (\tilde{A}_1(x)\nabla F_1(x)\bigr )\qquad x\in \varOmega , {\tag{1.23}} \end{equation}$$

with either the Dirichlet or periodic boundary condition on $\partial \varOmega$. Let $\varphi _2$ be a solution of 1.23 with $A_1,\tilde{A}_1$ and $F_1$ replaced by $A_2,\tilde{A}_2$ and $F_2$, respectively, and let $\varphi _2$ satisfy the same boundary condition as $\varphi _1$. Then

$$\begin{align} \lambda \|\nabla (\varphi _1-\varphi _2)\|_{0,\varOmega }&\le \max _{x\in \varOmega }\lvert (\tilde{A}_1-\tilde{A}_2)(x)\rvert \,\|\nabla F_1\|_{0,\varOmega }+\max _{x\in \varOmega }\lvert (A_1-A_2)(x)\rvert \,\|\nabla \varphi _2\|_{0,\varOmega }\\ &\quad +\max _{x\in \varOmega }\lvert \tilde{A}_2(x)\rvert \,\|\nabla (F_1-F_2)\|_{0,\varOmega }. {\tag{1.24}} \end{align}$$

Proof.

Inequality 1.24 is a direct consequence of

$$\begin{equation*} \lambda \|\nabla (\varphi _1-\varphi _2)\|_{0,\varOmega }^2\le \int \limits _\varOmega \nabla (\varphi _1-\varphi _2)\cdot \bigl ((\tilde{A}_2-\tilde{A}_1)\nabla F_1+(A_2-A_1)\nabla \varphi _2+\tilde{A}_2\nabla (F_2-F_1)\bigr ). \end{equation*}$$ ■

The following simple result underlies the stability of HMM for problem 1.1.

Lemma 1.9.

Let $\varphi$ be the solution of

$$\begin{equation} \left\{\begin{aligned} -\operatorname {div}\bigl (a\nabla \varphi \bigr )&=0\qquad &&\text{in  }\varOmega \subset \mathbb{R}^d,\\ \varphi &=V_\ell \qquad &&\text{on  }\in \partial \varOmega , \end{aligned}\right. {\tag{1.25}} \end{equation}$$

where $V_\ell$ is a linear function and $a=\bigl (a_{ij}\bigr )$ satisfies

$$\begin{equation*} \lambda I\le a\le \varLambda I. \end{equation*}$$

Then we have

$$\begin{equation} \|\nabla V_\ell \|_{0,\varOmega }\le \|\nabla \varphi \|_{0,\varOmega }\quad \text{and}\quad \Bigl (\int \limits _\varOmega \nabla \varphi \cdot \,a\nabla \varphi \Bigr )^{1/2}\le \Bigl (\int \limits _\varOmega \nabla V_\ell \cdot \,a\nabla V_\ell \Bigr )^{1/2}. {\tag{1.26}} \end{equation}$$

Proof.

Notice that $\varphi =V_\ell$ on the edges of $\varOmega$, using the fact that $\nabla V_\ell$ is a constant in $\varOmega$, and integration by parts leads to

$$\begin{equation*} \int \limits _\varOmega \nabla (\varphi -V_\ell )(x)\nabla V_\ell (x)\,dx=0, \end{equation*}$$

which implies

$$\begin{equation*} \int \limits _\varOmega \lvert \nabla \varphi (x)\rvert ^2\,dx=\int \limits _\varOmega \lvert \nabla V_\ell (x)\rvert ^2\,dx+\int \limits _\varOmega \lvert \nabla (\varphi -V_\ell )(x)\rvert ^2\,dx. \end{equation*}$$

This gives the first result in 1.26. Multiplying 1.25 by $\varphi (x)-V_\ell (x)$ and integrating by parts, we obtain

$$\begin{align*} \int \limits _\varOmega \nabla \varphi (x)\cdot a\,\nabla \varphi (x)\,dx&\;+\int \limits _\varOmega \nabla (\varphi -V_\ell )(x)\cdot a\,\nabla (\varphi -V_\ell )(x)\,dx\\ &=\int \limits _\varOmega \nabla V_\ell (x)\cdot a\,\nabla V_\ell (x)\,dx. \end{align*}$$

This gives the second part of 1.26.

■

Remark 1.10.

For this result, the coefficient $a=\bigl (a_{ij}\bigr )$ may depend on the solution, i.e., 1.25 may be nonlinear.

Remark 1.11.

The same result holds if we use instead a periodic boundary condition: $\varphi -V_\ell$ is periodic with period $\varOmega$.

2. Generalities

Here we prove Theorem 1.1. We will let $U_H=U_\mathrm{HMM}$ 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 $V\in H_0^1(D)$ as

$$\begin{equation*} A(V,V)=\int \limits _D\nabla V(x)\cdot \mathcal{A}(x)\nabla V(x)\,dx. \end{equation*}$$

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 $\varOmega =I_\delta (x_\ell )$ and 1.9, for any $V\in X_H$, we have

$$\begin{align} A_H(V,V)&\ge \lambda \sum _{K\in \mathcal{T}_H}\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell \intbar _{I_\delta (x_\ell )}\lvert \nabla V_\ell (x)\rvert ^2\,dx\\ &=\lambda \sum _{K\in \mathcal{T}_H}\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell \lvert \nabla V(x_\ell )\rvert ^2\\ &=\lambda \|\nabla V\|_0^2. {\tag{2.1}} \end{align}$$

Similarly, for any $V,W\in X_H$, we obtain

$$\begin{align} \lvert A_H(V,W)\rvert &\le \sum _{K\in \mathcal{T}_H}\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell \Bigl (\intbar _{I_\delta (x_\ell )}\nabla V_\ell \cdot a^{\,\varepsilon }\nabla V_\ell \Bigr )^{\frac{1}{2}} \Bigl (\intbar _{I_\delta (x_\ell )}\nabla W_\ell \cdot a^{\,\varepsilon }\nabla W_\ell \Bigr )^{\frac{1}{2}}\\ &\le \varLambda \sum _{K\in \mathcal{T}_H}\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell \lvert \nabla V(x_\ell )\rvert \,\lvert \nabla W(x_\ell )\rvert \\ &=\varLambda \sum _{K\in \mathcal{T}_H}\int _K\lvert \nabla V(x)\rvert \,\lvert \nabla W(x)\rvert \,dx\\ &\le \varLambda \|\nabla V\|_0\|\nabla W\|_0. {\tag{2.2}} \end{align}$$

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 Poincaré inequality.

To streamline the proof of Theorem 1.1, we introduce the following auxiliary bilinear form $\hat{A}_H$.

$$\begin{equation*} \hat{A}_H(V,W)=\sum _{K\in \mathcal{T}_H}\hat{A}_K(V,W)\quad \text{with}\quad \hat{A}_K(V,W)=\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell (\nabla W\cdot \mathcal{A}\nabla V)(x_\ell ). \end{equation*}$$

Classical results on numerical integration 11, Theorem 6 give for any $V,W\in X_H$,

$$\begin{equation} \left|\,\hat{A}_K(V,W)-\int \limits _K\nabla W\cdot \mathcal{A}\nabla V\,dx\,\right|\le CH^m\|V\|_{m,K}\|\nabla W\|_{0,K}\quad 1\le m\le k. {\tag{2.3}} \end{equation}$$

Moreover, for any $V,W\in X_H$, if $\|V\|_{k+1}$ and $\|W\|_2$ are bounded, we have 11, Theorem 8,

$$\begin{equation} \lvert \hat{A}_H(V,W)-A(V,W)\rvert \le CH^{k+1}\|V\|_{k+1}\|W\|_2. {\tag{2.4}} \end{equation}$$

Proof of Theorem 1.1.

Using the first Strang lemma 10, Theorem 4.1.1, we have

$$\begin{equation*} \|U_0-U_H\|_1\le C\inf _{V\in X_H}\Bigl (\|U_0-V\|_1+\sup _{W\in X_H}\frac{\lvert A_H(V,W)-A(V,W)\rvert }{\|W\|_1}\Bigr ). \end{equation*}$$

Let $V=\varPi U_0$ and using 1.19 with $m=1,p=q=2$, we have

$$\begin{equation} \inf _{V\in X_H}\|U_0-V\|_1\le \|U_0-\varPi U_0\|_1\le CH^k. {\tag{2.5}} \end{equation}$$

It remains to estimate $\lvert A_H(V,W)-A(V,W)\rvert$ for $V=\varPi U_0$ and $W\in X_H$. Using 2.3, we get

$$\begin{align} \lvert A_H(V,W)-A(V,W)\rvert &\le \lvert A_H(V,W)-\hat{A}_H(V,W)\rvert +\lvert \hat{A}_H(V,W)-A(V,W)\rvert \\ &\le \bigl (e(\mathrm{HMM})\|\nabla V\|_0+CH^k\|V\|_k\bigr )\|\nabla W\|_0. {\tag{2.6}} \end{align}$$

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 $w\in H_0^1(D)$ such that

$$\begin{equation} A(v,w)=(U_0-U_H,v)\quad \text{for all } v\in H_0^1(D). {\tag{2.7}} \end{equation}$$

The standard regularity result reads 24

$$\begin{equation} \|w\|_2\le C\|U_0-U_H\|_0. {\tag{2.8}} \end{equation}$$

Putting $v=U_0-U_H$ into the right-hand side of 2.7, we obtain

$$\begin{align} \|U_0-U_H\|_0^2&=A(U_0-U_H,w-\varPi w)+\bigl (A_H(U_H,\varPi w)-A(U_H,\varPi w)\bigr )\\ &=A(U_0-U_H,w-\varPi w)\\ &\;+\bigl (A_H(U_H-\varPi U_0,\varPi w)-A(U_H-\varPi U_0,\varPi w)\bigr )\\ &\;+\bigl (A_H(\varPi U_0,\varPi w)-A(\varPi U_0,\varPi w)\bigr ). {\tag{2.9}} \end{align}$$

Using 2.6 with $k=1$, we bound the first two terms in the right-hand side of the above identity as

$$\begin{equation*} \lvert A(U_0-U_H,w-\varPi w)\rvert \le C\|U_0-U_H\|_1\|w-\varPi w\|_1\le CH\|U_0-U_H\|_1\|w\|_2 \end{equation*}$$

and

$$\begin{equation*} \lvert A_H(U_H-\varPi U_0,\varPi w)-A(U_H-\varPi U_0,\varPi w)\rvert \le \bigl (e(\mathrm{HMM})+CH\bigr )\|U_0-U_H\|_1\|\varPi w\|_1. \end{equation*}$$

The last term in the right-hand side of 2.9 may be decomposed into

$$\begin{align*} A_H(\varPi U_0,\varPi w)-A(\varPi U_0,\varPi w)&=\bigl (A_H(\varPi U_0,\varPi w)-\hat{A}_H(\varPi U_0,\varPi w)\bigr )\\ &\;+\bigl (\hat{A}_H(\varPi U_0,\varPi w)-A(\varPi U_0,\varPi w)\bigr ). \end{align*}$$

It follows from 2.4 that

$$\begin{equation*} \lvert \hat{A}_H(\varPi U_0,\varPi w)-A(\varPi U_0,\varPi w)\rvert \le CH^{k+1}\|U_0\|_{k+1}\|w\|_2. \end{equation*}$$

By definition of $e(\mathrm{HMM})$ and using 1.20, we get

$$\begin{equation*} \lvert A_H(\varPi U_0,\varPi w)-\hat{A}_H(\varPi U_0,\varPi w)\rvert \le C e(\mathrm{HMM})\|\nabla \varPi U_0\|_0\|w\|_2. \end{equation*}$$

Combining the above estimates and using 2.8 lead to 1.12.

It remains to prove 1.13. As in 37, for any point $z\in D$, we define the regularized Green’s function $G^z\in H_0^1(D)$ and the discrete Green’s function $G_H^z\in X_H$ as

$$\begin{equation} \begin{alignedat} {2} A(G^z,V)&=(\delta _z,\partial V)\quad \text{for all } V\in H_0^1(D),\\ A(G_H^z,V)&=(\delta _z,\partial V)\quad \text{for all } V\in X_H, \end{alignedat} {\tag{2.10}} \end{equation}$$

where $\delta _z$ is the regularized Dirac-$\delta$ function defined in 37. It is well known that

$$\begin{equation} \|G^z-G_H^z\|_{1,1}\le C\quad \text{and}\quad \|G_H^z\|_{1,1}\le C\lvert \ln H\rvert . {\tag{2.11}} \end{equation}$$

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 $G_H^z$, a simple manipulation gives

$$\begin{align*} \partial (U_0-U_H)(z)&=A(G^z,U_0-\varPi U_0)+A(G^z,\varPi U_0-U_H)\\ &=A(G^z-G_H^z,U_0-\varPi U_0)+A(G_H^z,U_0-U_H)\\ &=A(G^z-G_H^z,U_0-\varPi U_0)+A_H(U_H,G_H^z)-A(U_H,G_H^z)\\ &=A(G^z-G_H^z,U_0-\varPi U_0)+\bigl (A_H(\varPi U_0,G_H^z)-A(\varPi U_0,G_H^z)\bigr )\\ &\quad +\bigl (A_H(U_H-\varPi U_0,G_H^z)-A(U_H-\varPi U_0,G_H^z)\bigr ). \end{align*}$$

Using 2.11, we obtain

$$\begin{align*} \|U_0-U_H\|_{1,\infty }&\le C\|U_0-\varPi U_0\|_{1,\infty }+\lvert A(\varPi U_0,G_H^z)-A_H(\varPi U_0,G_H^z)\rvert \\ &\quad +\lvert A(U_H-\varPi U_0,G_H^z)-A_H(U_H-\varPi U_0,G_H^z)\rvert . \end{align*}$$

Using 2.6, we get

$$\begin{align*} \lvert A(\varPi U_0,G_H^z)-A_H(\varPi U_0,&G_H^z)\rvert \le \bigl (e(\mathrm{HMM})+CH^k\bigr )\sum _{K\in \mathcal{T}_H}\|\varPi U_0\|_{k,K}\|\nabla G_H^z\|_{0,K}\\ &\le C\bigl (e(\mathrm{HMM})+H^k\bigr )\sum _{K\in \mathcal{T}_H}\|\varPi U_0\|_{k,\infty ,K}\|\nabla G_H^z\|_{L^1(K)}\\ &\le C\bigl (e(\mathrm{HMM})+H^k\bigr )\lvert \ln H\rvert \,\|U_0\|_{k+1,\infty }, \end{align*}$$

where we have used the inverse inequality 1.21.

Similarly, we have

$$\begin{align*} &\quad \lvert A(U_H-\varPi U_0,G_H^z)-A_H(U_H-\varPi U_0,G_H^z)\rvert \\ &\le \bigl (e(\mathrm{HMM})+CH\bigr )\sum _{K\in \mathcal{T}_H}\|U_H-\varPi U_0\|_{1,K}\|\nabla G_H^z\|_{0,K}\\ &\le C\bigl (e(\mathrm{HMM})+H\bigr )\sum _{K\in \mathcal{T}_H}\|U_H-\varPi U_0\|_{1,\infty ,K}\|\nabla G_H^z\|_{0,1,K}\\ &\le C\bigl (e(\mathrm{HMM})+H\bigr )\lvert \ln H\rvert \|U_0-U_H\|_{1,\infty }\\ &\quad +C\bigl (e(\mathrm{HMM})+H\bigr )\lvert \ln H\rvert \,H^k\|U_0\|_{k+1,\infty }. \end{align*}$$

A combination of the above three estimates yields

$$\begin{align*} \|U_0-U_H\|_{1,\infty }&\le CH^k+C\Bigl (e(\mathrm{HMM})+H\Bigr )\lvert \ln H\rvert \,\|U_0-U_H\|_{1,\infty }\\ &\quad +C\Bigl (e(\mathrm{HMM})+H^k\Bigr )\lvert \ln H\rvert \,\|U_0\|_{k+1,\infty }. \end{align*}$$

If $e(\mathrm{HMM})\lvert \ln H\rvert <C_0{:}=1/(2C)$, then there exits a constant $H_0$ such that for all $H\le H_0$,

$$\begin{equation*} C\Bigl (e(\mathrm{HMM})+H\Bigr )\lvert \ln H\rvert \le 1/2+CH\lvert \ln H\rvert <1. \end{equation*}$$

We thus obtain 1.13 and this completes the proof.

■

Combining the foregoing proof for the L$^2$ and W$^{1,\infty }$ estimates, using the Green’s function defined in 39, we obtain

Remark 2.1.

Under the same condition for the W$^{\,1,\infty }$ estimate in Theorem 1.1, we have

$$\begin{equation*} \|U_0-U_H\|_{L^\infty }\le C\bigl (e(\mathrm{HMM})+H^{k+1}\bigr )\lvert \ln H\rvert ^2. \end{equation*}$$

3. Estimating $e(\mathrm{HMM})$

In this section, we estimate $e(\mathrm{HMM})$ for problems with locally periodic coefficients. The estimate of $e(\mathrm{HMM})$ for problems with random coefficients can be found in the Appendix.

We assume that $a^{\,\varepsilon }(x)=a(x,x/\varepsilon )$, where $a^{\,\varepsilon }$ is smooth in $x$ and periodic in $y$ with period $I$. Define $\kappa =\lfloor \delta /\varepsilon \rfloor$, and we introduce $\hat{V}_\ell$ as

$$\begin{equation} \hat{V}_\ell (x)=V_\ell (x)+\varepsilon \chi ^k\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\frac{\partial V_\ell }{\partial x_k}(x), {\tag{3.1}} \end{equation}$$

where $\{\chi ^j\}_{j=1}^d$ is defined as: For $j=1$, …, $d$, $\chi ^j(x,y)$ is periodic in $y$ with period $I$ and satisfies

$$\begin{equation} -\frac{\partial }{\partial y_i}\Bigl (a_{ik}\frac{\partial \chi ^j}{\partial y_k}\Bigr )(x,y)=\frac{\partial }{\partial y_i}a_{ij}(x,y)\quad \text{in  } I,\qquad \int \limits _I\chi ^j(x,y)\,dy=0. {\tag{3.2}} \end{equation}$$

Given $\{\chi ^j\}_{j=1}^d$, the homogenized coefficient $\mathcal{A}=\bigl (\mathcal{A}_{ij}(x)\bigr )$ is given by

$$\begin{equation*} \mathcal{A}_{ij}(x)=\intbar _I\Bigl (a_{ij}+a_{ik}\frac{\partial \chi ^j}{\partial y_k}\Bigr )(x,y)\,dy. \end{equation*}$$

Note that $\{\chi ^j\}_{j=1}^d$ is smooth and bounded in all norms.

First let us consider the case when $I_\delta (x_\ell ) =x_\ell +\varepsilon I$, and 1.6 is solved with the periodic boundary condition. Denote by $\hat{v}^{\,\varepsilon }_\ell$ the solution of 1.6 with the coefficients $a^{\,\varepsilon }(x)$ replaced by $a(x_\ell ,x/\varepsilon )$. $\hat{v}^{\,\varepsilon }_\ell$ may be viewed as a perturbation of $v^{\varepsilon }_\ell$. Using Lemma 1.8, we get

$$\begin{equation} \|\nabla (v^{\varepsilon }_\ell -\hat{v}^{\,\varepsilon }_\ell )\|_{0,I_\varepsilon }\le C\varepsilon \|\nabla V_\ell \|_{0,I_\varepsilon }. {\tag{3.3}} \end{equation}$$

Observe that $\hat{v}^{\,\varepsilon }_\ell =\hat{V}_\ell$. A direct calculation yields

$$\begin{align*} \bigl (\nabla W\cdot (\mathcal{A}_H-\mathcal{A})\nabla V\bigr )(x_\ell )&= \intbar _{I_\varepsilon }\nabla w^{\varepsilon }_\ell \cdot \Bigl [a\Bigl (x,\frac{x}{\varepsilon }\Bigr )-a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\Bigr ]\nabla v^{\varepsilon }_\ell \,dx\\ &\quad +\intbar _{I_\varepsilon }\nabla w^{\varepsilon }_\ell \cdot a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\nabla (v^{\varepsilon }_\ell -\hat{v}^{\,\varepsilon }_\ell )\,dx. \end{align*}$$

Using 3.3, we get

$$\begin{equation} e(\mathrm{HMM})\le C\varepsilon . {\tag{3.4}} \end{equation}$$

Next we consider the more general case when $I_\delta$ is a cube of size $\delta$ not necessarily equal to $\varepsilon$. The following analysis applies equally well to the case when the period of $a(x,\cdot )$ is of general and even nonpolygonal shape. This situation arises in some examples of composite materials 30. We will show that if $\delta$ is much larger than $\varepsilon$, 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:

$$\begin{align} \bigl (\nabla W\cdot \mathcal{A}\nabla V\bigr )(x_\ell )&=\nabla W_\ell (x)\cdot \mathcal{A}(x_\ell )\nabla V_\ell (x)\\ &=\intbar _{I_{\kappa \varepsilon }(x_\ell )}\nabla W_\ell \cdot a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\nabla \hat{V}_\ell \,dx. {\tag{3.5}} \end{align}$$

We first establish some estimates on the solution of the cell problem 1.6. We will write $I_\delta$ instead of $I_\delta (x_\ell )$ if there is no risk of confusion.

Lemma 3.1.

There exists a constant $C$ independent of $\varepsilon$ and $\delta$ such that for each $\ell$,

$$\begin{equation} \|\nabla v^{\varepsilon }_\ell \|_{0,I_\delta \backslash I_{\kappa \varepsilon }}\le C\Bigl (\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{1/2}+\delta \Bigr )\|\nabla V_\ell \|_{0,I_\delta }. {\tag{3.6}} \end{equation}$$

Proof.

We still denote by $\hat{v}^{\,\varepsilon }_\ell$ the solution of 1.6 with the coefficient $a^{\,\varepsilon }(x)$ replaced by $a(x_\ell ,x/\varepsilon )$. Using Lemma 1.8, we get

$$\begin{equation} \|\nabla (v^{\varepsilon }_\ell -\hat{v}^{\,\varepsilon }_\ell )\|_{0,I_\delta }\le C\delta \|\nabla V_\ell \|_{0,I_\delta }. {\tag{3.7}} \end{equation}$$

Define $\theta ^{\,\varepsilon }_\ell =\hat{v}^{\,\varepsilon }_\ell -\hat{V}_\ell$, which obviously satisfies

$$\begin{equation} \left\{ \begin{aligned} -\operatorname {div}\Bigl (a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\nabla \theta ^{\,\varepsilon }_\ell (x)\Bigr )&=0\qquad &&x\in I_\delta (x_\ell ),\\ \theta ^{\,\varepsilon }_\ell (x)&=-\varepsilon \chi ^k\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\frac{\partial V_\ell }{\partial x_k}(x)\qquad &&x\in \partial I_\delta (x_\ell ). \end{aligned}\right. {\tag{3.8}} \end{equation}$$

Note that $\theta ^{\,\varepsilon }_\ell$ 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 $x'=x/\delta$ over $I_\delta$ and $\varepsilon '=\varepsilon /\delta$.

$$\begin{equation} \|\nabla \theta ^{\,\varepsilon }_\ell \|_{0,I_\delta }\le C\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{1/2}\|\nabla V_\ell \|_{0,I_\delta }. {\tag{3.9}} \end{equation}$$

This together with 3.7 gives

$$\begin{equation} \|\nabla (v^{\varepsilon }_\ell -\hat{V}_\ell )\|_{0,I_\delta }\le C\Bigl (\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{1/2}+\delta \Bigr )\|\nabla V_\ell \|_{0,I_\delta }. {\tag{3.10}} \end{equation}$$

A straightforward calculation gives

$$\begin{equation} \|\nabla \hat{V}_\ell \|_{0,I_\delta \backslash I_{\kappa \varepsilon }}\le C\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{1/2}\|\nabla V_\ell \|_{0,I_\delta }, {\tag{3.11}} \end{equation}$$

which together with 3.10 leads to

$$\begin{align*} \|\nabla v^{\varepsilon }_\ell \|_{0,I_\delta \backslash I_{\kappa \varepsilon }}&\le \|\nabla \hat{V}_\ell \|_{0,I_\delta \backslash I_{\kappa \varepsilon }}+ \|\nabla (v^{\varepsilon }_\ell -\hat{V}_\ell )\|_{0,I_\delta \backslash I_{\kappa \varepsilon }}\\ &\le \|\nabla \hat{V}_\ell \|_{0,I_\delta \backslash I_{\kappa \varepsilon }} +\|\nabla (v^{\varepsilon }_\ell -\hat{V}_\ell )\|_{0,I_\delta }\\ &\le C\Bigl (\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{1/2}+\delta \Bigr )\|\nabla V_\ell \|_{0,I_\delta }. \end{align*}$$

This gives 3.6.

■

As in 3.6, we also have

$$\begin{equation} \|\nabla v^{\varepsilon }_\ell \|_{0,I_\delta \backslash I_{(\kappa -2)\varepsilon }}\le C\Bigl (\Bigl (\dfrac{\varepsilon }{\delta }\Bigr )^{1/2}+\delta \Bigr )\|\nabla V_\ell \|_{0,I_\delta }. {\tag{3.12}} \end{equation}$$

Theorem 3.2.

$$\begin{equation} e(\mathrm{HMM})\le C\Bigl (\frac{\varepsilon }{\delta }+\delta \Bigr ). {\tag{3.13}} \end{equation}$$

Proof.

Note that $v^{\varepsilon }_\ell =(v^{\varepsilon }_\ell -\hat{v}^{\,\varepsilon }_\ell )+\theta ^{\,\varepsilon }_\ell +\hat{V}_\ell$. We have

$$\begin{equation*} \bigl (\nabla W\cdot (\mathcal{A}_H-\mathcal{A})\nabla V\bigr )(x_\ell )={:}I_1+I_2+I_3, \end{equation*}$$

where

$$\begin{align*} I_1&=\intbar _{I_\delta }\nabla w^{\varepsilon }_\ell \cdot a\Bigl (x,\frac{x}{\varepsilon }\Bigr )\nabla (v^{\varepsilon }_\ell -\hat{v}^{\,\varepsilon }_\ell )\,dx,\quad I_2=\intbar _{I_\delta }\nabla w^{\varepsilon }_\ell \cdot a\Bigl (x,\frac{x}{\varepsilon }\Bigr )\nabla \theta ^{\,\varepsilon }_\ell \,dx,\\ I_3&=\intbar _{I_\delta }\nabla w^{\varepsilon }_\ell \cdot a\Bigl (x,\frac{x}{\varepsilon }\Bigr )\nabla \hat{V}_\ell \,dx-\nabla W_\ell \cdot \mathcal{A}(x_\ell )\nabla V_\ell . \end{align*}$$

Using 3.7 and 2.2, we bound $I_1$ as

$$\begin{align*} \lvert I_1\rvert &\le \varLambda \delta ^{-d}\|\nabla (v^{\varepsilon }_\ell -\hat{v}^{\,\varepsilon }_\ell )\|_{0,I_\delta }\|\nabla w^{\varepsilon }_\ell \|_{0,I_\delta }\\ &\le C\delta ^{1-d}\|\nabla V_\ell \|_{0,I_\delta }\|\nabla W_\ell \|_{0,I_\delta }=C\delta \lvert \nabla V_\ell \rvert \,\lvert \nabla W_\ell \rvert . \end{align*}$$

Using the symmetry of $a^{\,\varepsilon }$, $I_2=\intbar _{I_\delta }\nabla \theta ^{\,\varepsilon }_\ell \cdot a\Bigl (x,\frac{x}{\varepsilon }\Bigr )\nabla w^{\varepsilon }_\ell \,dx$ and

$$\begin{align*} I_2&=\intbar _{I_\delta }\nabla \Bigl (\theta ^{\,\varepsilon }_\ell +\varepsilon \chi ^k\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\frac{\partial V_\ell }{\partial x_k} (1-\rho ^{\varepsilon })\Bigr )\cdot a\Bigl (x,\frac{x}{\varepsilon }\Bigr )\nabla w^{\varepsilon }_\ell \,dx\\ &\quad -\intbar _{I_\delta }\nabla \Bigl (\varepsilon \chi ^k\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\frac{\partial V_\ell }{\partial x_k}(1-\rho ^{\varepsilon })\Bigr ) \cdot a\Bigl (x,\frac{x}{\varepsilon }\Bigr )\nabla w^{\varepsilon }_\ell \,dx, \end{align*}$$

where $\rho ^{\varepsilon }(x)\in \mathcal{C}_0^{\infty }(I_\delta ),\lvert \nabla \rho ^{\varepsilon }\rvert \le C/\varepsilon$, and

$$\begin{equation} \rho ^{\varepsilon }(x)=\begin{cases} 1&\text{if dist}(x,\partial I_\delta )\ge 2\varepsilon ,\\ 0&\text{if dist}(x,\partial I_\delta )\le \varepsilon . \end{cases} {\tag{3.14}} \end{equation}$$

Using 1.6 and $\theta ^{\,\varepsilon }_\ell +\varepsilon \chi ^k\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\frac{\partial V_\ell }{\partial x_k}(1-\rho ^{\varepsilon })\in H_0^1(I_\delta )$, integrating by parts makes the first term in the right-hand side of $I_2$ vanish; therefore we write $I_2$ as

$$\begin{equation*} I_2=-\intbar _{I_\delta }a_{ij}\Bigl (x,\frac{x}{\varepsilon }\Bigr )\frac{\partial w^{\varepsilon }_\ell }{\partial x_i} \frac{\partial \chi ^k}{\partial y_j}\frac{\partial V_\ell }{\partial x_k}(1-\rho ^{\varepsilon })\,dx+\varepsilon \intbar _{I_\delta }a_{ij}\Bigl (x,\frac{x}{\varepsilon }\Bigr )\frac{\partial w^{\varepsilon }_\ell }{\partial x_i} \chi ^k\frac{\partial V_\ell }{\partial x_k}\frac{\partial \rho ^{\varepsilon }}{\partial x_j}\,dx. \end{equation*}$$

Using 3.12, we bound $I_2$ as

$$\begin{equation*} \lvert I_2\rvert \le C\delta ^{-d}\|\nabla w^{\varepsilon }_\ell \|_{0,I_\delta \backslash I_{(\kappa -2)\varepsilon }}\|\nabla V_\ell \|_{0,I_\delta \backslash I_{(\kappa -2)\varepsilon }}\le C\Bigl (\frac{\varepsilon }{\delta }+\delta ^2\Bigr )\lvert \nabla W_\ell \rvert \,\lvert \nabla V_\ell \rvert . \end{equation*}$$

Using 3.2 and integrating by parts, we obtain

$$\begin{equation*} \intbar _{I_\delta }\nabla w^{\varepsilon }_\ell \cdot a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\nabla \hat{V}_\ell \,dx=\intbar _{I_\delta }\nabla W_\ell \cdot a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\nabla \hat{V}_\ell \,dx, \end{equation*}$$

which together with 3.5 gives

$$\begin{align*} I_3&=\intbar _{I_\delta }\nabla w^{\varepsilon }_\ell \Bigl [a\Bigl (x,\frac{x}{\varepsilon }\Bigr )-a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\Bigr ]\nabla \hat{V}_\ell \,dx\\ &\quad +\frac{1}{\delta ^d}\int \limits _{I_\delta \backslash I_{\kappa \varepsilon }}\nabla W_\ell \cdot a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\nabla \hat{V}_\ell \,dx+(\lvert \kappa \varepsilon /\delta \rvert ^d-1)\nabla W_\ell \cdot \mathcal{A}(x_\ell )\nabla V_\ell . \end{align*}$$

The last term of $I_3$ is bounded by

$$\begin{equation*} \left|\,\lvert \kappa \varepsilon /\delta \rvert ^d-1\,\right|\,\lvert \nabla W_\ell \cdot \mathcal{A}(x_\ell )\nabla V_\ell \rvert \le C\frac{\varepsilon }{\delta }\lvert \nabla V_\ell \rvert \lvert \nabla W_\ell \rvert , \end{equation*}$$

where we have used $\left|\,\lvert \kappa \varepsilon /\delta \rvert ^d-1\,\right|\le C\varepsilon /\delta$. Using 3.11, we get

$$\begin{align*} \delta ^{-d}\left|\,\int \limits _{I_\delta \backslash I_{\kappa \varepsilon }}\nabla W_\ell \cdot a\Bigl (x_\ell ,\frac{x}{\varepsilon }\Bigr )\nabla \hat{V}_\ell \,dx\,\right|&\le C\frac{1}{\delta ^d}\|\nabla \hat{V}_\ell \|_{0,I_\delta \backslash I_{\kappa \varepsilon }}\|\nabla W_\ell \|_{0,I_\delta \backslash I_{\kappa \varepsilon }}\\ &\le C\Bigl (\frac{\varepsilon }{\delta }+\delta ^2\Bigr )\lvert \nabla V_\ell \rvert \lvert \nabla W_\ell \rvert . \end{align*}$$

Consequently, we obtain

$$\begin{align*} \lvert I_3\rvert &\le C\delta ^{1-d}\|\nabla w^{\varepsilon }_\ell \|_{0,I_\delta }\|\nabla \hat{V}_\ell \|_{0,I_\delta } +C\Bigl (\dfrac{\varepsilon }{\delta }+\delta ^2\Bigr )\lvert \nabla V_\ell \rvert \lvert \nabla W_\ell \rvert \\ &\le C\Bigl (\dfrac{\varepsilon }{\delta }+\delta \Bigr )\lvert \nabla V_\ell \rvert \lvert \nabla W_\ell \rvert . \end{align*}$$

Combining the estimates for $I_1,I_2$ and $I_3$ gives the desired result 3.13.

■

Remark 3.3.

An explicit expression for $v^{\varepsilon }_\ell$ is available in one dimension, from which we may show that the bound for $e(\mathrm{HMM})$ is sharp.

4. Reconstruction and compression

4.1. Reconstruction procedure

Next we consider how to construct better approximations to $u^{\varepsilon }$ 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

$$\begin{equation*} \left\{\begin{aligned} -\operatorname {div}\bigl (a^{\,\varepsilon }(x)\nabla (\tilde{u}^{\,\varepsilon }-u^{\varepsilon })(x)\bigr )&=0\qquad && x\in \varOmega _\eta ,\\ (\tilde{u}^{\,\varepsilon }-u^{\varepsilon })(x)&=U_H(x)-u^{\varepsilon }(x)\qquad &&x\in \partial \varOmega _\eta . \end{aligned}\right. \end{equation*}$$

Using classical interior estimates for elliptic equation 24, we have

$$\begin{equation*} \|\nabla (\tilde{u}^{\,\varepsilon }-u^{\varepsilon })\|_{0,\varOmega }\le \frac{C}{\eta }\|\tilde{u}^{\,\varepsilon }-u^{\varepsilon }\|_{0,\varOmega _\eta }. \end{equation*}$$

Using the Hopf maximum principle, we get

$$\begin{align*} \frac{1}{\eta ^2}\intbar _{\varOmega _\eta }\lvert (\tilde{u}^{\,\varepsilon }-u^{\varepsilon })(x)\rvert ^2\,dx&\le \frac{C}{\eta ^2}\|\tilde{u}^{\,\varepsilon }-u^{\varepsilon }\|_{L^\infty (\varOmega _\eta )}^2 \le \frac{C}{\eta ^2}\|u^{\varepsilon }-U_H\|_{L^\infty (\partial \varOmega _\eta )}^2\\ &\le \frac{C}{\eta ^2}\Bigl (\|U_0-U_H\|_{L^\infty (\varOmega _\eta )}^2 +\|u^{\varepsilon }-U_0\|_{L^\infty (\varOmega _\eta )}^2\Bigr ). \end{align*}$$

A combination of the above two results implies Theorem 1.4.

■

Proof of Theorem 1.5.

Denote $I_\varepsilon (x_K)=x_K+\varepsilon I$ and define $\hat{u}^{\,\varepsilon }$ as the solution of

$$\begin{equation} -\operatorname {div}\Bigl (a\bigl (x_K,\dfrac{x}{\varepsilon }\bigr )\nabla \hat{u}^{\,\varepsilon }(x)\Bigr )=0\qquad \text{in  }I_\varepsilon (x_K), {\tag{4.1}} \end{equation}$$

with the boundary condition that $\hat{u}^{\,\varepsilon }-U_H$ is periodic on $\partial I_\varepsilon (x_K)$ and

$$\begin{equation*} \int _{I_\varepsilon (x_K)}(\hat{u}^{\,\varepsilon }-U_H)\,dx=0, \end{equation*}$$

where $x_K$ is the barycenter of $K$.

It is easy to verify that $\hat{u}^{\,\varepsilon }$ takes the explicit form

$$\begin{equation} \hat{u}^{\,\varepsilon }(x)=U_H(x)+\varepsilon \chi ^k\Bigl (x_K,\frac{x}{\varepsilon }\Bigr )\frac{\partial U_H}{\partial x_k}(x). {\tag{4.2}} \end{equation}$$

Note that the periodic extension of $\hat{u}^{\,\varepsilon }-U_H$ is still $\varepsilon \chi ^k\bigl (x_K,\frac{x}{\varepsilon }\bigr )\frac{\partial U_H}{\partial x_k}(x)$. This means that $\hat{u}^{\,\varepsilon }$ is also well defined for the whole of $K$ and takes the same explicit form as 4.2.

Using $\int _I\chi ^k(x_K,y)\,dy=0$ for $k=1$, …, $d$ and that $\nabla U_H$ is a piecewise constant on $K$, we obtain

$$\begin{equation} \int \limits _{I_\varepsilon (x_K)}(\hat{u}^{\,\varepsilon }-U_H)(x)\,dx=\int \limits _{I_\varepsilon (x_K)}\varepsilon \chi ^k\bigl (x_K,\frac{x}{\varepsilon }\bigr )\frac{\partial U_H}{\partial x_k}(x)\,dx=0. {\tag{4.3}} \end{equation}$$

As in 3.7, we have

$$\begin{equation*} \|\nabla (\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon })\|_{0,I_\varepsilon (x_K)}\le C\varepsilon \|\nabla U_H\|_{0,I_\varepsilon (x_K)}. \end{equation*}$$

From the construction of $\tilde{u}^{\,\varepsilon }$, we have for any $x_1 \in K$,

$$\begin{equation*} \|\nabla (\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon })\|_{0,I_\varepsilon (x_1)}=\|\nabla (\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon })\|_{0,I_\varepsilon (x_K)}. \end{equation*}$$

Since $\nabla U_H$ is constant over $K$, we get

$$\begin{equation} \|\nabla (\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon })\|_{0,K}\le C\varepsilon \|\nabla U_H\|_{0,K}. {\tag{4.4}} \end{equation}$$

Adding up for all $K\in \mathcal{T}_H$ and using the a priori estimate $\|\nabla U_H\|_0\le C\|f\|_0$, we obtain

$$\begin{equation} \Bigl (\sum _{K\in \mathcal{T}_H}\|\nabla (\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon })\|_{0,K}^2\Bigr )^{1/2}\le C\varepsilon \|\nabla U_H\|_0\le C\varepsilon . {\tag{4.5}} \end{equation}$$

Using 4.2, a straightforward calculation gives

$$\begin{equation*} \frac{\partial \hat{u}^{\,\varepsilon }}{\partial x_i}=\frac{\partial U_H}{\partial x_i}+\frac{\partial \chi ^k}{\partial y_i}\Bigl (x_K,\frac{x}{\varepsilon }\Bigr )\frac{\partial U_H}{\partial x_k}. \end{equation*}$$

Define the first order approximation of $u^{\varepsilon }$ as

$$\begin{equation*} u^{\varepsilon }_1(x)=U_0+\varepsilon \chi ^k\Bigl (x,\frac{x}{\varepsilon }\Bigr )\frac{\partial U_0}{\partial x_k}, \end{equation*}$$

where $\{\chi ^k\}_{k=1}^d$ is the solutions of 3.2. Obviously,

$$\begin{equation*} \frac{\partial u^{\varepsilon }_1}{\partial x_i}=\frac{\partial U_0}{\partial x_i}+\Bigl (\varepsilon \frac{\partial \chi ^k}{\partial x_i}+\frac{\partial \chi ^k}{\partial y_i}\Bigr )\Bigl (x,\frac{x}{\varepsilon }\Bigr )\frac{\partial U_0}{\partial x_i}+\varepsilon \chi ^k\Bigl (x,\frac{x}{\varepsilon }\Bigr )\frac{\partial ^2U_0}{\partial x_i\partial x_k}. \end{equation*}$$

A combination of the above estimates leads to

$$\begin{align*} \|\nabla (\hat{u}^{\,\varepsilon }-u^{\varepsilon }_1)\|_{0,K}&\le C\|\nabla (U_H-U_0)\|_{0,K}+C\varepsilon \lvert U_0\rvert _{1,K}\\ &\;+\left\|\Bigl (\frac{\partial \chi ^k}{\partial y _i}\Bigl (x,\frac{x}{\varepsilon }\Bigr )-\frac{\partial \chi ^k}{\partial y_i}\Bigl (x_K,\frac{x}{\varepsilon }\Bigr )\Bigr )\frac{\partial U_0}{\partial x_k}\right\|_{0,K}+C\varepsilon \lvert U_0\rvert _{2,K}\\ &\le C\lvert U_0-U_H\rvert _{1,K}+C(\varepsilon +H)\|U_0\|_{2,K}. \end{align*}$$

Summing up for all $K\in \mathcal{T}_H$ and using Theorem 1.1 for $k=1$ and Theorem 1.2 for the case $I_\delta =x_K+\varepsilon I$, we get

$$\begin{equation*} \Bigl (\sum _{K\in \mathcal{T}_H}\|\nabla (\hat{u}^{\,\varepsilon }-u^{\varepsilon }_1)\|_{0,K}^2\Bigr )^{1/2}\le C(\varepsilon +H), \end{equation*}$$

which together with 4.5 and the classical estimate for $u^{\varepsilon }-u^{\varepsilon }_1$ 53245, i.e.,

$$\begin{equation*} \|u^{\varepsilon }-u^{\varepsilon }_1\|_1\le C\sqrt \varepsilon , \end{equation*}$$

gives

$$\begin{align*} \Bigl (\sum _{K\in \mathcal{T}_H}\|\nabla (u^{\varepsilon }-\tilde{u}^{\,\varepsilon })\|_{0,K}^2\Bigr )^{1/2}&\le \|u^{\varepsilon }-u^{\varepsilon }_1\|_1+\Bigl (\sum _{K\in \mathcal{T}_H}\|\nabla (u^{\varepsilon }_1-\hat{u}^{\,\varepsilon })\|_{0,K}^2\Bigr )^{1/2}\\ &\quad +\Bigl (\sum _{K\in \mathcal{T}_H}\|\nabla (\hat{u}^{\,\varepsilon }-\tilde{u}^{\,\varepsilon })\|_{0,K}^2\Bigr )^{1/2}\\ &\le C(\sqrt {\varepsilon }+H). \end{align*}$$ ■

Corollary 4.1.

$$\begin{equation} \|\tilde{u}^{\,\varepsilon }-u^{\varepsilon }\|_0\le C(\varepsilon +H^2). {\tag{4.6}} \end{equation}$$

Proof.

Using the definition of $\tilde{u}^{\,\varepsilon }$, we have $\int _{I_\varepsilon (x_K)}(\tilde{u}^{\,\varepsilon }-U_H)(x)\,dx=0$. Together with 4.3, we have

$$\begin{equation*} \int \limits _{I_\varepsilon (x_K)}(\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon })(x)\,dx=0. \end{equation*}$$

An application of the Poincaré inequality gives

$$\begin{equation*} \|\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon }\|_{0,I_\varepsilon (x_K)}\le C\varepsilon \|\nabla (\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon })\|_{0,I_\varepsilon (x_K)}\le C\varepsilon ^2\|\nabla U_H\|_{0,I_\varepsilon (x_K)}. \end{equation*}$$

As before for any $I_\varepsilon (x_1)$, $\|\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon }\|_{0,I_\varepsilon (x_1)}=\|\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon }\|_{0,I_\varepsilon (x_K)}$, note that $\nabla U_H$ is a constant on $K$. We obtain

$$\begin{equation} \|\tilde{u}^{\,\varepsilon }-\hat{u}^{\,\varepsilon }\|_{0,K}\le C\varepsilon ^2\|\nabla U_H\|_{0,K}. {\tag{4.7}} \end{equation}$$

On each element $K$, we have

$$\begin{equation*} \|\hat{u}^{\,\varepsilon }-U_H\|_{0,K}\le C\varepsilon \|\nabla U_H\|_{0,K}. \end{equation*}$$

Combining the above and summing up for all $K\in \mathcal{T}_H$, we get

$$\begin{equation*} \|\tilde{u}^{\,\varepsilon }-U_H\|_0\le C\varepsilon \|\nabla U_H\|_0\le C\varepsilon , \end{equation*}$$

which together with

$$\begin{equation*} \|u^{\varepsilon }-U_H\|_0\le \|u^{\varepsilon }-U_0\|_0+\|U_0-U_H\|_0\le C(\varepsilon +H^2) \end{equation*}$$

leads to 4.6, where we have used the estimate for $U_0$ 53245, i.e.,

$$\begin{equation*} \|u^{\varepsilon }-U_0\|_0\le C\varepsilon . \end{equation*}$$■

4.2. 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 $Qu^{\varepsilon }-U_H$. We first list some natural properties of the compression operator.

•

For any $\phi \in X$, $Q\phi \in X_H$.

•

There exists a constant $C$ such that$$\begin{equation*} \|Q\phi \|_0\le C\|\phi \|_0. \end{equation*}$$

•

For any $k\ge 1$, if $\phi \in H^{k+1}(\Omega )\cap H_0^1(\Omega )$, then$$\begin{equation*} \|\phi -Q\phi \|_0\le CH^{k+1}\lvert \phi \rvert _{k+1}. \end{equation*}$$

Theorem 4.2.

Assume that $Q$ satisfies all three requirements and $U_0\in H^{k+1}(D)$ for any $k\ge 1$. Then

$$\begin{equation} \|Qu^{\varepsilon }-U_H\|_0\le C(\varepsilon +H^{k+1}). {\tag{4.8}} \end{equation}$$

Moreover, if $\mathcal{T}_H$ is quasi-uniform, then

$$\begin{equation} \|Qu^{\varepsilon }-U_H\|_1\le C\Bigl (\frac{\varepsilon }{H}+H^k\Bigr ). {\tag{4.9}} \end{equation}$$

Proof.

We decompose $Qu^{\varepsilon }-U_H$ into

$$\begin{equation} Qu^{\varepsilon }-U_H=Q(u^{\varepsilon }-U_0)+(QU_0-U_0)+(U_0-U_H). {\tag{4.10}} \end{equation}$$

Using the fact that $Q$ is bounded in L$^2_{\mathrm{norm}}$, we obtain

$$\begin{equation*} \|Q(u^{\varepsilon }-U_0)\|_0\le C\|u^{\varepsilon }-U_0\|_0\le C\varepsilon . \end{equation*}$$

Using the third property of $Q$, we have

$$\begin{equation*} \|QU_0-U_0\|_0\le CH^{k+1}. \end{equation*}$$

Using Theorem 1.1 and the first estimate in Theorem 1.2, we have

$$\begin{equation*} \|U_0-U_H\|_0\le C(\varepsilon +H^{k+1}). \end{equation*}$$

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 of operators meet all three requirements:

•

the L$^2$-projection operator onto $X_H$,

•

the Clément-type interpolation operator 12.

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 $\varepsilon /H$ 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 623:

$$\begin{equation} \left\{\begin{aligned} -\operatorname {div}\bigl (a^{\,\varepsilon }\bigl (x,u^{\varepsilon }(x)\bigr )\nabla u^{\varepsilon }(x)\bigr )&=f(x)\qquad &&x\in D,\\ u^{\varepsilon }(x)&=0\qquad &&x\in \partial D. \end{aligned} \right. {\tag{5.1}} \end{equation}$$

In this section, we define $X{:}=W^{1,p}_0(D)$ with $p>1$ and $X_H$ is defined as the $\mathcal{P}_k$ finite element subspace of $X$.

We assume that $a^{\,\varepsilon }(x,u^{\varepsilon })$ satisfies

$$\begin{equation*} \lambda \lvert \xi \rvert ^2\le a^{\,\varepsilon }_{ij}\xi _i\xi _j\le \varLambda \lvert \xi \rvert ^2\quad \text{for all }\xi \in \mathbb{R}^d, \end{equation*}$$

with $0<\lambda \le \varLambda$. Moreover, we assume that $a^{\,\varepsilon }(x,z)$ is equi-continuous in $z$ uniformly with respect to $x$ and $\varepsilon$.

The homogenized problem, if it exists, is of the following form:

$$\begin{equation*} \left\{ \begin{aligned} \mathcal{L}U_0{:}=-\operatorname {div}\bigl (\mathcal{A}\bigl (x,U_0(x)\bigr )\nabla U_0(x)\bigr )&=f(x)\qquad &&x\in D,\\ U_0(x)&=0\qquad &&x\in \partial D. \end{aligned} \right. \end{equation*}$$

If we let

$$\begin{equation*} A(v,w)=\bigl (\mathcal{A}(x,v)\nabla v,\nabla w\bigr )\quad \text{for all } v,w\in X, \end{equation*}$$

then

$$\begin{equation} A(U_0,v)=(f,v)\quad \text{for all } v\in X', {\tag{5.2}} \end{equation}$$

where $X'$ is the dual space of $X$.

The linearized operator of $\mathcal{L}$ at $U_0$ is defined for any $v\in H_0^1(D)$ by

$$\begin{equation*} \mathcal{L}_{\text{lin}}(U_0)v=-\operatorname {div}\bigl (\mathcal{A}(x,U_0)\nabla v+\mathcal{A}_p(x,U_0)\nabla U_0\,v\bigr ), \end{equation*}$$

where $\mathcal{A}_p(x,z){:}=\nabla _z\mathcal{A}(x,z)$. $\mathcal{L}_{\text{lin}}$ induces a bilinear form through

$$\begin{equation*} \hat{A}(u;v,w){:}=\bigl (\mathcal{A}(x,u)\nabla v,\nabla w\bigr )+\bigl (\mathcal{A}_p(x,u)\nabla u\,v,\nabla w\bigr )\quad \text{for all } v,w\in H_0^1(D). \end{equation*}$$

Our basic assumption is that the linearized operator $\mathcal{L}_{\text{lin}}$ is an isomorphism from $H_0^1(D)$ to $H^{-1}(D)$, so $U_0$ must be an isolated solution of 5.2.

To formulate HMM, for each quadrature point $x_\ell$, define $v^{\varepsilon }_\ell$ to be the solutions of

$$\begin{equation} \left\{ \begin{aligned} -\operatorname {div}\bigl (a^{\,\varepsilon }(x,v^{\varepsilon }_\ell )\nabla v^{\varepsilon }_\ell (x)\bigr )&=0\qquad && x\in I_\delta (x_\ell ),\\ v^{\varepsilon }_\ell (x)&=V_\ell (x)\qquad &&x\in \partial I_\delta (x_\ell ). \end{aligned}\right. {\tag{5.3}} \end{equation}$$

We can define $w^{\varepsilon }_\ell$ similarly.

For any $V,W\in X_H$, define

$$\begin{equation*} \nabla W(x_\ell )\cdot \mathcal{A}_H\bigl (x_\ell ,V(x_\ell )\bigr )\nabla V(x_\ell )=\intbar _{I_\delta (x_\ell )} \nabla w^{\varepsilon }_\ell (x)\cdot a^{\,\varepsilon }\bigl (x,v^{\varepsilon }_\ell (x)\bigr )\nabla v^{\varepsilon }_\ell (x)\,dx \end{equation*}$$

and

$$\begin{equation*} A_H(V,W){:}=\sum _{K\in \mathcal{T}_H}\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell \nabla W(x_\ell )\cdot \mathcal{A}_H\bigl (x_\ell ,V(x_\ell )\bigr )\nabla V(x_\ell ). \end{equation*}$$

The HMM solution is given by the problem:

Problem 5.1.

Find $U_H\in X_H$ such that

$$\begin{equation} A_H(U_H,V)=(f,V)\quad \text{for all } V\in X_H. {\tag{5.4}} \end{equation}$$

For any $v,v_H,w\in X$, define

$$\begin{equation} R(v,v_H,w){:}=A(v_H,w)-A(v,w)-\hat{A}(v;v_H-v,w). {\tag{5.5}} \end{equation}$$

It is easy to see that for any $v$ and $v_H$ satisfying $\|v\|_{1,\infty }+\|v_H\|_{1,\infty }\le M$,

$$\begin{equation} \lvert R(v,v_H,w)\rvert \le C(M)(\|e_H\|_{0,2p}^2+\|e_H\nabla e_H\|_{0,p})\|\nabla w\|_{0,q} {\tag{5.6}} \end{equation}$$

for $e_H{:}=v-v_H$ and $\frac{1}{p}+\frac{1}{q}=1,p,q\ge 1$ (see 42, Lemma 3.1 for a similar result). Therefore we have

Lemma 5.2.

$U_H\in X_H$ is the solution of Problem 5.1 if and only if

$$\begin{align} \hat{A}(U_0;U_0-U_H,V)&=R(U_0,U_H,V)\\ &\quad +A_H(U_H,V)-A(U_H,V)\quad \text{for all } V\in X_H. {\tag{5.7}} \end{align}$$

For any $V,W\in X_H$, define

$$\begin{equation} E(V,W){:}=\nabla W(x_\ell )\cdot (\mathcal{A}_H-\mathcal{A})\bigl (x_\ell ,V(x_\ell )\bigr )\nabla V(x_\ell ). {\tag{5.8}} \end{equation}$$

Define $e(\mathrm{HMM})$ as

$$\begin{equation} e(\mathrm{HMM})=\max _{\substack{x_\ell \in K,K\in \mathcal{T}_H\\V\in X_H\cap W^{1,\infty }(D),\,W\in X_H\\ }}\frac{\lvert E(V,W)\rvert }{\lvert \nabla V_\ell \rvert \,\lvert \nabla W_\ell \rvert }. {\tag{5.9}} \end{equation}$$

The existence and uniqueness of the solution of Problem 5.1 are proved in the following lemma.

Lemma 5.3.

Assume that $U_0\in W^{2,p}(D)$ with $p>d$ and $\mathcal{L}_{\text{lin}}$ is an isomorphism from $H_0^1(D)$ to $H^{-1}(D)$. If $\,e(\mathrm{HMM})$ is uniformly bounded and there exist constants $H_0>0$ and $\mathcal{M}_1>0$ such that for $0<H\le H_0$ and

$$\begin{equation} e(\mathrm{HMM})^{1/2}\lvert \ln H\rvert \le \mathcal{M}_1, {\tag{5.10}} \end{equation}$$

then Problem 5.1 has a solution $U_H$ satisfying

$$\begin{equation} \begin{alignedat} {1} \|U_H-P_HU_0\|_{1,\infty }&\le e(\mathrm{HMM})^{1/2}+H^{1-d/p},\\ \|U_0-U_H\|_{1,\infty }&\le C\bigl (e(\mathrm{HMM})^{1/2}+H^{1-d/p}\bigr ), \end{alignedat} {\tag{5.11}} \end{equation}$$

where $P_HU_0\in X_H$ is defined as

$$\begin{equation} \hat{A}(U_0;P_HU_0,V)=\hat{A}(U_0;U_0,V)\quad \text{for all } V\in X_H. {\tag{5.12}} \end{equation}$$

Moreover, if there exists a constant $\eta (M)$ with $0<\eta (M)<1$ such that

$$\begin{equation} \sum _{K\in \mathcal{T}_H}\lvert K\rvert \sum _{x_\ell \in K}\omega _\ell \lvert E(V,Z)-E(W,Z)\rvert \le \eta (M)\|V-W\|_1\|Z\|_1 {\tag{5.13}} \end{equation}$$

for all $V,W\in X_H\cap W^{1,\infty }(D)$ and $Z\in X_H$, satisfying $\|V\|_{1,\infty }+\|W\|_{1,\infty }\le M$, then there exists a constant $H_1>0$ such that for $0<H\le H_1$, the HMM solution $U_H$ satisfying 5.11$_1$ is locally unique.

Proof.

Since $\mathcal{L}_{\text{lin}}$ is an isomorphism from $H_0^1(D)$ to $H^{-1}(D)$, there exists a constant $C$ such that

$$\begin{equation*} \sup _{W\in H_0^1(D)} \frac{\hat{A}(U_0;V,W)}{\|W\|_1}\ge C\|V\|_1 \quad \text{for all }V\in H_0^1(D). \end{equation*}$$

Using 42, Lemma 2.2, we conclude that there exists a constant $H_2>0$ such that for $0<H\le H_2$,

$$\begin{equation} \sup _{W\in X_H} \frac{\hat{A}(U_0;V,W)}{\|W\|_1}\ge C\|V\|_1 \quad \text{for all }V\in X_H. {\tag{5.14}} \end{equation}$$

Therefore there is a unique solution $P_HU_0\in X_H$ satisfying 5.12 and

$$\begin{equation} \|U_0-P_HU_0\|_{1,\infty }\le CH^{1-d/p}. {\tag{5.15}} \end{equation}$$

Moreover, let $\hat{G}_H^z$ be the finite element approximation of the regularized Green’s function associated with $\hat{A}(U_0;\cdot ,\cdot )$. Using 42, equation 2.11, or using 5.14, similarly to 2.11, we have

$$\begin{equation} \|\hat{G}_H^z\|_{1,1}\le C\lvert \ln H\rvert . {\tag{5.16}} \end{equation}$$

Define a nonlinear mapping $T{:}\;\;X_H\rightarrow X_H$ by